Quantum Link Homology via Trace Functor I
Anna Beliakova, Krzysztof Karol Putyra, Stephan Martin Wehrli
aa r X i v : . [ m a t h . G T ] S e p Quantum Link Homology via Trace Functor I
Anna Beliakova Krzysztof K. Putyra Stephan M. Wehrli
Abstract
Motivated by topology, we develop a general theory of traces and shadows foran endobicategory, which is a pair: bicategory C and endobifunctor Σ : C C .For a graded linear bicategory and a fixed invertible parameter q , we quantize thistheory by using the endofunctor Σ q such that Σ q α := q − deg α Σ α for any 2-morphism α and coincides with Σ otherwise.Applying the quantized trace to the bicategory of Chen–Khovanov bimoduleswe get a new triply graded link homology theory called quantum annular link ho-mology . If q = 1 we reproduce Asaeda–Przytycki–Sikora (APS) homology for linksin a thickened annulus. We prove that our homology carries an action of U q ( sl ),which intertwines the action of cobordisms. In particular, the quantum annularhomology of an n –cable admits an action of the braid group, which commutes withthe quantum group action and factors through the Jones skein relation. This pro-duces a nontrivial invariant for surfaces knotted in four dimensions. Moreover,a direct computation for torus links shows that the rank of quantum annular ho-mology groups does depend on the quantum parameter q . Contents
Quantum Hochschild homology 41 K –theoretic invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 An application to Hochschild homology of algebras . . . . . . . . . . . . 46 R . . . . . . . . . . . . . . . . . . . . . 515.3 Annular link homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.4 Homology for links in a thickened M¨obius band . . . . . . . . . . . . . . 545.5 Chen–Khovanov homology for tangles . . . . . . . . . . . . . . . . . . . . 55 sl revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Deformation of the annular skein category . . . . . . . . . . . . . . . . . 616.3 Quantum annular homology . . . . . . . . . . . . . . . . . . . . . . . . . 656.4 Homology for (2 , n ) torus links . . . . . . . . . . . . . . . . . . . . . . . . 67 A Background survey 76
A.1 Representations of U q ( sl ) . . . . . . . . . . . . . . . . . . . . . . . . . . 76A.2 Knots and tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.3 Constructions on categories . . . . . . . . . . . . . . . . . . . . . . . . . 78A.4 Bicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Background and Overview
Trace are an important source of topological invariants. Given a category C , a traceis a collection of functions t x defined on the endomorphism spaces C ( x, x ), indexed byobjects x ∈ C , such that for any pair of composable morphisms x g y f x in C the relation t x ( g ◦ f ) = t y ( f ◦ g )holds. When C is the category of vector spaces, then the above relation determinesthe trace uniquely (up to a scalar) as the sum of the diagonal entries of a matrix repre-senting the endomorphism. The well-known Reshetikhin-Turaev package of link invari-ants is obtained by applying traces to linear endomorphisms associated with tangles. Inthis context, the cyclicity relation is interpreted topologically as passing to the annularclosure of a tangle. 2very category admits the universal trace , defined asTr( C ) := a x ∈ Ob( C ) C ( x, x ) . g ◦ f ∼ f ◦ g (1.1)where f and g run through all pairs of composable morphisms. Any trace on C factorizesthrough Tr( C ). If C is an additive category, then it receives the Chern character mapfrom the Grothendieck group K ( C ), and hence can be considered as an alternativedecategorification functor [BGHL14, BHLZ14]. Another important feature of the traceis its functoriality.Let us consider the category T an of tangles, objects of which are points on the x -axisin R and morphisms are tangles in R × I . Any functor F : T an A induces a mapTr( F ) : Tr( T an ) Tr( A ) , which is a universal annular link invariant associated with A . In the Reshetikhin–Turaevcase, A is a category of representations of a quantum group.For quantum sl the Reshetikhin–Turaev construction was categorified by Chen andKhovanov in [CK14]. For this purpose T an was extended to a bicategory Tan with2–morphisms given by tangle cobordisms. Chen and Khovanov defined a projective bi-functor e F CK : Tan
Com b/h ( gBirep )valued in the homotopy category of graded sweet bimodules Birep . Here, ‘projective’means that the bifunctor is defined on 2–morphisms only up to a sign.In this paper, we develop a theory of traces in endobicategories and apply it to theChen–Khovanov construction to obtain a new triply graded quantum annular link homol-ogy theory. The sign issue will be resolved in a second paper [BHPW18] by redefining ourand Chen–Khovanov’s constructions using gl –foams. We will use it in [BHPW] to showthat all known definitions of the colored Khovanov homology coincide in the quantum annular setting when the quantization parameter is not a root of unity.
Traces in Endobicategories
Let us first present our categorical results. There are two ways to define traces on a bi-category: applying (1.1) to morphism categories results in the vertical trace, and a moregeneral horizontal trace was defined in [BHLZ14]. Let us visualize this construction forthe bicategory
Cob , the objects of which are points on a line, 1–morphisms are flat tan-gles in R × I , and 2–morphisms are cobordisms in ( R × I ) × I . The horizontal trace ofa cobordism between two different (2 , shadow , and was definedby Ponto and Shulman [PS13]. In this paper, inspired by topology, we further gener-alize the notions of the universal trace of a category, as well as vertical and horizontaltraces of a bicategory. Recall that a surface bundle M over a circle with monodromy Following [Kh02] we say that a bimodule is sweet if it is finitely generated and projective whenconsidered as a left and as a right module separately. −→ ≈ Figure 1: A horizontal trace of a cobordism with corners. φ is constructed from a thickened surface F × I by gluing F × { } to F × { } alongthe diffeomorphism φ : M := F × I . ( p, ∼ ( φ ( p ) , . (1.2)To mimic this construction on a more abstract level, we consider a pair: a category C and an endofunctor Σ : C C . Then a Σ –twisted trace of C is defined asTr( C , Σ) := a x ∈ Ob( C ) C ( x, Σ x ) . Σ f ◦ g ∼ g ◦ f (1.3)where f and g run through all pairs of composable morphisms Σ x g y f x in C . Thisconstruction is functorial with respect to ( C , Σ) and universal. We recover the usualtrace of C when Σ is the identity functor.Let T an be the category of isotopy classes of tangles in R × I . The rotation ofthe plane induces an endofunctor Σ on T an , for which Tr( T an , Σ) recovers links in thethickened M¨obius band.We observe that any natural transformation of the endofunctor on an endocategoryinduces a new trace function. For example, fixing a framed (1 , τ we can definea natural transformation of the identity functor of T an by sending a collection of n points to the n –cabling τ n , viewed as a morphism in T an . It is a natural transformationbecause of the equality τ m ◦ T = T ◦ τ n , which holds for any ( m, n )–tangle T . The sameeuqality also implies that the map sending an ( n, n )–tangle T to the annular closure of thecomposition T ◦ τ n is well-defined trace function on T an . In particular, one can composeany ( n, n )–tangle with a full twist before closing. Hence, traces on endocategories oftangles encode information not only about a simple annular closure b T of any ( n, n )-tangle T , but also on all satellites of b T .Triangulated categories are treated separately. They admit triangulated traces, whichare additive with respect to homomorphisms of distinguished triangles, see (2.14). Weshow that twisted triangulated traces on the homotopy endocategory of bounded com-plexes Com b/h ( C ) are Lefschetz traces, which are unique extensions of twisted traceson C . In particular, there is an isomorphism between the universal triangulated traceTr △ ( Com b/h ( C ) , Σ) and Tr( C , Σ).Twisted traces admit a quantization when C is pregraded. For that fix an invertibleparameter q and consider the endofunctor Σ q that coincides with Σ on objects, butΣ q f := q −| f | Σ f for a homogeneous morphism f of degree | f | . This results in the quantumuniversal trace Tr q ( C , Σ) := Tr( C , Σ q ) = a x ∈ Ob( C ) C ( x, Σ x ) . Σ f ◦ g − q | f | g ◦ f . A category is pregraded if its sets of morphisms are graded modules. K –theoretic equivalence, gener-alizing that of Keller [Ke98]. Roughly speaking, it states that the homotopy class ofthe chain map (between quantum Hochschild–Mitchell chain complexes) induced by a bi-module M depends only on the image of M in the Grothendieck group, see Proposition4.9. This very powerful tool is used later to find the quantum Hochschild homology ofChen–Khovanov algebras.To categorify the constructions described above, we define the notions of a Σ –twistedpreshadow and a Σ –twisted horizontal trace for a bicategory C and its endobifunctorΣ, generalizing those introduced in [PS13] and [BHLZ14]. The pair ( C , Σ) is calledan endobicategory . We prove that any twisted preshadow factors through the twistedhorizontal trace under the mild assumption that C has left duals. Hence, for bicategorieswith duals the horizontal trace hTr( C , Σ) is a universal preshadow.When C is pregraded, a deformation of Σ, defined by setting Σ q f := q −| f | Σ f for anyhomogeneous 2-morphism f of degree | f | , leads to a theory of quantum preshadows anda quantum horizontal trace.The horizontal trace can be viewed as a categorification of the trace of an endocategoryin the following sense. Let Π : n Cat ( n − Cat be the decategorification functor,which forgets the n –morphisms and identifies isomorphic ( n − C , Σ)) ≈ Tr(Π C , ΠΣ) for any small endobicategory( C , Σ) with both left and right duals (see Section 3.5).Applying the construction (1.3) to morphism categories in C results in the verticaltrace vTr( C , Σ). There is a functor from the vertical to the horizontal trace of an en-dobicategory, which is full and faithful, but not necessarily surjective on objects. Quiteoften a preshadow can be restricted to a collection of traces on morphisms sets of the ver-tical trace, for instance the shadow computing coinvariants in bimodules restricts tothe Hattori–Stallings trace (see Section 3.8.2).Of particular interest to us are preshadows on bicategories of complexes. We extendevery preshadow on ( C , Σ) to a preshadow on
Com b/h ( C , Σ), which—by the analogy totraces—we call the
Lefschetz preshadow . These satisfy a higher analogue of additivitywith respect to distringuished triangles (see Proposition 3.17 for a precise statement).Again we provide many examples of preshadows on endobicategories. Among themare twisted spaces of coinvariants, quantum Hochschild homology, and component-wiseHochschild homology of a complex of bimodules (which is an example of a Lefschetzpreshadow mentioned above). Furthermore, we compute explicitly twisted horizontaltraces of
Tan ( F ), the bicategory of oriented tangles in a thickened surface F and orientedtangle cobordisms (see Appendix A.2 for a precise definition). Theorem A.
Let M be a surface bundle with fiber F and monodromy φ ∈ Diff ( F ) .There is an equivalence of categories hTr( Tan ( F ) , φ ∗ ) ≃ L inks ( M ) (1.4) where φ ∗ ( S ) := ( φ × id × id)( S ) for a cobordism S ⊂ F × I × I . Here L inks ( M ) stands for the category of oriented links in M and oriented link cobordismsin M × I . In this paper we are interested in the case of when M is a solid torus,5een as a thickened annulus A × I or as a thickened M¨obius band M e × I . We writeshortly L inks ( A ) and L inks ( M ) respectively for the categories of links. Both arise byconsidering tangles in a thickened plane with φ being the identity map or rotation by 180degrees. Elements of L inks ( A ) are called annular links .An analogous result holds for Cob , the bicategory of flat tangles embedded in a stripe R × I , and for its famous quotient by local relations listed in Theorem 5.1, known asthe Bar-Natan bicategory BN . Since a real line admits only two diffeomorphisms up toisotopy, the identification space in this case is either an annulus A or as a M¨obius band M . Hence, the horizontal traces of BN with the corresponding endofunctors recoverthe Bar-Natan categories B N ( A ) and B N ( M ). The identification B N ( A ) ∼ = hTr( BN )is also proven in [QR15].Notice that 2-morphisms in Tan ( F ) are graded by the Euler characteristic, so that L inks ( M ) can be quantized. Namely, we get the category L inks q ( M ) := hTr q ( k Tan ( F ) , φ ∗ )where k is a fixed ring containing the value q , and k Tan ( F ) is the linear extension of Tan ( F ), which 2–morphisms are formal linear combinations of cobordisms with coeffi-cients from k . This category it admits the following graphical description. Representthe image of F × { } in M as a cooriented membrane, with the coorientation inducedfrom the orientation of I . The objects of L inks q ( M ) are oriented links in M that inter-sect the membrane transversely, whereas morphisms are link cobordisms up to isotopies,where an isotopy moving a cobordism of Euler characteristic d through the membranescales the cobordism by q ± d := q = q − Because the local relations defining BN are homogeneous, we construct analogouslythe quantum annular Bar-Natan category as the additive closure of the quantum hori-zontal trace B N q ( A ) := hTr ⊕ q ( BN ) . (1.5)In important feature of B N q ( A ) is that a torus wrapped once along the annulus evaluatesto q + q − instead of 2, see (6.5). Link Homologies via Traces
Our next goal is to construct new functorial invariants of links in a solid torus and ina M¨obius band. The first annular link homology theory was constructed by Asaeda,Przytycki, and Sikora [APS04], and it can be rephrased as applying the APS TQFTfunctor F A : B N ( A ) M od ( k )to the formal Khovanov bracket [BN05]. Grigsby, Licata, and Wehrli observed in [GLW15]that the annular link homology admits an action of sl commuting with the maps inducedby annular cobordisms. This motivates the search of a quantized annular homology, onwhich the quantum sl acts. 6ur aim is to use B N q ( A ) to deform the APS construction. Even though it is im-mediate that a torus evaluates in B N q ( A ) to q + q − when it intersects the membranein a circle, this does not extend naively to a TQFT functor. Instead, we use the wholealgebraic machinery developed in the first part of the paper to construct such a TQFT,especially Theorem A, the universality of the horizontal trace, and the Lefschetz pre-shadow on the homotopy bicategory of Chen-Khovanov bimodules, induced by the quan-tum Hochschild homology.For this purpose, let us examine the homology of tangles constructed by Chen andKhovanov. In [CK14] they defined a family of diagrammatic algebras A n , commonlycalled arc algebras , associated a graded ( A m , A n )–bimodule with each flat ( m, n )–tangle,and constructed a bimodule map for any cobordism of flat ( m, n )–tangles. This givesthe bifunctor F CK : BN gBirep valued in the bicategory of sweet graded bimodules. A precomposition with the formalKhovanov bracket J − K results in e F CK : Tan
Com b/h ( gBirep ) , which assigns to an ( m, n )-tangle T a chain complex C CK ( T ) of graded ( A m , A n )–bimodules,the homotopy class of which is an isotopy invariant of T .The algebra A n categorifies the n –th tensor power of the fundamental representation V of U q ( sl ). It was first introduced by Braden [Bra02] using generators and relations todescribe the category of perverse sheaves on Grassmannians. The arc algebras and theirrepresentations were independently studied by Brundan and Stroppel [BS11, BS10].We can now use our previous results to extend any given preshadow on gBirep tothe homotopy category of complexes, and then pull it back along e F CK to obtain a pre-shadow on Tan . Since all involved bicategories have duals, such a preshadow factorsthrough hTr q ( Tan ) defining an annular link homology theory.An immediate choice of a preshadow on gBirep is the Hochschild homology HH • ofChen–Khovanov bimodules. However, we can also utilize the grading and use the quantumHochschild homology qHH • , a one parameter deformation of HH • that factors throughthe quantum horizontal trace of gBirep . This results in the following commuting diagram Tan
Com b/h ( BN ) Com b/h ( gBirep )hTr q ( Tan ) Com b/h (hTr q ( BN )) Com b/h ( U q ( sl )) J − KJ b − K ∗ e F CK F A q d ( − ) d ( − ) qHHH (1.6)where the horizontal maps are functors and the vertical one are preshadows. The notation qHHH means that we apply qHH • to each bimodule in the Chen–Khovanov complexseparately.To identify the bottom right corner of this diagram we apply the K –theoretic invari-ance to arc algebras. We deduce that the Chern character map K ( A n ) qHH • ( A n )is actually an isomorphism. Hence, the higher Hochschild homology of A n vanishes and qHH ( A n ) ∼ = V ⊗ n . This relates hTr ⊕ q ( Birep ) =
B N q ( A ) with the graded representationcategory g R ep ( U q ( sl )). Finally, to construct our TQFT functor F A q : B N q ( A ) g R ep ( U q ( sl ))7e use the naturality of the Chern character map to identify B N q ( A ) with (the gradedextension of) the Temperley–Lieb category TL and F A q with the faithful functor from TL to g R ep ( U q ( sl )).This produces a well-defined homology for annular links. To any annular closure L ofan ( n, n )-tangle T it assigns a chain complex C Kh A q ( L ) := qHHH • ( A n , C CK ( T )) = qHHH ( A n , C CK ( T )) (1.7)and to any annular cobordism an induced chain map. In addition to the homological andquantum grading, C Kh A q ( L ) admits a third grading, called the annular grading, whichcomes from the weigh decomposition of the Chen–Khovanov invariant.The chain maps induced by annular cobordisms are defined only up to multiplicationby ± q ± : the sign comes from the fact that e F CK is merely a projective functor, andthe overall power of q is not well-defined, because it depends on the presentation ofan annular cobordism as a horizontal closure of a surface with corners. We call thisbehavior ‘ q –projective’. Theorem B.
The quantum annular homology Kh A q ( L ) is a triply graded invariant ofan annular link L , which is q –projectively functorial with respect to annular link cobor-disms. Moreover, it admits an action of the quantum group U q ( sl ) that commutes withthe differential and the maps induced by annular link cobordisms intertwine this action. It was conjectured by Auroux, Grigsby, and the third author in [AGW15] that the ASPhomology of the braid closure coincides with the Hochschild homology of the Chen–Khovanov complex associated to that braid. They checked this conjecture in the next-to-top annular grading.Observe that the quantum annular homology arises as the second page of the spec-tral sequence associated to qCH • ( A n , C CK ( T )), the bicomplex computing the quantumHochschild complex for a complex of bimodules. Using the vanishing of the higher quan-tum Hochschild homology groups for arc algebras, we can actually show that this spectralsequence collapses at the second page. Hence we have the following. Theorem C.
Let b T be the annular closure of an ( n, n ) –tangle T . Then there is an iso-morphism Kh A q ( b T ) ∼ = qHH • ( A n ; C CK ( T )) , (1.8) natural with respect to chain maps associated to tangle cobordisms. The annular gradingin Kh A ( b T ) corresponds to the weight decomposition of C CK ( T ) . When q = 1, Theorem C proves Conjecture 1.1 from [AGW15] and motivates us tocall our new link invariant the quantum annular link homology .We also show by explicit computation that the rank of the quantum annular linkhomology for (2 , n ) torus links does depend on the quantum parameter q . Hence, thequantized theory is richer than APS annular link homology (which is the case q = 1).There are many papers devoted to fixing the sign issue to get a strictly functorialKhovanov homology [Vo15, Bla10, CMW09, Ca07]. In our next paper [BHPW18] wewill reconstruct the Chen–Khovanov invariant using Blanchet foams to obtain a strictlyfunctorial quantum annular link homology theory with an intertwining U q ( gl )-action.8 pplications and Generalizations Let us first argue why the quantum annular link homology is actually more sensible to4D topology than APS.A link cobordism W : L L induces a (projective) map W ∗ : Kh ( L ) Kh ( L ) onKhovanov homology of L . The trace class of this endomorphism is an invariant of theannular closure c W of W and is characterised by its Lefschetz traceΛ( W ∗ ) = X i,j ( − i q j tr W i,j ∗ , where W i,j ∗ is the component of W ∗ in homological grading i and quantum grading j ,and tr is the Hattori–Stallings trace (which is the usual linear trace when we work overa field). We prove the following result. Theorem D.
Let c W ⊂ S × R be a closed surface obtained as an annular closure ofa link cobordism W : L L with L ⊂ R . Then Kh A q ( c W ) = Λ( W ∗ ) is the gradedLefschetz trace of W ∗ : Kh ( L ) Kh ( L ) , the endomorphism of the Khovanov homologyof L . In particular, Kh A q ( S × L ) coincides with the Jones polynomial J ( L ) . Note that the APS invariant is trivial for closed surfaces. Therefore, our invariant isa nontrivial deformation.Next, we establish a nontrivial braid group action on the quantum annular homologyof cablings of a framed long knot. Consider a framed annular knot K ⊂ A × I . It definesan embedding ν K : A × I A × I with the tubular neighborhood of K as its image,and hence induces a functor K ∗ : T an L inks ( A ) mapping oriented points B ⊂ R to a collection of circles K B : = ν K ( S × B ), and an oriented tangle T ⊂ R × I tothe oriented cobordism K T := ν K ( S × T ) between these circles. Applying the quantumannular homology produces a map of homology Kh A q ( K T ) : Kh A q ( K B ) Kh A q ( K B ′ )for any oriented tangle T ∈ T an ( B, B ′ ), defined up to an overall power of q (we work incharacteristic 2 here to avoid the sign issue). This gives rise to a q –projectively functorialaction of T an , i.e. Kh A q ( K T ′ T ) = q k Kh A q ( K T ′ ) ◦ Kh A q ( K T ) , for any composable tangles T , T ′ , and some k ∈ Z . The action was first observed in[GLW15] in the non-quantized setting. We compute this action and show that it factorsthrough the Jones skein relation. Theorem E.
Let K be a framed annular link, considered as an object in L inks q . Thereis a functorial action of T an on the quantum annular homology of oriented cablings of K , that takes a tangle T to the chain map Kh A q ( K T ) , and which intertwines the actionof U q ( sl ) . It factors through the Jones skein relation q Kh A q ( K ) − q − Kh A q ( K ) = ( q − q − ) Kh A q ( K ) (7.5) if K intersects the membrane in one point. twisted quantum Hochschild homology, defined fora pair of a graded algebra A and its automorphism. Here, we consider the automorphismof arc algebras induced by the horizontal flip of diagrams. Again, setting q = 1 recoversthe original APS homology.We have already observed that any preshadow applied to C CK ( T ) produces an annularlink invariant. In particular, we can precompose an ( n, n )–tangle T with and n –cablingof a framed (1 , R . In the other generalizationwe again fix a (1 , T and we assign the quantum annular homology of either b T or its mirror image with an essential circle in an annulus. We then use the dualitybetween the homology of a link and its mirror image to define the differential in the chaincomplex.In a follow up [BHPW] we will define a quantum homology for colored annular links. Avery surprising feature of this construction is that the Cooper–Krushkal infinite complexcategorifying the n -th Jones–Wenzl idempotent becomes finite in the annular closure fora generic value of q , and it does coincide with the annular n -colored Khovanov complex. Strategy for the Proof of Theorem B
To construct our new homology theory we follow the general recipe for Khovanov homol-ogy as described in [BN05]. There, a link L in a thickened surface F is assigned a formalcomplex J L K in the Bar-Natan skein category B N ( F ), whose objects are non-intersectingcurves in F and morphisms are cobordisms in F × I . To construct actual homology onethen applies to J L K a certain TQFT functor F F : B N ( F ) M od ( k ), where k is a fixedring of scalars. The main body of this paper is devoted to the construction of such a func-tor when F = A . The theory of preshadows and horizontal traces on endobicategories isneeded to guarantee the existence of the TQFT functor F A q in (1.6).Let us first explain why the two bottom arrows in (1.6) are dashed. Strictly speaking,a quantum preshadow on Com b/h ( BN ) factorizes through hTr q ( Com b/h ( BN )) rather than Com b/h (hTr q ( BN )). Lemma 3.16 resolves this problem by showing that we actually canapply qHH • to each bimodule separately, rather than to the complex of bimodules, anddeal with flat tangles rather than formal complexes of them. Hence, the pullback shadow( F CK ) ∗ qHH defined on BN factors through B N q ( A ), inducing a linear functor F A q : B N q ( A ) M od ( k ) . To identify the target of this functor with R ep ( U q ( sl )) we need to find an isomorphism V ⊗ n ∼ = qHH ( A n ) (1.9)where A n is the bimodule associated by F CK with the trivial ( n, n )–tangle, and qHH ( A n )is the module assigned to n parallel essential circles in A .10hen q = 1, the existence of the isomorphism (1.9) follows from the invariance resultfor Hochschild homology due to Keller [Ke98]: if A is a finite dimensional algebra, E ⊂ A is a separable subalgebra such that A = E ⊕ rad( A ), each simple A –module is one-dimensional, and A has finite global dimension, then HH • ( E ) ∼ = HH • ( A ). It was provenby Brundan and Stroppel that arc algebras have finite global dimension when k is a field[BS11], and other conditions hold after setting E = A n , the degree zero part of A n .The latter consists of idempotents, and hence is isomorphic to k n . This provides theisomorphism (1.9) for q = 1.The 0th quantum Hochschild homology of A n can be computed by hands, which isall one needs to understand the construction of our invariant. However, computation ofhigher Hochschild homology is needed to identify the invariant with the total Hochschildhomology of Chen–Khovanov bimodules, as conjectured in [AGW15]. For that we reprovethe Keller’s result for quantum and—more generally—for twisted Hochschild homologyof an algebra A by identifying the latter with quantum Hochschild–Mitchell homologyof the category of finite dimensional representations of A , twisted by an appropriateendofunctor. The advantage of replacing algebras and bimodules with representationcategories and functors is that the latter provides a more flexible framework, in whichthe action of bimodules on homology, and so the K –theoretic invariance, is easier tounderstand, see Section 4.3.Further we observe that the canonical embedding of vTr q ( BN ) into hTr q ( BN ) isan equivalence of categories. This implies that every cobordism in hTr q ( BN ) is a linearcombination of those of the form S × T , where T is a Temperley-Lieb diagram. Therefore,the isomorphism (1.9) determines F A q completely, leading to a commuting diagram offunctors TL B B N q ( A )g R ep ( U q ( sl )) ≃ F TL F A q (1.10)where the horizontal functor sends a flat tangle T from TL to the annular cobordism S × T , and F TL is the Reshetikhin–Turaev realization of the Temperley–Lieb diagramsas U q ( sl ) intertwiners between tensor powers of V , the fundamental representation ofthe quantum group. The category B B N q ( A ) is a quotient of B N q ( A ) by a certain localrelation defined in [Boe08].It follows from that our annular quantum TQFTs functor F A q , when restricted to B B N q ( A ), is faithful. Once the relation between F A q and F TL is established, we candirectly see that setting q = 1 recovers the APS TQFT functor. Outline
The paper is organized as follows. In Section 2 we develop the theory of traces in en-docategories. We discuss fundamental properties of the universal twisted trace, suchas the universal property, functoriality, and connection to the additive Grothendieckgroup. The special cases of triangulated and graded endocategories are treated sepa-rately. The section ends with a number of examples.Section 3 is devoted to the construction of categorical traces in endobicategories. Wefirst introduce the notion a (pre)shadow and construct the twisted horizontal trace. Itsuniversal property and functoriality are shown in Sections 3.2 and 3.3. Then we discuss11he connection with the vertical trace and prove that the horizontal trace is a categorifi-cation of the universtal trace of an endocategory. Triangulated and quantum preshadowsare discussed further. Again, all examples are gathered at the end of the section. Theo-rem A is proven in Section 3.8.7.After the theory of categorified traces is established we take a closer look at Hochschildhomology and its quantization in Section 4. We show that it is a preshadow on the cat-egory of small linear categories and prove its K –theoretic invariance.With Section 5 we move to the topological part of the paper. Here we review linkhomology theories used in this paper: the formal bracket of Bar-Natan, the Asaeda–Przytycki–Sikora homology for links in a thickened annulus and in a thickened M¨obiusband, and the Chen–Khovanov homology for tangles.We quantize the APS homology in Section 5. Theorems B and C are proven in Section6.3, whereas 6.4 presents a computation of the invariant for torus links.Section 7 contains further applications and generalizations: an extension to annularlink cobordisms, the action of oriented tangles on cablings, quantization of the APShomology for links in a thickened M¨obius band, and two generalizations of the annularhomology.Finally, we gathered in Appendix basic definitions and results concerning links andtangles, representations of U q ( sl ), and elements of the (bi)category theory. Basic conventions and notation
Throughout the paper k is a fixed unital commutative ring or field such as Z , Z p , or C ,and and linearity means linearity over k . An algebra means a k –algebra that is projectiveover k and likewise for modules and bimodules.Graded means always Z –graded. We denote by { d } the upwards degree shift, i.e. M { d } i = M i − d for a graded module M = L d M d .A differential in a complex has homological degree +1 increases the homological de-gree. Thus we follow the cohomological notation and put indices as superscripts. The onlyexception is the Hochschild homology. The homological degree shift [ d ] moves a complexdownwards, i.e. C [ d ] i = C d + i .Ordinary categories are typed with calligraphc letters ( C , V ect , etc.), whereas boldletters are reserved for bicategories ( C , Rep , etc.). We usually use small latin letters forobjects ( x , y , etc.) and for morphisms ( f , g , etc.), whereas 2–morphisms are named bygreek letters ( α , β , etc.), with the exception of canonical isomorphisms in bicategories,for which gothic letters are used ( a , l , m , etc.). Capital letters are mostly reserved forfunctors. Acknowledgements
The authors are grateful to Adrien Brochier, Matthew Hogancamp, Mikhail Khovanov,Slava Krushkal, Aaron Lauda, David Rose, and Paul Wedrich for stimulating discus-sions. During an early stage of the research Robert Lipshitz suggested to look on higherHochschild homology of the arc algebras and Ben Webster pointed a connection betweenHochschild homology and the global dimension. The first two authors are supported bythe NCCR SwissMAP founded by the Swiss National Science Foundation. The thirdauthor was supported by the NSF grant DMS-1111680.12
Generalized traces
We start with a discussion on twisted traces in general categories. Section 2.2 dealswith traces on homotopy categories of complexes, whereas 2.3 describes how twistedtraces can be deformed in case of graded categories. The section ends with a list ofexamples, the most important of which are the twisted Hattori–Stallings trace (2.4.6)and the annular closure of a tangle (2.4.8). For a brief list of constructions on categoriessee Appendix A.3.
Choose a category C with an endofunctor Σ : C C . The following definition extendsthe usual notion of a symmetric trace.
Definition 2.1.
A collection t = { t x } x ∈ C of functions t x : C ( x, Σ x ) S valued in a set S is a Σ –twisted trace or a trace on C with a monodromy Σ if t y (Σ f ◦ g ) = t x ( g ◦ f ) forevery pair of morphisms Σ x g y f x .We shall sometimes refer to traces defined above as right traces , whereas a left trace is defined dually as a collection of morphisms from C (Σ x, x ); the trace condition takesthe form t y ( f ◦ Σ g ) = t x ( g ◦ f ). The two definitions are clearly equivalent when Σis invertible. The naming convention is motivated by traces in pivotal categories, seeExample 2.4.7. Lemma 2.2.
A trace t with monodromy Σ is Σ –invariant, i.e. t Σ x (Σ f ) = t x ( f ) for any f ∈ C ( x, Σ x ) .Proof. Take y = Σ x and g = id Σ x in the trace relation.We often write t : ( C , Σ) S for a twisted trace on C , despite that it is not definedfor all morphisms. Further natural conditions on the components of twisted traces areimposed if ( C , Σ) has an additional structure. For instance, when both C and Σ are linear,then S is assumed to be a module over the ring of coefficients k and each component t x : C ( x, Σ x ) S to be a linear homomorphism.From the point of view of Category Theory, a twisted trace is a dinatural transforma-tion from C ( − , Σ( − )) to the constant bifunctor ∆ S that assigns S to any pair of objectsand id S to any pair of morphisms. It follows that there exists a universal Σ -twisted trace tr Σ : ( C , Σ) Tr( C , Σ) when C is small: the coend of C ( − , Σ( − )) [ML98, ChapterIX.6]. Explicitly, Tr( C , Σ) := a x ∈ Ob( C ) C ( x, Σ x ) . Σ f ◦ g ∼ g ◦ f (2.1)or, when C is k –linear,Tr( C , Σ) := M x ∈ Ob( C ) C ( x, Σ x ) . span k { Σ f ◦ g − g ◦ f } (2.2)where f and g run through all pairs of morphisms Σ x g y f x in C . Each componentof tr Σ takes a morphism f ∈ C ( x, Σ x ) to its equivalence class tr Σ ( f ) ∈ Tr( C , Σ), calledthe trace class of f . The universality means that every twisted trace t : ( C , Σ) S Σ , i.e. there is a unique (linear) function u : Tr( C , Σ) S such that the following triangle C ( x, Σ x ) S Tr( C , Σ) t x tr Σ x ∃ ! u (2.3)commutes for each object x ∈ C . Explicitly, u (tr Σ ( f )) = t x ( f ) for f ∈ C ( x, Σ x ).We shall often refer to Tr( C , Σ) as the
Σ-twisted trace of C , keeping in mind the com-ponents tr Σ x . Moreover, following the standard convention, we omit Σ from the notationif it is the identity functor, writing simply Tr( C ).The trace Tr( C , Σ) can be seen as a coend parametrized by Σ. Therefore, by theParameter Theorem for coends [ML98, cp. Theorem IX.7.2], a natural transformation ofendofunctors induces a map between traces. In fact, such a map exists in a more generalsituation when C can vary as well. First let us create the appropriate framework. Definition 2.3. An endocategory is a pair ( C , Σ) consisting of a category C and a func-tor Σ : C C . Functors between endocategories ( C , Σ) and ( C ′ , Σ ′ ) are pairs ( F, ω )consisting of a functor F : C C ′ and a natural transformation ω : F Σ Σ ′ F . It isfinally understood that a natural transformation η : ( F, ω ) ( F ′ , ω ′ ) between two suchfunctors intertwines ω with ω ′ .The above describes E ndo C at , the 2-category of endocategories. The condition fora natural transformation η can be rephrased by saying that the square F Σ x F ′ Σ x Σ ′ F x Σ ′ F ′ x η Σ x ω x ω ′ x Σ ′ η x (2.4)commutes for every x ∈ C . In the following we argue that isomorphism classes of functorsbetween endocategories induce maps between traces.Choose a functor ( F, ω ) : ( C , Σ) ( C ′ , Σ ′ ) and a trace t : ( C ′ , Σ ′ ) S . We checkdirectly that F ∗ t x ( f ) := t F x ( ω x ◦ F f ) defines a Σ–twisted trace on C ; we call it the pullbackof t along F . Proposition 2.4.
Choose a trace t : ( C ′ , Σ ′ ) S . Then ( GF ) ∗ t = F ∗ ( G ∗ t ) for a pairof functors ( F, ω ) : ( C , Σ) ( C ′ , Σ ′ ) and ( G, ω ) : ( C ′ , Σ ′ ) ( C ′′ , Σ ′′ ) . Morevoer, F ∗ t = F ′∗ t if ( F, ω ) and ( F ′ , ω ′ ) are isomorphic functors.Proof. The first equality follows from a direct computation F ∗ ( G ∗ t ) x ( f ) = ( G ∗ t ) F x ( ω x ◦ F f ) = t GF x ( ω ′ F x ◦ Gω x ◦ GF f ) = ( GF ) ∗ t x ( f ) , (2.5)while for the second statement we use (2.4) to computeΣ ′ η x ◦ ω x ◦ F f ◦ η − x = ω ′ x ◦ η Σ x ◦ F f ◦ η − x = ω ′ x ◦ F ′ f (2.6)where η : ( F, ω ) ( F ′ , ω ′ ) is a natural isomorphism. Corollary 2.5.
A trace t on ( C , Σ) can be twisted by a natural transformation η : Σ Σ to ( η ∗ t ) x := t x ( η x ◦ f ) . roof. The trace η ∗ t is a pullback of t along the functor of endocategories (Id , η ).Let E ndo C at be the truncation of E ndo C at to a category obtained by forgettingnatural transformation and identifying isomorphic functors. Proposition 2.4, when com-bined with the universal property of Tr, implies functoriality of the universal trace. Corollary 2.6.
There is a functor
Tr : E ndo C at S et that sends an endocategory ( C , Σ) to its universal trace Tr( C , Σ) and a functor ( F, ω ) : ( C , Σ) ( C ′ , Σ ′ ) to the map Tr(
F, ω ) : Tr( C , Σ) Tr( C ′ , Σ ′ ) that takes tr Σ ( f ) into tr Σ ′ ( ω ◦ F f ) . Recall that the split Grothendieck group K ( C ) of an additive category is generatedby isomorphism classes [ x ] of objects of C modulo the relation [ x ⊕ y ] = [ x ] + [ y ]. Thereis a natural additive map h : K ( C ) Tr( C ) called the Chern character [BHLZ14]that takes [ x ] to the trace class of id x . This map is no longer well-defined for twistedtraces—identity morphisms does not represent trace classes in Tr( C , Σ). However, themap can be ‘twisted’ in certain cases.
Theorem 2.7.
Let ( C , Σ) be an additive endocategory and η : Id Σ a natural trans-formation of functors. Then there is a natural homomorphism of abelian groups h η : K sp ( C ) Tr( C , Σ) , [ x ] ∼ = tr Σ ( η x ) , (2.7) called the twisted Chern character .Proof. By Corollary 2.6 there is a map of traces η ∗ : Tr( C ) Tr( C , Σ), which takesthe trace class tr( f ) of f ∈ C ( x, x ) into tr Σ ( η x ◦ f ). Compose it with the usual Cherncharacter to obtain h η .Being defined for not necessarily additive categories is a big advantage of traces whencompared to the Grothendieck group. In fact, Tr( C , Σ) can be often computed by con-sidering only a certain subset of objects, which may not be preserved by Σ. For a precisestatement, define the trace of C restricted to B ⊂ Ob( C ) as the quotientTr( C | B, Σ) := M x ∈ B C ( x, Σ x ) . span k { Σ f ◦ g − g ◦ f } (2.8)where f and g run through all pairs of morphisms Σ x g y f x in C with x, y ∈ B (but Σ x may not belong to B ). There is a natural homomorphismTr( C | B, Σ) Tr( C , Σ) . (2.9)We say that B generates additively C if every object of C is a direct summand of an objectof the form b ⊕ · · · ⊕ b r with b i ∈ B . For example, Ob( C ) generates additively boththe additive closure C ⊕ and the idempotent completion Kar( C ) of C (see Section A.3).Both categories come with natural endofunctors induced by Σ, for which we use the samesymbol. Proposition 2.8.
Suppose that C is generated additively by B . Then (2.9) is an iso-morphism. In particular, the inclusions of endocategories ( C , Σ) ( C ⊕ , Σ) , x ( x ) (2.10)( C , Σ) (Kar( C ) , Σ) , x id x . (2.11) induce natural isomorphisms of the universal traces. Remark . Although we discuss here only right traces, all the results can be easilytranslated to left traces. In particular, an endocategory ( C , Σ) admits the universal left trace Tr ℓ ( C , Σ). Unless stated otherwise, the universal trace always means the righttrace.
Let T be a triangulated category and Σ its triangulated endofunctor, meaning that thereis a natural isomorphism α x : Σ( X [1]) ∼ = (Σ X )[1] and for each distinguished triangle X f Y g Z h X [1] (2.12)the triangle Σ X Σ f Σ Y Σ g Σ Z α X ◦ Σ h Σ X [1] (2.13)is also distinguished. We say that a trace t on ( T , Σ) is triangulated if it is additive withrespect to homomorphisms of distinguished triangles, i.e. t ( b ) = t ( a ) + f ( c ) (2.14)for any commutative diagramΣ XX Σ YY Σ ZZ Σ X [1] X [1] Σ ff Σ gg α X ◦ Σ hha b c a [1] (2.15)with distinguished triangles as rows. Lemma 2.10.
Assume that t is a triangulated trace on ( T , Σ) . Then t ( f [1]) = − t ( f ) for any f ∈ T ( x, Σ x ) .Proof. Apply the additivity of t to the diagram (2.15) with X X [1] − id X [1]as the top triangle and ( a, b, c ) = ( f, , f [1]).Likewise for usual traces, triangulated traces are stable under pullbacks along trian-gulated functors, and there is the universal triangulated trace Tr △ ( T , Σ), through whichall triangulated Σ-twisted traces on T factor.A particular example of a triangulated category is Com b/h ( C ), the homotopy categoryof bounded formal complexes over C , i.e. sequences of objects and morphisms . . . C i d i C i +1 d i +1 C i +2 . . . with d i +1 ◦ d i = 0 and C i = 0 except finitely many indices. The degree shift functor shiftsthe complex leftwards and negates its differential, whereas the distinguished triangles areof the form C • f • D • in • cone • ( f ) pr • C [1] • , where cone • ( f ) stands for the mapping cone of f , see Section A.3 for the details. It isimplicitly assumed that C is additive (if not, take its additive closure).16 efinition 2.11. A Lefschetz trace Λ t on Com b/h ( C ) induced by a trace t on C is definedby the formula Λ tx ( f • ) := X i ( − i t x ( f i ) . (2.16)It is immediate that a Lefschetz trace satisfies the Σ-twisted trace relation if it isinduced by a Σ-twisted trace. A little longer computation shows that it is triangulated.In fact, as explained below, all triangulated traces on Com b/h ( C ) are Lefschetz traces. Inwhat follows we consider C as a subcategory of Com b/h ( C ) by understanding an object X as the complex (0 X
0) with X at homological degree 0. Proposition 2.12.
Every triangulated Σ -twisted trace on Com b/h ( C ) is a Lefschetz trace,induced by its restriction to C . In particular, Tr( C , Σ) ∼ = Tr △ ( Com b/h ( C ) , Σ) .Proof. Observe that a bounded complex C • is a mapping cone of a map between smallercomplexes: (0 C min { i : C i = 0 } ,and C > min = (0 C min +1 C min +2 . . . ). The first statement follows thusfrom the induction on the length of a complex together with additivity of triangulatedtraces (2.14). In particular, the universal triangulated Σ-twisted trace on Com b/h ( C )is the Lefschetz trace induced by the universal Σ-twisted trace on C , which impliesthe second statement. A linear category C is graded if it admits an autoequivalence { } : C C , calledthe degree shift . The Grothendieck group K sp ( C ) of a graded category has a naturalstructure of a Z [ q ± ]–module by setting q · [ x ] := [ x { } ]. Motivated by Theorem 2.7 wewant to introduce a similar relation on the trace. There are two ways to do it: by definingthe action of q explicitly or by deforming the trace relation. In what follows we exploreboth approaches.A graded endocategory is a quadruple ( C , Σ , { } , α ) consisting of a category C , end-ofunctors Σ and { } , where the latter is an equivalence, and a natural isomorphism α : Σ( − ) { } ∼ = Σ( −{ } ). The equivalence { } is called the degree shift functor. We useit to equip the universal trace Tr( C , Σ) with an action of Z [ q ± ] by setting q · tr Σ ( f ) := tr Σ ( α x ◦ f { } ) (2.17)for any f ∈ C ( x, Σ x ). The following is an immediate consequence of this definition. Corollary 2.13.
Let ( C , Σ , { } , α ) be a graded additive endocategory and η : Id Σ a natural transformation satisfying η x { } = α x ◦ η x { } . Then the twisted Chern character h η : K sp ( C ) Tr( C , Σ) is Z [ q ± ] –linear. We shall now introduce the action differently by deforming the trace relation. Forthat recall the notion of a pregraded category : it is a category C , morphism sets of whichare graded k –modules C ( x, y ) = M d ∈ Z C ( x, y ) d , (2.18)and the degree is additive with respect to the composition. Naturally, we say that ( C , Σ)is a pregraded category when Σ preserves the grading, i.e. Σ( C ( x, y ) d ) ⊂ C (Σ x, Σ y ) d forall objects x and y . 17 pregraded endocategory ( C , Σ) can be transformed into a graded one ( C gr , Σ gr )by introducing a formal degree shift and taking degree zero morphisms only. Thence,objects of C gr are symbols x { d } with x ∈ Ob( C ) and d ∈ Z , morphisms sets are C gr ( x { a } , y { b } ) := C ( x, y ) a − b , and the degree shift functor increases the number inbrackets by one. The endofunctor Σ gr takes x { a } to (Σ x ) { a } .Dually, a graded endocategory ( C , Σ) can be extended to a pregraded ( C pre , Σ pre ) byforgetting the degree shift functor, while introducing morphisms of nonzero degrees bysetting C pre ( x, y ) d := C ( x { d } , y ) and extending the composition with the formula (cid:0) z g y { d ′ } (cid:1) ◦ (cid:0) y f x { d } (cid:1) := (cid:0) z g y { d ′ } f { d ′ } x { d + d ′ } (cid:1) (2.19)To define Σ pre we use the natural isomorphism α :Σ pre x := Σ x for x ∈ Ob( C pre ) = Ob( C )Σ pre f := Σ f ◦ α dx for f ∈ C pre ( x, y ) d = C ( x { d } , y )The two construction described above are clearly inverse to each other. Definition 2.14.
Choose a k [ q ± ]–module S . A quantum trace on a pregraded endo-category ( C , Σ) is a linear trace t : ( C , Σ) S satisfying the deformed trace condition t y (Σ f ◦ g ) = q | f | t x ( g ◦ f ) for every pair of homogeneous morphisms Σ x g y f x , wherewe write | f | for the degree of f .Write C q for the category C with coefficients extended to k [ q ± ]. There exists a uni-versal Σ–twisted quantum traceTr q ( C , Σ) := M x ∈ Ob( C ) C q ( x, Σ x ) . span k [ q ± ] { Σ f ◦ g − q | f | g ◦ f } (2.20)with the obvious components tr Σ q : C ( x, Σ x ) Tr q ( C , Σ), through which every quantumtrace on ( C , Σ) factorizes. It is a graded k [ q ± ]–module, because the defining relation ishomogeneous. One can see it as a one parameter deformation of Tr( C , Σ). As before, weshall write Tr q ( C ) when Σ is the identity functor.Quantum traces arise from a deformation of the endofunctor Σ. Indeed, a quantumtrace on ( C , Σ) is precisely a trace on ( C q , Σ q ) as defined in the previous section, whereΣ q f := q −| f | Σ f for a homogeneous morphism f . In particular,Tr q ( C , Σ) ∼ = Tr( C q , Σ q ) (2.21)so that all the properties of traces can be easily translated to the quantum setting. Proposition 2.15.
A functor of pregraded endocategories ( F, ω ) : ( C , Σ) ( C ′ , Σ ′ ) induces a map of universal quantum traces Tr q ( F, ω ) : Tr q ( C , Σ) Tr q ( C ′ , Σ ′ ) , whichtakes tr Σ q ( f ) into tr Σ ′ q ( ω ◦ F f ) . Write as before Tr q ( C | B, Σ) for the quantum trace restricted to B ⊂ Ob( C ). We saythat C is generated additively by B in the graded sense if it is generated additively by˜ B := { x { d } | x ∈ B, d ∈ Z } . The following is a special case of Proposition 4.2. Proposition 2.16.
The map Tr q ( C | B, Σ) Tr q ( C , Σ) is an isomorphism if C isgenerated additively by B in a graded sense. In particular, Tr q ( C , Σ) ∼ = Tr q ( C ⊕ , Σ) and Tr q ( C , Σ) ∼ = Tr q (Kar( C ) , Σ) . d componentTr dq ( C , Σ) := M x ∈ Ob( C ) C q ( x, Σ x ) d . span k [ q ± ] { Σ f ◦ g − q | f | g ◦ f } (2.22)is annihilated by (1 − q d ), which follows immediately from Lemma 2.2.Traces on graded endocategories and quantum traces on pregraded endocategories areclosely related to each other. Proposition 2.17.
Let ( C , Σ) be a pregraded extension of a graded endocategory ( C , Σ ) .Then Tr( C , Σ ) and Tr q ( C , Σ) are isomorphic k [ q ± ] –modules.Proof. As the notation suggest, C ( x, Σ x ) are precisely the degree zero morphisms in C ( x, Σ x ). Therefore, there is an obvious surjective map i : Tr( C , Σ ) Tr q ( C , Σ).We check first that it intertwines the action of k [ q ± ]. For that let ι x ∈ C ( x, x { } ) correspond to id x { } ∈ C ( x { } , x { } ) and notice the equality f { } = ι Σ x ◦ f ◦ ι − x forany morphism f : x Σ x . Hence,tr q ( α x ◦ f { } ) = tr q ( α x ◦ ι Σ x ◦ f ◦ ι − x ) = q · tr q (Σ ι − x ◦ α x ◦ ι Σ x ◦ f ) = q · tr q ( f )in Tr q ( C , Σ). To show that i is injective, and hence an isomorphism, take any morphisms f ∈ C ( x, y ) d and g ∈ C ( y, Σ x ) − d , and recall how Σ and the composition of morphismsof nonzero degree is defined in C : g ◦ f := g ◦ f {− d } Σ f ◦ g := Σ f ◦ α d ◦ g { d } where the left and right sides use the composition in C and C respectively. Therefore,tr(Σ f ◦ α d ◦ g { d } ) = tr (cid:0) α d ◦ ( g ◦ f {− d } ) { d } (cid:1) = q d tr( g ◦ f {− d } )in Tr( C , Σ ), revealing that the quantum trace relation in Tr q ( C , Σ) is a consequence ofthe module structure on Tr( C , Σ ). In particular, the kernel of i is trivial. Remark . Triangulated traces also admit deformations as long as the triangulatedcategory, on which they are defined, has an additional grading compatible with the tri-angulated structure. An example is the homotopy category
Com /h ( C ) of a pregradedcategory C , objects of which are formal complexes with a graded differential, and formalchain maps have graded components. In what follows we provide a number of (twisted) traces that appear in algebraic ortopological contexts. Of particular interest to us are the Hattori–Stallings trace (togetherwith its twisted version) and annular closures of tangles.19 .4.1 The trace of a linear map
Let V ect be the category of finite dimensional vector spaces over a field F . It is a classicalresult from Linear Algebra that there is a unique collection of maps tr V : End( V ) F satisfying the trace relation and such that tr V (id) = dim V . Explicitly,tr( f ) := X i f ii ∈ F (2.23)where ( f ij ) is a matrix representation of f in some basis of V . This is the universal trace: V ect is generated additively by B = { F } , so that Tr( V ect ) ∼ = Tr( V ect |{ F } ) = F . In case of Z –graded vector spaces the above trace admits a deformationtr q ( f ) := X n q n tr( f | V n ) ∈ F [ q ± ] , (2.24)where V n ⊂ V is the subspace generated by homogeneous vectors of degree n . It satisfiesthe quantum trace relation: q | f | tr q ( g ◦ f ) = X n ∈ Z q n + | f | tr( g ◦ f | V n ) = X n ∈ Z q n + | f | tr( f ◦ g | W n + | f | ) = tr q ( f ◦ g ) (2.25)for any pair of homogeneous maps V g W f V . Again, Tr q ( g V ect ) ∼ = F [ q ± ] as B = { F } generates the category additively in a graded sense. Recall that the
Lefschetz number Λ( f • ) of an endomorphism f • of a chain complex ofvector spaces is the alternating sumΛ( f • ) := X i ( − i tr( f i ) . (2.26)It is straightforward to check that Λ satisfies the trace relation. Even more, Lefschetznumbers of homotopic maps are equal, so that Λ descends to a trace on Com b/h ( F ),the homotopy category of complexes. It is the universal triangulated trace as stated inProposition 2.12. When complexes of graded vector spaces are considered, one constructsthe graded Lefschetz number by replacing tr with tr q in (2.26). The spectrum of an endomorphism f : V V of a vector space is the set of nonzeroeigenvalues. It is an example of a non-linear trace on V ect . Indeed,( f ◦ g )( x ) = λx ⇒ ( g ◦ f )( g ( x )) = λg ( x )and g ( x ) = 0 if λ = 0. Scaling a homomorphism scales elements of its spectrum accord-ingly, but the spectum of f + g cannot be expressed in general in terms of eigenvalues of f and g . 20 .4.5 Hattori–Stallings trace Let R ep ( A ) be the category of finitely generated projective right modules over a ring A .The Hattori–Stallings trace [Ha65, Sta65] is a homomorphism t H SP : End A ( P ) A/ [ A, A ]defined as the compositionEnd A ( P ) ∼ = P ⊗ A P ∗ ǫ P A/ [ A, A ] , (2.27)where P ∗ = Hom A ( P, A ) is the left A –module dual to P and A/ [ A, A ] is the quotientof k –modules with [ A, A ] generated by differences ab − ba for all a, b ∈ A . The leftisomorphism takes an endomorphism f to ( f ⊗ id)( coev (1)) and the right map is given as ǫ P ( x ⊗ α ) := α ( x ). In particular, ǫ P ◦ ( g ⊗ id) = ǫ Q ◦ (id ⊗ g ∗ ) for any g ∈ Hom A ( Q, P ),which guarantees that the trace relation is satisfied. When P is free of finite rank, t H SP ( f ) = P i f ii for f ∈ End A ( P ) represented by the matrix ( f ij ).Projective modules are direct summands of free modules. Hence, R ep ( A ) is generatedadditively by A , seen as a right A –module, and the universal traceTr( R ep ( A )) ∼ = Tr( R ep A |{ A } ) = A/ [ A, A ] (2.28)coincides with the Hattori–Stallings trace.
Choose an (
A, A )–bimodule M that is finitely generated and projective as a right module,so that ( − ) ⊗ A M is an endofunctor of R ep ( A ). We construct a twisted Hattori–Stallingstrace tr M as the compositionHom A ( P, P ⊗ A M ) c P P ⊗ A M ⊗ A P ∗ ǫ P M . [ A, M ] =: coInv( M ) , (2.29)where the k –module [ A, M ] is generated by differences am − ma with a ∈ A and m ∈ M , c P ( f ) = ( f ⊗ id)( coev (1)), and ǫ P ( x ⊗ m ⊗ α ) := α ( x ) m . The quotient coInv( M ) iscalled the space of coinvariants in M . Again, the trace condition is satisfied because ǫ P ◦ ( g ⊗ id ⊗ id) = ǫ Q ◦ (id ⊗ id ⊗ g ∗ ) for any g ∈ Hom A ( Q, P ).The twisted Hattori–Stallings trace is the universal trace on ( R ep ( A ) , M ). Indeed,Tr( R ep ( A ) , M ) ∼ = Tr( R ep ( A ) |{ A } , M ) ∼ = coInv( M ) (2.30)by Proposition 2.8, where the right isomorphism arises from the identifications of End A ( A )and Hom A ( A, M ) with A and M respectively, each evaluating a function at 1 ∈ A .Given an algebra endomorphism ϕ ∈ End
Alg ( A ) one constructs a twisted trace on R ep ( A ) as follows. For a right module V define its ϕ –twisting V ϕ := V ⊗ A ϕ A , where ϕ A is obtained from A by redefining the action as a · x := ϕ ( a ) x . In case ϕ is invertible, thereis an isomorphism V ϕ ∼ = V ϕ − that takes v ⊗ a to vϕ − ( a ), where V ϕ − is obtained from V simply by twisting the action of A . If so, a homomorphism f : W V ϕ is preciselya k –linear map f : W V satisfying f ( wϕ ( a )) = f ( w ) a . This leads to a twistedHattori–Stallings trace tr ϕ : R ep ( A ) A . [ A, A ] ϕ , (2.31) Notice the difference between ǫ and the evaluation map ev : P ∗ ⊗ P A , the latter being a ho-momorphism of (
A, A )–bimodules.
A, A ] ϕ = span k { ϕ ( a ) b − ba | a, b ∈ A } . Notice that tr ϕ ( f ) = P i f ii for anyendomorphism f of a free A –module of finite rank. This is the same formula as inthe untwisted case, but computed in a different quotient of A .Choose an invertible scalar q ∈ k . If A = L d ∈ Z A d is a graded algebra and M a graded bimodule, then the twisted Hattori–Stallings trace can be deformed to takevalues in the quantum space of coinvariants of M coInv q ( M ) := M/ [ A, M ] q , (2.32)where [ A, M ] q = span k { am − q d ma | a ∈ A d , m ∈ M } . The details are left to the reader. A pivotal category is a monoidal category C with left duals and a natural isomorphism δ x : x ∗∗ x for any x ∈ C . Particular examples are • the category V ect of finite dimensional vector spaces together with δ = δ vect the standard isomorphism, and • the representation category R ep ( H ) of a Hopf algebra H with a pivot , a group-like element u ∈ H satisfying S ( h ) = uhu − ; the isomorphism δ V is given bythe formula δ V ( x ) = u · δ vect ( x ).It is common to represent C diagrammatically: products of objects as sequences of pointslabeled by the factors, and morphisms as graphs with coupons, read from bottom to topand understood as follows: x ∗ x ≡ x ⊗ ∗ xI ev ∗ x x ≡ I ∗ x ⊗ x coev xyf ≡ xy f The relations between evaluation and coevaluation translates into straightening zig-zags= =so that isotopic diagrams represent the same morphism. The left and right pivotal traces t ℓ ( f ) and t r ( f ) of an endomorphism f ∈ C ( x, x ) are defined as the compositions t ℓ ( f ) := fδ x ∗ x ∗∗ xxx t r ( f ) := fδ − x ∗ x ∗∗ xxx (2.33)They satisfy left and right version of the trace relation (see the discussion below Defini-tion 2.1) t ℓ ( f ◦ g δ ) = t ℓ ( g ◦ f ) t r ( f δ ◦ g ) = t r ( g ◦ f ) (2.34)with ( − ) δ := δ ◦ ∗∗ ( − ) ◦ δ − . The two traces coincide when C is spherical , but in generalthey are different. See [BBG18] for more details.22hen C = R ep ( H ) is the representation category of a pivotal Hopf algebra, one cancompute the universal right traceTr( R ep ( H ) , δ ) ∼ = H . span k { S ( y ) x − xy | x, y ∈ H } (2.35)using the same methods as before. The resemblence with the twisted trace from Sec-tion 2.4.6 is not a coincidence: there is a natural isomorphism of functors on R ep ( H ) η : ( − ) δ ( − ) u , η V ( x ) = xu − , (2.36)where u acts on H by conjugation. Indeed, η V ( xh ) = ( xu − )( uhu − ) = η V ( x ) · h . Analo-gous statments hold for the left trace. Let q : R × I R × S be the quotient map that identifies ( x, y,
0) with ( x, y,
1) forany ( x, y ) ∈ R . If the tangle T ⊂ R × I has the same endpoints on both boundaryplanes, then q ( T ) is a link in R × S . It is called the annular closure of T , because it canvisualized at the level of tangle diagrams by connecting the endpoints inside an annulus,see Figure 2. We consider it as a link b T in R by using the standard embedding of R × S ∼ = D × S into the 3-space. d ( − ) Figure 2: The annular closure of a tangle.The annular closure described above is a trace function on T an ( R ) valued in Links ( R ).Indeed, the links d T ′ T and d T T ′ are isotopic by rotation. Embeddings of R × S into R are parametrized by framed knots—the tubular neighbor-hood of a knot is homeomorphic to R × S , but it is the framing that fixes the isotopyclass of the homeomorphism. Let h : R × S R be the embedding corresponding toa framed knot K . The image h ( L ) of a link L ⊂ R × S is called a satellite link with companion K .Write K as an annular closure of a framed (1 , J . The tangle J describesa tangled embedding of R × I into itself. Let J B be the image under this embedding ofthe trivial tangle B × I , where B is any set of points on R . Then J B ′ T and T J B areisotopic for any tangle T with B and B ′ as its bottom and top boundary, where the isotopyslides T along J . In other words, J • is a natural transformation of the identity functoron T an ( R ). Using Proposition 2.4 we construct a new trace function d ( − ) J on T an ( R ).It takes a tangle T with same top and bottom boundary to b T J := h ( b T ), a satellite knotwith compagnion K (see Figure 3). 23 d ( − ) J T Figure 3: The closure of a (3 , Choose a surface F with a diffeomorphism φ ∈ Diff ( F ). The mapping torus M := F × I . ( p, ∼ ( φ ( p ) ,
0) (2.37)is called the surface bundle with fiber F and monodromy φ . The annular closure oftangles can be generalized to a procedure producing links in M .Given a tangle T ⊂ F × I with input φ ( B ) and output B we construct its closure b T := π ( T ) in M , where π : F × I M is the quotient map. This operation satisfiesthe left trace relation with respect to the endofunctor φ ∗ on T an ( F ) that takes a tangle T ⊂ F × I into ( φ × id)( T ). In another words, closing tangles describes a twisted trace d ( − ) : ( T an ( F ) , φ ∗ ) Links ( M ). Theorem 2.19.
The closure d ( − ) is the universal left trace on ( T an ( F ) , φ ∗ ) .Proof. Links ( M ) is a quotient set of Tr( T an ( F ) , φ ∗ ), because each link in M is a closureof a certain tangle. Suppose that closures b T and b T ′ are isotopic in M . The isotopycan be expressed as a sequence of isotopies supported in small 3–balls, so that we mayassume that it fixes some fiber F ′ ⊂ M . Then the cuts b T and b T ′ along F ′ are isotopicin F × I and the trace relation implies that the images of T and T ′ in Tr ℓ ( T an ( F ) , φ ∗ )coincide. The Kauffman Bracket Skein Module S ( M ) of an oriented 3–manifold M is a Z [ A ± ]–module generated by isotopy classes of framed tangles in M (we require the isotopies tofix ∂M if nonempty) modulo the local relations= A + A − (2.38)= − A − A − (2.39)where the diagrams represent pieces of tangles picked by some small ball, outside of whichthe tangles coincide. In particular, the circle in the second relation bounds a disk. When F is a surface, then S ( F × I ) is a category and any φ ∈ Diff ( F ) induces an endofunctor φ ∗ as in the previous example. The proof of Theorem 2.19 can be easily modified toshow that the universal left trace Tr ℓ ( S ( F × I ) , φ ∗ ) computes S ( M ), the skein module ofthe surface bundle M with fiber F and monodromy φ .24 Traces of bicategories
In this section we categorify twisted traces. We begin with the definition of a preshadowand the construction of the (twisted) horizontal trace of an endobicategory. The universalproperty and functoriality of the latter are shown in Sections 3.2 and 3.3 respectively,whereas Sections 3.4 and 3.5 discuss two ways to obtain traces from shadows: by re-stricting to the vertical trace or by decategorifying. These are followed by a materialon Lefschetz preshadows and a quantization of this framework. Examples are listed atthe end of the section.
Choose a bicategory C together with a strong bifunctor Σ : C C . These come togetherwith natural 2-isomorphisms, the coherence isomorphisms a : h ◦ ( g ◦ f ) ∼ = ( h ◦ g ) ◦ f, m : Σ g ◦ Σ f ∼ = Σ( g ◦ f ) , l : id y ◦ f ∼ = f, i : id Σ x ∼ = Σ(id x ) , r : f ◦ id x ∼ = f, which are often omitted for clarity, see Appendix A.4. Definition 3.1.
A Σ –twisted preshadow or a preshadow with monodromy
Σ on C valuedin a category T is a collection of functors hh−ii x : C (Σ x, x ) T , one per object x ∈ C ,and natural morphisms θ g,f : hh f ◦ Σ g ii y hh g ◦ f ii x in T , one for each pair of 1–morphisms x g y f Σ x , such that the following diagrams commute hh f ◦ Σ( h ◦ g ) ii y hh ( h ◦ g ) ◦ f ii xθ hh ( f ◦ Σ h ) ◦ Σ g ii y hh g ◦ ( f ◦ Σ h ) ii z hh ( g ◦ f ) ◦ Σ h ii z hh h ◦ ( g ◦ f ) ii x hh a ∗ ( ◦ m − ) ii θ hh a ii θ hh a ii (3.1) hh k ◦ id Σ x ii x hh k ◦ Σ(id x ) ii x hh id x ◦ k ii x hh k ii x hh ◦ i ii θ hh r ii hh l ii (3.2)for all x h z g y f Σ x and x k Σ x . A preshadow is a shadow if each θ g,f isan isomorphism. It is symmetric when θ f,g ◦ θ g,f = id. Remark . The naturality of θ means that θ g ′ ,f ′ ◦ hh α ◦ Σ β ii y = hh β ◦ α ii x ◦ θ g,f for any2-morphisms α : f f ′ and β : g g ′ .We often write hh−ii : ( C , Σ) T for a twisted preshadow on C , despite that it isnot a bifunctor. It is understood that when morphism categories C ( x, y ) are linear, thenso are both T and the functors hh−ii x . 25wisted shadows are categorified versions of traces in the sense, that the equality t ( f ◦ Σ g ) = t ( g ◦ f ) defining a (left) twisted trace is replaced by a choice of a naturalisomorphism hh f ◦ Σ g ii ∼ = hh g ◦ f ii . When C are Σ are strict, then the diagrams (3.1) and(3.2) simplify to θ h ◦ g,f = θ h,g ◦ f θ g,f ◦ Σ h and θ id ,k = id k .The definition of a preshadow coincides with the one of a categorified trace from[HPT15] when Σ = Id and C has only one object, i.e. it is a monoidal category. Incomparison to [PS13] we do not require θ f,g ◦ θ g,f = id.In what follows we construct a twisted preshadow for every bicategory—a generaliza-tion of the horizontal trace [BHLZ14]. The next section is devoted to show its functorialityas well as universality. Definition 3.3.
The Σ -twisted horizontal trace of a bicategory C is the category hTr( C , Σ),objects of which are 1–morphisms f ∈ C (Σ x, x ), morphisms from f ∈ C (Σ x, x ) to g ∈ C (Σ y, y ) are equivalence classes [ p, α ] of squaresΣ x x Σ y y f Σ p pgα (3.3)modulo the relation Σ x x Σ y y f pg Σ p ′ Σ p α Σ τ ∼ Σ x x Σ y y f Σ p ′ g p ′ pα τ (3.4)with [id x , f ] the identity on f , and the composition[ q, β ] ◦ [ p, α ] := [ q ◦ p, ( β ◦ Σ p ) ∗ ( q ◦ α )] (3.5)can be visualized as stacking squares one on top of the other:Σ y y Σ z z g Σ q qhβ ◦ Σ x x Σ y y f Σ p pgα := Σ x x Σ y y Σ z z f Σ p pg Σ q qhαβ (3.6)Unitarity and associativity follows from (3.4) with an appropriate composition of associ-ators and unitors as τ . Proposition 3.4.
The horizontal trace is the target of a preshadow hh−ii h on ( C , Σ) withcomponents hh−ii h x : C (Σ x, x ) hTr( C , Σ) the obvious functors and θ h g,f := [ g, a ] . It isa shadow when C has right duals.Proof. Commutation of both (3.1) and (3.2) is proven by a direct computation, which isleft to the reader. When C has right duals, consider a morphism represented byΣ x y x Σ y Σ x y f g Σ g ∗ g ∗ Σ g f id y id Σ x coev ev (3.7)26here the triangles are filled with the coevaluation and evaluation 2-morphisms, andthe middle parallelogram with a suitable composition of unitors. It is a two-sided inverseof θ h g,f due to the relations (A.17) between ev and coev .The idea underlying the definition of the horizontal trace is to represent 1-endomor-phisms by boundary circles of a cylinder, the interior of which carries a 2-morphism.Indeed, such a cylinder can be obtained by gluing the vertical boundaries of the square(3.3) together, but keeping the seam line, see Figure 4. The seam is cooriented and Σ p pfgα −→ α fgp Σ p Figure 4: Visualization of the horizontal trace.a 2-morphism can “slide” through it, in which case it is acted upon by Σ as describedby (3.4). The morphism θ h is represented in this picture by the cylinder obtained fromthe identity on g ◦ f by applying a half-twist to its top boundary circle, see Figure 5. g fg f id g idΣ id twist the top circleby π clockwise f Σ gg f id g Σ g Σ id
Figure 5: Visualization of the cyclicity morphism in hTr.The symmetric horizontal trace (i.e. for Σ = Id) is also called the annularizationfunctor [MW10]. Its generalization to any surface is known as factorization homology [BZBJ15].
Let hh−ii : ( C , Σ) T and hh−ii ′ : ( C , Σ) T ′ be two preshadows. We say that hh−ii ′ factorizes through hh−ii if there exist a functor T : T T ′ and a collection ofnatural isomorphisms η x : T ◦ hh−ii x hh−ii x ′ such that η x ◦ T ( θ g,f ) = θ ′ g,f ◦ η y for any x g y f Σ x . Factorizations ( T, η ) and ( T ′ , η ′ ) are equivalent if there exists a naturalisomorphism ǫ : T T ′ such that η x = η ′ x ∗ ( ǫ ◦ hh−ii ), i.e. C (Σ x, x ) TT ′hh−ii x hh−ii x ′ TT ′ ǫη ′ = C (Σ x, x ) TT ′hh−ii x hh−ii x ′ Tη (3.8)The following result states that the horizontal trace twisted by Σ is the universal Σ–twisted preshadow. 27 heorem 3.5. If C has left duals, then every preshadow on ( C , Σ) factorizes through hTr( C , Σ) uniquely up to an equivalence.Proof. Given a preshadow hh−ii : ( C , Σ) T we construct a functor T : hTr( C , Σ) T by taking f : Σ x x to hh f ii x and a morphism [ p, α ] : f g to the composition hh f ii x hh coev ◦ ii x hh ∗ p ◦ p ◦ f ii x hh ◦ α ii y hh ∗ p ◦ g ◦ Σ p ii xθ p, ∗ p ◦ g hh p ◦ ∗ p ◦ g ii y hh ev ◦ ii y hh g ii y , (3.9)where associators and unitors are omitted for clarity. Notice that hh−ii = T ◦ hh−ii h and θ g,f = T ( θ h g,f ). For uniqueness, suppose that ( T ′ , η ) : hh−ii h hh−ii is anotherfactorization of a morphism of shadows with each η x an isomorphism. Then η is a naturalisomorphism of functors η : T ′ T , because hh f ii h = f and T f = hh f ii ; naturality followsfrom (3.9) and the observation that a similar sequence determines T ′ ([ p, α ]). Corollary 3.6.
Suppose C has both left and right duals. Then any preshadow on C isa shadow.Proof. It follows from Theorem 3.5 that a preshadow ( hh−ii , θ ) factorizes through the hor-izontal trace. In particular, θ is an image of θ h , which is an isomorphism by Proposi-tion 3.4. To discuss functoriality of hTr we begin with constructing a suitable 3-category of endo-bicategories.A morphism of endobicategories ( F , ω ) : ( C , Σ) ( C ′ , Σ ′ ) consists of a strong bi-functor F : C C ′ , so that the coherence 2-morphisms m F : F g ◦ F f F ( g ◦ f ) and i F : id F x F (id x ) are invertible, and a weak natural transformation ω : Σ ′ F F
Σ.The latter comes with a collection of 2-morphismsΣ ′ F x F Σ x Σ ′ F y F Σ y ω x Σ ′ F f F Σ fω y ω f (3.10)parametrized with 1-morphisms f ∈ C ( x, y ). The 2-morphisms ω f are not required to beinvertible. If they are, we say that ( F , ω ) is strong .Suppose that ( F , ω ) and ( F ′ , ω ′ ) are morphisms between the same endobicategories.A natural transformation ( η, n ) : ( F , ω ) ( F ′ , ω ′ ) consists of a strong natural transfor-mation of bifunctors η : F F ′ and a natural 2-morphism n x : η Σ x ◦ ω x ω ′ x ◦ Σ ′ η x that—in addition to the usual coherence conditions—fits into the commutative hexagon F ′ Σ f ◦ η Σ x ◦ ω x ω ′ y ◦ Σ ′ ( η y ◦ F f ) F ′ Σ f ◦ ω ′ x ◦ Σ η x ω ′ y ◦ Σ ′ F ′ f ◦ Σ ′ η x η Σ y ◦ F Σ f ◦ ω x η Σ y ◦ ω y ◦ Σ ′ F f ◦ n x ω ′ f ◦ ◦ Σ ′ η f η Σ f ◦ ◦ ω f n y ◦ (3.11)28or any 1-morphism f ∈ C ( x, y ), where canonical isomorphisms are omitted.Finally, we require that a modification Γ : η η ′ of natural transformations ( η, n )and ( η ′ , n ′ ) satisfies ( ◦ ΣΓ x ) ∗ n x = n ′ x ∗ (Γ Σ ′ x ◦ ) in addition to the usual coherencecondition. Definition 3.7.
We write
EndoBicat for the 3-category of endobicategories, morphisms,natural transformations and modifications, as described above. Its restriction to endobi-categories with left duals is denoted by
EndoBicat ∗ .Preshadows can be pulled back along morphisms of endobicategories. Indeed, givena preshadow hh−ii on ( C ′ , Σ ′ ) and a morphism ( F , ω ) : ( C , Σ) ( C ′ , Σ ′ ) we define a itspullback ( F ∗ hh−ii , F ∗ θ ) on ( C , Σ) as follows: F ∗ hh f ii x := hh F f ◦ ω x ii F x F ∗ hh α ii x := hh F α ◦ ii F x with the cyclicity morphism ( F ∗ θ ) g,f the composition F ∗ hh f ◦ Σ g ii y = hh F ( f ◦ Σ g ) ◦ ω y ii F y hh m − F ◦ ii hh F f ◦ F Σ g ◦ ω y ii F y hh ◦ ω g ii hh F f ◦ ω x ◦ Σ ′ F g ii F yθ hh F g ◦ F f ◦ ω x ii F x hh m F ◦ ii hh F ( g ◦ f ) ◦ ω x ii F x = F ∗ hh g ◦ f ii x . Notice that the second arrow is invertible when ( F , ω ) is a strong morphism. Proposition 3.8.
The datum ( F ∗ hh−ii , F ∗ θ ) is a preshadow on ( C , Σ) . It is a shadowwhen ( F , ω ) is a strong morphism and ( hh−ii , θ ) is a shadow.Proof. Left as an exercise.Let hh−ii , hh−ii ′ : ( C , Σ) T be two preshadows. A collection of natural transforma-tions η x : hh−ii x hh−ii x ′ is a morphism of preshadows if it is coherent with the cyclicitymorphisms: θ ′ g,f ◦ η y = η x ◦ θ g,f . There is a morphism of pullback shadows associated toany natural transformation of morphisms of endobicategories( C , Σ) ( C ′ , Σ ′ ) ( F ,ω )( F ′ ,ω ′ )( η, n ) (3.12)Indeed, there is a morphism in hTr( C ′ , Σ ′ )Σ ′ F x F Σ x F x Σ ′ F ′ x F ′ Σ x F ′ x ω x F f Σ ′ η x η Σ x η x ω ′ x F ′ f n x η − f (3.13)which, assuming that C ′ has left duals, descend to a morphism ( η, n ) f in T between F ∗ hh f ii and F ′∗ hh f ii for any shadow hh−ii on ( C ′ , Σ ′ ). Lemma 3.9.
The collection ( η, n ) ∗ is a morphism of shadows from F ∗ hh−ii to F ′∗ hh−ii .Morphisms induced by isomorphic transformations are equal. roof. The naturality and compatibility with cyclicity morphisms can be checked inhTr( C ′ , Σ ′ ). They follow directly from the coherence conditions of η f and n x . The detailsare left as an exercise.Everything together implies functoriality of the horizontal trace. Let EndoBicat ∗ , be the restriction of EndoBicat ∗ to a 2-category, obtained by forgetting modificationsand identifying isomorphic natural transformations. Theorem 3.10 (Functoriality of the horizontal trace) . The horizontal trace hTr extendsto a strict 2-functor hTr :
EndoBicat ∗ , Cat , which assigns to a morphism of endo-bicategories ( F , ω ) : ( C , Σ) ( C ′ , Σ ′ ) the functor hTr( F , ω ) : hTr( C , Σ) hTr( C ′ , Σ ′ ) that takes an object f : Σ x x into F f ◦ ω x and a morphism [ p, α ] to the class of the di-agram Σ ′ F x F Σ x F x Σ ′ F y F Σ y F y ω x F f Σ ′ F p F Σ p F pω y F gω p F α (3.14) and a natural transformation ( η, n ) is sent to the natural transformation hTr( η, n ) withcomponents (3.13) .Proof. Apply Theorem 3.5 to the pullback along ( F , ω ) of the universal preshadow on( C ′ , Σ ′ ) to construct hTr( F , ω ). The coherence condition for a natural transformationguarantees that hTr( η, n ) is a morphism of shadows. Corollary 3.11.
Let ( F , ω ) : ( C , Σ) ( C ′ , Σ ′ ) be a local equivalence, i.e. ω is invertibleand F restricts to equivalences of morphism categories C ( x, y ) ≈ C ′ ( F x, F y ) for allobjects x, y ∈ C . Then hTr( F , ω ) is full and faithful.Proof. By replacing C ′ with the image of F we may assume that the bifunctor is a biequiv-alence. Then hTr( F , ω ) is an equivalence of categories by Theorem 3.10 and Lemma 3.9,hence, full and faithful. Assume now that Σ fixes objects, i.e. Σ x = x for any object x ∈ C . Then Σ canbe restricted to each morphism category C ( x, y ). Replacing each of them by its traceTr( C ( x, y ) , Σ) results in the vertical trace vTr( C , Σ): a category with the same objectsas C , morphisms twisted trace classes of 2-morphisms α : f Σ f , and compositioninduced by the horizontal composition of 2-morphisms. The composition in vTr( C , Σ)is both unital and associative, because the trace class of a morphism is invariant underconjugation.There is a natural functor vTr( C , Σ) hTr( C , Σ), defined by expanding objects toidentity morphisms. Explicitly, x ( x id x x ) and xy f Σ f σ x xy y id x Σ f f id y σ It is clearly full and faithful, but not necessarily surjective on objects.30hen C has left duals, then the above functor can be used to restrict a preshadow hh−ii : ( C , Σ) T to a trace t hh−ii : ( C ( x, y ) , Σ) T ( hh id x ii , hh id y ii ) using the formula t hh−ii ( p α Σ p ) := hh [ p, α ] ii , (3.15)where we identify hh−ii with the corresponding functor on hTr( C , Σ). The following isan immediate consequence of this construction.
Proposition 3.12.
The restriction to C ( x, y ) of the universal Σ –twisted preshadow isthe universal Σ –twisted trace. An n –category C can be truncated to an ( n − n − n Cat ( n − Cat . Clearly, this construction applies to endo- n -categories as well.We are mostly interested in the case n = 1 , hh−ii : C T results in a (left) trace Π hh−ii : Π C Π T .Conversely, any left trace on Π C can be lifted to a shadow if C is small. Lemma 3.13.
Choose a small endobicategory ( C , Σ) and a (left) trace tr : (Π C , ΠΣ) S for some set S . There exists a shadow T : ( C , Σ) S , such that Π S = S and Π T = tr .Proof. In what follows we write dom( α ) := f and cod( α ) := f ′ for the domain andcodomain of a 2–morphism α : f f ′ . Let ˜ S be the category with Ob( ˜ S ) = S andmorphisms ˜ S ( s, t ) the finite sequences ( α n , . . . , α ) of 2–morphisms α i ∈ C (Σ x i , x i ) sat-isfying tr(dom( α i +1 )) = tr(cod( α i )) for i = 1 , . . . , n −
1, such that s = tr(dom( α )) and t = tr(cod( α n )). Composition of morphisms is defined as concatenation of sequences.The category S is a quotient of ˜ S by the relations( . . . , β, α, . . . ) ∼ ( . . . , β ∗ α, . . . ) whenever β ∗ α exists, and (3.16)( . . . , , . . . ) ∼ ( . . . , . . . ) . (3.17)They allow us to reduce a given sequence of 2–morphism to a sequence of noncomposablemorphisms, none of which is the identity. An easy application of the Bergman DiamondLemma [Be78] shows that this reduced sequence is unique. In particular,( β m , . . . , β ) ◦ ( α , . . . , α n ) ∼ () (3.18)implies m = n and each β i ◦ α i = if both sequences are reduced. Hence, each α i is a 2–isomorphism in C if ( α , . . . , α n ) is an isomorphism in S , which implies thattr(dom( α i )) = tr(cod( α i )) anddom( α n , . . . , α ) = tr(dom( α )) = tr(cod( α n )) = cod( α n , . . . , α ) . (3.19)Therefore, S does not have isomorphisms between different objects, and Π S = S . The de-sired shadow T : ( C , Σ) S is defined as T ( f ) := tr( f ) and T ( α ) := ( α ).The above result allows us to formally prove that the horizontal trace is a categorifi-cation of the universal trace of a category. Despite proving it only for small bicategories,we believe that with a slight modification of our argument the result can be generalizedto locally small bicategories. A bicategory is small if its 2-morphisms (and so objects as well as 1-morphisms) form a set. heorem 3.14. There is a natural bijection
Π(hTr( C , Σ)) ≈ Tr ℓ (Π C , ΠΣ) for each smallendobicategory ( C , Σ) with left and right duals.Proof. Because C has right duals, hTr( C , Σ) is a shadow and we can apply Π to geta trace Π(hTr( C , Σ)). We must show that it is the universal left trace, i.e. that every lefttrace t : (Π C , ΠΣ) S factorizes through it. For that use Lemma 3.13 to find a shadow T : ( C , Σ) S that lifts t . According to Theorem 3.5, T factorizes through hTr( C , Σ),and applying Π results in a desired factorization of t .Thence, the horizontal trace is a categorification of the universal left trace. Righttraces are categorified by a dual version of shadows, defined on morphism categories C ( x, Σ x ). The convention for the twisting of hTr used in this paper is justified bythe relation of hTr with the vertical trace discussed in Section 3.4. Let
Com b/h ( C ) be a bicategory obtained from a locally additive bicategory C by replacingeach morphism category C ( x, y ) with the corresponding homotopy category of boundedcomplexes Com b/h ( C ( x, y )). It is locally triangulated, i.e. the functors f ◦ ( − ) and ( − ) ◦ f preserve distinguished triangles and commute with homological degree shifts. Havingchosen an endobifunctor Σ on C we extend it naturally over Com b/h ( C ). Definition 3.15.
A preshadow hh−ii : (
Com b/h ( C ) , Σ) T is triangulated if T is a tri-angulated category and the components hh−ii x are triangulated functors.A twisted preshadow hh−ii : ( C , Σ) T can be extended to a twisted triangulatedpreshadow hhh−iii : Com b/h ( C )(Σ x, x ) Com b/h ( T ), which we call the Lefschetz preshadowinduced by hh−ii . For each object x ∈ C the functor hhh−iii x is constructed by applying hh−ii x component-wise: hhh f • iii x = (cid:0) . . . hh f i ii x hh d ii x hh f i +1 ii x hh d ii x hh f i +2 ii x . . . (cid:1) (3.20)and likewise for 2-morphisms. Notice that hhh g • ◦ f • iii ix = L p + q = i hh g p ◦ f q ii x for a pair of1-morphisms x g • y f • Σ x . We check below that the sumsΘ ig • ,f • := X p + q = i ( − pq θ g p ,f q (3.21)are components of a formal chain map Θ g • ,f • : hhh f • ◦ Σ g • iii x hhh g • ◦ f • iii y . It is invertibleif ( hh−ii , θ ) is a shadow. Lemma 3.16.
The datum ( hhh−iii , Θ) is a triangulated preshadow on Com b/h ( C ) . It isa shadow if so is hh−ii .Proof. The functors hhh−iii x are additive and, as such, they preserve mapping cones. Hence,they are triangulated. To check that Θ g • ,f • is a chain map, we compute d ig • ◦ f • ◦ Θ ig • ,f • = X p + q = i ( − pq hh d pg • ◦ id +( − p id ◦ d qf • ii ◦ θ g p ,f q = X p + q = i ( − pq (cid:0) θ g p +1 ,f q ◦ hh id ◦ d p Σ g • ii + ( − p θ g p ,f q +1 ◦ hh d qf • ◦ id ii (cid:1) X p + q = i +1 ( − pq θ g p ,f q ◦ hh ( − q id ◦ d p Σ g • + d qf • ◦ id ii = Θ i +1 g • ,f • ◦ d if • ◦ Σ g • and the naturality follows from the naturality of θ :Θ ig • ,f • ◦ hhh α • ◦ Σ β • iii i = X p + q = i ( − pq θ g p ,f q ◦ hh α p ◦ Σ β q ii = X p + q = i ( − pq hh β q ◦ α p ii ◦ θ g p ,f q = hhh β • ◦ α • iii i ◦ Θ ig • ,f • for any 2-morphisms α • : f • f • and β • : g • g • . Finally, commutativity of (3.1)and (3.2) can be easily checked component-wise.Recall that a triangulated trace is additive with respect to (twisted) endomorphismsof distinguished triangles. An analoguous statement holds for Lefschetz preshadows. Proposition 3.17.
Let C have left duals and choose a commuting diagram in Com b/h ( C ) p • ◦ f • r • ◦ f • s • ◦ f • ( p • ◦ f • )[1] g • ◦ Σ p • g • ◦ Σ r • g • ◦ Σ s • ( g • ◦ Σ p • )[1] α ◦ idid ◦ Σ α β ◦ idid ◦ Σ β γ ◦ idid ◦ Σ γπ ρ σ π [1] (3.22) where f • ∈ Com b/h ( C )(Σ x, x ) , g • ∈ Com b/h ( C )(Σ y, y ) , and p • α r • β s • γ p • [1] isa distinguished triangle in Com b/h ( C )( x, y ) . Then hhh [ r, ρ ] iii = hhh [ p, π ] iii + hhh [ s, σ ] iii .Proof. The 1-morphism s • is homotopy equivalent to the mapping cone of α : p • r • .In particular, σ n : ( s • ◦ f • ) n ( g • ◦ Σ s • ) n consists of components σ p,p : p j +1 ◦ f i g i ′ ◦ Σ p j ′ +1 σ r,p : r j ◦ f i g i ′ ◦ Σ p j ′ +1 σ p,r : p j +1 ◦ f i g i ′ ◦ Σ r j ′ σ r,r : r j ◦ f i g i ′ ◦ Σ r j ′ (3.23)for i + j = i ′ + j ′ = n . A simple diagram chasing shows that σ p,p and σ r,r are chain mapshomotopic to π [1] and ρ respectively. We can thus take σ • p,p = π [1] • and σ • r,r = ρ • .Recall that the left dual to ( C • , d ) is given by the complex . . . ∗ C − i − ∗ d − i ∗ C − i ∗ d − i +1 ∗ C − i +1 . . . (3.24)with ∗ C − i in homological degree i . The evaluation and coevaluation maps do not vanishonly in homological degree zero, where they are given by the obvious maps A P coev Ci M i ∗ C i ⊗ C i = ( ∗ C • ⊗ C • ) (3.25)and ( C • ⊗ ∗ C • ) = M i C i ⊗ ∗ C i P ev Ci A. (3.26)In particular, s j = p j +1 ⊕ r j leads to the decomposition( ∗ s ) − j ◦ s j = ( ∗ p j +1 ◦ p i + j ) ⊕ ( ∗ r j ◦ r j ) ⊕ ( ∗ p j +1 ◦ r j ) ⊕ ( ∗ r j ◦ p j +1 ) . (3.27)33he coevaluation takes values only in the first two terms, which are also the only compo-nents on which the evaluation does not vanish. Hence, the only components of hhh [ s, σ ] iii that contribute are hh f i ii hh coev ◦ ii hh ∗ p j +1 ◦ p j +1 ◦ f i ii hh ◦ σ p,p ii hh ∗ p j +1 ◦ g i Σ ◦ p j +1 ii ( − ( i + j ) j θ hh p j +1 ◦ ∗ p j +1 ◦ g i ii hh ev ◦ ii hh g i ii , (3.28)and those with p j +1 replaced by r j . Because σ • p,p = π [1] • and σ • r,r = ρ • , the compositions(3.28) (resp. those with r j instead of p j +1 ) coincide with the components of hhh [ p [1] , π [1]] iii (resp. hhh [ r, ρ ] iii ). Hence, hhh [ s, σ ] iii = hhh [ r, ρ ] iii + hhh [ p [1] , π [1]] iii = hhh [ r, ρ ] iii − hhh [ p, π ] iii . Corollary 3.18.
Assume that C has left duals and Σ fixes objects of C . Then the Lef-schetz preshadow hhh−iii on Com b/h ( C ) induced by hh−ii restricts to the Lefschetz trace onCom b/h ( C ( x, y )) induced by t hh−ii , the trace defined as in (3.15) .Proof. The restricted trace is triangulated due to Proposition 3.17 applied to the case f = id x and g = id y . Hence, the thesis follows from Proposition 2.12. A bicategory C is locally pregraded if each morphism category C ( x, y ) is pregraded,the horizontal composition preserves the degree of 2-morphisms, and the canonical 2-isomorphisms a , l , and r are homogeneous of degree 0. As usual, we write | α | for the degreeof a homogeneous 2-morphism α . We say that C is locally graded if each morphismcategory C ( x, y ) comes with a translation functor { } that is an equivalence, such thatfor any pair of composable 1-morphism x f y g z the horizontal compositions g ◦ f { } , g { }◦ f , and ( g ◦ f ) { } are naturally isomorphic. Likewise for categories, both frameworksare equivalent: a locally pregraded bicategory can be formally extended to a locally gradedone, and vice versa. In what follows we choose the pregraded framework.Let ( C , Σ) be a locally pregraded endobicategory. This means that C is locally pre-graded and Σ is graded, i.e. it preserves the degree of 2-morphisms and the structural2-isomorphisms m and i are homogeneous in degree 0. Definition 3.19.
A Σ –twisted quantum preshadow on C valued in a pregraded k [ q ± ]–linear category T is a collection of graded functors hh−ii x : C (Σ x, x ) T together withgraded morphisms θ g,f : hh f ◦ Σ g ii y hh g ◦ f ii x in T , one for each pair of 1–morphisms x g y f Σ x , which are natural in the graded sense, i.e. θ g ′ ,f ′ ◦ hh α ◦ Σ β ii = q | β | hh β ◦ α ii ◦ θ g,f (3.29)for homogeneous 2-morphisms α : f f ′ and β : g g ′ , and such that the definingdiagrams for preshadows (3.1) and (3.2) commute. We say that ( hh−ii , θ ) is a quantumshadow if each θ f,g is invertible.Likewise in the case of traces, quantum preshadows arise by extending coefficients of C to k [ q ± ] and deforming Σ into Σ q by redefining it on homogeneous 2-morphisms asΣ q ( α ) := q −| α | Σ( α ) . (3.30)Quantum preshadows on ( C , Σ) are then precisely preshadows on ( C , Σ q ). It follows thatthere is a deformed twisted horizontal trace hTr q ( C , Σ), the quantum Σ –twisted horizontal race of C , obtained by deforming the defining relation intoΣ x x Σ y y f pg Σ p ′ Σ p α Σ τ ∼ q | τ | Σ x x Σ y y f Σ p ′ g p ′ pα τ (3.31)where α and τ are homogeneous. When C is locally graded, then, in an analogy to tracesof categories, the k [ q ± ]–linear structure on hTr q ( C , Σ) satisfies q · [ p, α ] := [ p { } , α { } ].Diagrammatically, q · Σ x x Σ y y f Σ p pgα := Σ x x Σ y y f Σ p { } p { } gα { } (3.32)The following is a direct generalization from the undeformed framework. In particu-lar, every Σ–twisted quantum preshadow factorizes through hTr q ( C , Σ) uniquely up toan equivalence.
Proposition 3.20.
Assume that C has left duals. Then hTr q ( C , Σ) with hh−ii h x the ob-vious functors is the universal Σ –twisted quantum preshadow. It is a quantum shadow if C has right duals. Locally pregraded endobicategories form a 3-category if we require all bifunctor to pre-serve degrees of 2-morphisms and all structural 2-morphisms (such as ω f in the definitionof a morphism or η f and n x in the definition of a natural transformation) to be graded.A pullback of a twisted quantum preshadow along a morphism of locally pregraded endo-bicategories is again a twisted quantum preshadow, and natural transformations of suchmorphisms induce morphisms of preshadows. The following is an immediate consequenceof that and the universality of hTr q . Theorem 3.21 (Functoriality of the quantum horizontal trace) . The quantum horizontaltrace hTr q extends to a strict 2-functor from the 2-category of locally pregraded endobi-categories with duals to pregraded categories. We omit the explicit construction of the 2-functor, because it is a verbatim copy ofthe one from Theorem 3.10.
Choose a pair of functors C F D G C . The pair ( G, id) is a morphism of endocate-gories, from ( C , F G ) to ( D , GF ), and as such it induces a map between their universaltraces Tr( G, id) : Tr( C , F G ) Tr( D , GF ) , tr( f ) tr( Gf ) . To show naturality, choose natural transformations η : F F ′ and µ : G G ′ . Theydetermine morphisms of endocategories(Id , µ ◦ F η ) : ( C , F G ) ( C , F ′ G ′ )35Id , η ◦ Gµ ) : ( D , GF ) ( D , G ′ F ′ )and hence maps on traces. Using µ ◦ F η = F ′ η ◦ µ we check directly that(Tr( G ′ , Id) ◦ Tr(Id , µ ◦ F η ))(tr( f )) = tr( G ′ F ′ η ◦ G ′ µ ◦ G ′ f ) =tr( G ′ ( µ ◦ f ) ◦ η ) = tr( η ◦ G ( µ ◦ f )) = (Tr(Id , η ◦ Gµ ) ◦ Tr( G, id))(tr( f )) . Because tr(
F Gf ) = tr( f ) by Lemma 2.2, the above upgrades Tr( − ) to a symmetricshadow on Cat , the 2-category of small categories. The same argument shows thatTr q ( − ) is a quantum shadow on gCat , the 2-category of small graded categories, withthe cyclicity map satisfying tr( F Gf ) = tr(
F Gf ◦ id) = q | f | tr( f ) for a homogeneous f .Although not every functor has a left dual (left dualizable functors are exactly thosewith a left adjoint), Tr( − ) factors through hTr( Cat ) (resp. Tr q ( − ) factors throughhTr q ( gCat )). Indeed, a representant ( F, ω ) of a morphism in the horizontal trace isprecisely a morphism of endocategories, and it can be checked directly that Tr(
F, ω )(resp. Tr q ( F, ω )) depends only on the image of (
F, ω ) in the horizontal trace; see alsoTheorem 4.5.
Write
Bimod for the bicategory of k –algebras, bimodules, and bimodule maps; the hor-izontal composition of bimodules N ∈ Bimod ( A, B ) and N ′ ∈ Bimod ( B, C ) is definedas their tensor product: N ′ ◦ N := N ⊗ B N ′ . It admits a symmetric shadow that as-signs to an ( A, A )–bimodule M the space of coinvariants coInv( M ) := M/ [ A, M ] andthe cyclicity morphism of which is the standard twist: θ N ′ ,N ([ n ′ ⊗ n ]) = [ n ⊗ n ′ ], where N ∈ Bimod ( A, B ) and N ′ ∈ Bimod ( B, A ).Likewise, the quantum space of coinvariants coInv q ( − ) defined in (2.32) determinesa shadow on gBimod , the locally graded bicategory of graded algebras, graded bimod-ules, and graded bimodule maps. The cyclicity morphism is a noninvolutive deformationof the standard twist: θ N ′ ,N ([ n ′ ⊗ n ]) = q | n ′ | [ n ⊗ n ′ ] for homogeneous n ′ ∈ N ′ and n ∈ N .Let gRep be the restriction of gBimod to those bimodules that are finitely generatedand projective as right modules. It can be realized as a sub-2-category of gCat byinterpreting an algebra A as its representation category g R ep ( A ), and an ( A, B )–bimodule N as the functor ( − ) ⊗ A N . Therefore, it admits a preshadow as described in the previousexample, which factorizes through hTr q ( gRep ). Explicitly, it assigns to A AB B
MP PM ′ f (3.33)the linear map [ m ] tr Pq ( p f ( m ⊗ p )), where tr Pq is the quantum deformationof the twisted Hattori–Stallings trace from Section 2.4.5. We claim that this shadowcoincides with the one restricted from Bimod . Indeed, given N ∈ gRep ( A, B ) and N ′ ∈ gRep ( B, A ) with homogeneous n ∈ N and n ′ ∈ N ′ , the cyclicity map takesthe class of [ n ′ ⊗ n ] totr N ′ q ( n ′ ⊗ n ⊗ id N ′ ) = ǫ q,N ′ ( n ′ ⊗ n ⊗ coev N ′ (1)) == q | n ′ | [ n ⊗ (id N ′ ⊗ ev N ′ )( coev N ′ (1) ⊗ n ′ )] = q | n ′ | [ n ⊗ n ′ ] = θ N ′ ,N ([ n ′ ⊗ n ]) . .8.3 Twisted coinvariants Let f Bimod be an extension of
Bimod , objects of which are pairs (
A, ϕ ) consisting ofan algebra A and its automorphism ϕ , and whose 1- and 2-morphisms are bimodulesand bimodule maps respectively with no restriction imposed. It can be upgraded toan endobicategory via an endobifunctor Σ that affects only 1-morphisms:Σ N := ϕ N ψ ∼ = A ϕ ⊗ A N ⊗ B ψ B for N ∈ f Bimod ( A, ϕ ; B, ψ ), where A ϕ is the ( A, A )–bimodule A with the right action x · a := xϕ ( a ), and likewise for ψ B . For any object ( A, ϕ ) there is an isomorphism ofbimodules A ∼ = ϕ A ϕ that takes a to ϕ ( a ). Hence, Σ( N ⊗ B N ′ ) ∼ = Σ N ⊗ A Σ N ′ as desired.Twisted spaces of coinvariants from Section 2.4.6 constitute a shadow on f Bimod withthe cyclicity mapcoInv ψ (Σ N ′ ⊗ A N ) = coInv( ψ B ψ ⊗ B N ′ ⊗ A ϕ A ⊗ A N ) ∼ =coInv( ϕ A ⊗ A N ⊗ B N ′ ) = coInv ϕ ( N ⊗ B N ′ ) (3.34)for N ∈ f Bimod ( A, ϕ ; B, ψ ) and N ′ ∈ f Bimod ( B, ψ ; A, ϕ ). Recalling that Σ N ′ and N ′ coincide as k –modules, we find out that (3.34) is the twist [ n ′ ⊗ n ] [ n ⊗ n ′ ].Let f Rep ⊂ f Bimod be the restriction to bimodules that are finitely generated and pro-jective as right modules. Such an (
A, B )–bimodule N has a left dual ∗ N := Hom B ( N, B ),the (
B, A )–bimodule of right B –linear functions on N . Thus the above shadow can berestricted to a trace on vTr( f Rep ( A, ϕ ; B, ψ ) , Σ) that takes f ∈ Hom( N, Σ N ) to the com-positioncoInv ϕ ( A ) coev coInv ϕ ( N ⊗ B ∗ N ) f ⊗ id coInv ψ (Σ N ⊗ B ∗ N ) θ coInv ψ ( ∗ N ⊗ A N ) ev coInv ψ ( B ) , (3.35)which, when evaluated at [ a ] ∈ coInv ϕ ( A ), recovers the ψ –twisted Hattori–Stallings traceof f a ( n ) := f ( an ). Let M be an ( A, A )–bimodule and write R • ( A ) for the bar resolution of A [Lo, 1.1.11].The Hochschild homology of A with coefficients in M or, shortly, the Hochschild homologyof M is the homology of the chain complex CH • ( A, M ) := coInv( M ⊗ A R • ( A )), to whichwe refer as the Hochschild complex . Explicitly, CH n ( A, M ) = M ⊗ A ⊗ n and the Hochschilddifferential is the alternating sum ∂ ( m ⊗ a ⊗ · · · ⊗ a n ) = ma ⊗ a ⊗ · · · ⊗ a n + n − X i =1 ( − i m ⊗ a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +( − n a n m ⊗ a ⊗ · · · ⊗ a n − . (3.36) Notice that ∗ N is projective as a left module, but not necessarily as a right one. Hence, in general,it is not a 1-morphism in f Rep .
37n particular, HH ( A, M ) ∼ = coInv( M ) is the ordinary space of coinvariants.Hochschild homology is not directly a shadow on Bimod : for that one has to extend itto derived categories of bimodules and use the derived tensor product as the composition.It is not necessary in the case of
Rep , because an (
A, A )–bimodule of the form N ⊗ B N ′ has two projective resolutions P • ( N, N ′ ) := R • ( A ) ⊗ A ( N ⊗ B N ′ ) (3.37)and R • ( N, N ′ ) := Tot( R • ( A ) ⊗ A N ⊗ B R • ( B ) ⊗ B N ′ ) , (3.38)which, by the uniqueness of the resolution, are homotopy equivalent. Therefore, there isa sequence of homotopy equivalences CH • ( B, N ′ ⊗ A N ) ≃ coInv( R • ( N ′ , N )) ∼ = coInv( R • ( N, N ′ )) ≃ CH • ( A, N ⊗ B N ′ ) , (3.39)where the middle map is the standard twist [ b ⊗ n ′ ⊗ a ⊗ n ] [ a ⊗ n ⊗ b ⊗ n ′ ], in which a stands for a tensor product of elements of A , and likewise for b . Doing this twice resultsin a map chain homotopic to the identity, so that CH • is a symmetric shadow on Rep .Let ϕ ∈ Aut( A ). The ϕ –twisted Hochschild complex CH ϕ • ( A, M ) of an (
A, A )–bimo-dule M is obtained by replacing M with ϕ M (or, equivalently, coInv with coInv ϕ ): CH ϕ • ( A, M ) := coInv( ϕ M ⊗ A R • ( A )) . It coincides with CH • ( A, M ) except that the last term in the differential is replaced with( − n ϕ ( a n ) m ⊗ a ⊗ · · · ⊗ a n − . (3.40)Hochschild chains can be visualized by oriented circles divided into segments, one labeledwith m ∈ M and the others with a , . . . , a n . Each of the terms of the differential mergestwo segments multiplying their labels: m a a ma a − m a a + a m a In the twisted case add a mark on the circle between segments labeled m and a n . Tomerge these two segments, one has to move the mark over a n , acting upon it with ϕ : m a a m a ϕ ( a ) ϕ ( a ) ma Defined a priori for the untwisted case, the cyclicity map (3.39) can be easily extendedto the twisted homology following (3.34): CH ψ • ( B, Σ N ′ ⊗ A N ) = CH • ( B, ψ B ψ ⊗ B N ′ ⊗ A ϕ N ) ≃ CH • ( A, ϕ N ⊗ B ψ B ψ ⊗ B N ′ ) ∼ = CH • ( A, ϕ N ⊗ B N ′ ) = CH ϕ • ( A, N ⊗ B N ′ ) , which upgrades the twisted homology to a symmetric shadow on f Rep .38 .8.5 Quantum Hochschild homology
Fix an invertible elements q ∈ k . We define the quantum Hochschild complex of a graded( A, A )–bimodule M by replacing coInv( M ) with its deformation: qCH • ( A, M ) := coInv q ( M ⊗ A R • ( A )) . (3.41)It has the same chain groups as CH • ( A, M ) with coefficients extended to k ′ , but the lastterm of the differential is replaced with( − n q −| a n | a n m ⊗ a ⊗ · · · ⊗ a n − . (3.42)The quantum Hochschild homology of M , denoted by qHH • ( A, M ), is the homology ofthis complex.The quantum Hochschild complex is a nonsymmetric shadow on gRep . Indeed, re-placing the middle isomorphism in (3.39) with [ b ⊗ n ′ ⊗ a ⊗ n ] q | b ⊗ n ′ | [ a ⊗ n ⊗ b ⊗ n ′ ]results in a homotopy equivalence θ N ′ ,N : qCH • ( B, N ′ ⊗ A N ) ≃ qCH • ( A, N ⊗ B N ′ ) (3.43)that is no longer involutive: θ N,N ′ ◦ θ N ′ ,N scales a chain of degree d by q d . The twistedHochschild homology can be deformed likewise, leading to the twisted quantum Hochschildcomplex qCH ϕ • ( A, M ) of a graded (
A, A )–bimodule M and its homology qHH ϕ • ( A, M ).The details are left to the reader.
HHH
The construction of Hochschild homology extends naturally to complexes of bimodules,by letting CH • ( A, C • ) := coInv( C • ⊗ A R • ( A )) for a complex of ( A, A )–bimodules. It canbe checked that it is a triangulated shadow on
Com b/h ( Rep ).Instead of computing the total Hochschild homology of the complex C • one can apply HH • component-wise. This leads to a Lefschetz shadow HHH on Com b/h ( Rep ) thatreplaces C • with . . . HH • ( A, C i ) HH • ( A, C i +1 ) HH • ( A, C i +2 ) . . . (3.44)It is the second page of the spectral sequence associated to coInv( C • ⊗ A R • ( A )) seen asa bicomplex. This functor appears quite often in categorification of link invariants, whenlinks are considered as closures of braids.Naturally, when dealing with complexes of graded bimodules, one can replace HH • with qHH • to obtain qHHH , which is a quantum shadow. According to Corollary 3.18, itrestricts to a quantum trace on Com b/h ( gRep )( A, B ) for any graded algebras A and B . Ittakes a particularly simple form when A = B = k , because then qHH ( k ) = k and higherhomology vanishes. Corollary 3.22.
Let C • ∈ Com b/h ( gRep ( k , k )) be a bounded complex and f • ∈ End( C • ) its graded endomorphism. The endomorphism of qHH ( k ) ∼ = k induced by ( C • , f • ) isthe multiplication by Λ q ( f • ) = P i,j ( − i q j tr( f i,j ) , where i and j are the homological andinternal gradings in C • respectively, and tr is the Hattori–Stallings trace on R ep ( k ) . .8.7 Cobordisms of links in surface bundles It is generally a hard problem to identify the horizontal trace hTr( C , Σ) for a givenbicategory C , but the answer is very natural for Tan ( F ), the bicategory of points ina surface F , tangles in F × I , and tangle cobordisms in F × I × I . Recall that the quotientspace M := F × I . ( p, ∼ ( ϕ ( p ) , , (3.45)where ϕ is a diffeomorphism of F , is a manifold called the surface bundle with fiber F andmonodromy ϕ . Theorem A.
Let M be a surface bundle with fiber F and monodromy φ ∈ Diff ( F ) .There is an equivalence of categories hTr( Tan ( F ) , φ ∗ ) ≃ L inks ( M ) (3.46) where φ ∗ ( S ) := ( φ × id × id)( S ) for a cobordism S ⊂ F × I × I .Proof. Let π : F × I M be the quotient map and consider the fiber F := π ( F × { } )along which M can be cut open to F × I . Objects of hTr( Tan ( F ) , φ ∗ ) can be identifiedwith links in M transverse to F , while morphisms are represented by link cobordisms in M × I transverse to the 3-dimensional membrane F × I . The cobordism can be deformedby an ambient isotopy that fixes the membrane, and the trace relation allows us to isotopethe embedding of the membrane (although F × { i } is fixed for i = 0 , Tan ( F ) , φ ∗ ) L inks ( M ) that forgets the mem-brane. By the transversality argument it is essentially surjective on objects (each linkis isotopic to a link transverse to F ) and full on morphisms (each surface between linkstransverse to F can be isotoped to be transverse to the standard membrane F × I ). Itremains to show that if two surfaces S, S ′ ⊂ F × I × I represent isotopic cobordisms b S and b S ′ in M × I , then their images in hTr( Tan ( F ) , φ ∗ ) coincide.Assume there is an isotopy ϕ t of M × I taking b S to b S ′ with support supp( ϕ ) disjointfrom M × ∂I and a membrane F ′ × I for some fiber F ′ ⊂ M . It is enough to consider onlysuch isotopies, because every two isotopic surfaces in M × I are connected by a sequenceof them. If F ′ = F , then S and S ′ are already isotopic in F × I × I and we are done.Otherwise, let p ∈ S be the point over which F ′ lives and consider a bump function β : I S with β (0) = β (1) = 1 and β ( t ) = p for t ∈ [ ǫ, − ǫ ] for some ǫ > ϕ ) ⊂ M × [ ǫ, − ǫ ], see Fig. 6. The preimage in M × I of the graph of β is a membrane isotopic to F × I ; it can be visualized as pushing the interior of F × I onto F ′ × I . Because the new membrane is disjoint from the support of ϕ , the cuts of b S and b S ′ along it are isotopic. This proves the faithfulness, because the cuts represent inhTr( Tan ( F ) , φ ∗ ) the same morphisms as the surfaces S and S ′ .Each orientation preserving diffeomorphism φ of R is isotopic to identity, so thathTr( Tan , φ ∗ ) is equivalent to the category of links in a solid torus. Because tangles ina thickened plane R × I can be represented by diagrams on the stripe { } × R × I , is itworth to consider those diffeomorphisms that preserve the line { } × R . There are twoof them: • the identity, in which case the stripe is closed to an annulus A , and • the rotation by 180 degrees, for which the image of the stripe is a M¨obius band M . A support of an isotopy is the closure of the set of points that are not stationary under the isotopy. F ′ Figure 6: A visualization of the isotopy pushing F × I onto F ′ × I . Each point ofthe cylinder represents a fiber of the F –bundle M × I S × I . The thick straight lineis the standard membrane F × I , whereas the curve is its isotopic deformation. The flatpart of the curve corresponds to the piece of the deformed membrane contained in F ′ × I .The solid torus is a trivial line bundle over A and a twisted one over M respectively.Hence, given an invariant of tangles computed from their diagrams, there are two waysto get invariants of links in a solid torus. Corollary 3.23.
There are equivalences of categories hTr(
Tan ) ≃ L inks ( S × R ) ≃ hTr( Tan , ρ ∗ ) , where ρ ∈ Diff ( R ) is the half–rotation. Thence, a bifunctor I : Tan C inducesinvariants of links in a solid torus hTr( I ) : L inks ( R × S ) hTr( C ) , and hTr( I , ρ ∗ ) : L inks ( R × S ) hTr( C , Σ) , where Σ is an endofunctor of C satisfying Σ ◦ I ∼ = I ◦ ρ ∗ . In particular, hh hTr( I ) ii is a linkinvariant for any symmetric preshadow hh−ii on C . Here we develop a machinery to show that higher quantum Hochschild homology of Chen–Khovanov algebras vanish, which is used to proof the conjecture of Auroux, Grigsby, andWehrli. In particular, we prove the invariance of quantum Hochschild homology under K –theoretic equivalences. This section can be skipped by a reader interested only inthe construction of the quantum link homology. Hereafter we fix an invertible q ∈ k . Choose a small pregraded endocategory ( C , Σ). The (twisted) quantum Hochschild–Mitchell complex of ( C , Σ) is the chain complex q C H • ( C , Σ) with components q C H n ( C , Σ) := M x ,...,x n ∈ Ob( C ) C ( x , Σ x n ) ⊗ C ( x , x ) ⊗ · · · ⊗ C ( x n , x n − ) (4.1)and differential the alternating sum ∂ n = P ni =0 ( − i d in , where d in ( f ⊗ . . . ⊗ f n ) := ( f ⊗ . . . ⊗ ( f i ◦ f i +1 ) ⊗ . . . ⊗ f n if i < n,q −| f n | (Σ f n ◦ f ) ⊗ f ⊗ . . . ⊗ f n − if i = n. (4.2)41ts homology is called the (twisted) quantum Hochschild–Mitchell homology of ( C , Σ) anddenoted by q H H • ( C , Σ). We also use the symbols
C H • and H H • when q = 1, drop-ping the adjective ‘quantum’. In particular, H H • ( C , Id) recovers the usual Hochschild–Mitchell homology of a category [Mit72]. Furthermore,
H H ( C , Σ) and q H H ( C , Σ) areprecisely the universal Σ–twisted trace of C and its quantum deformation.Expressing the differential as an alternating sum makes it evident that the Hochschild–Mitchell complex arises from a presimplicial module [Lo, 1.0]. Despite not working inthis framework, we will often use the following characterisation of chain homotopies. Lemma 4.1.
Choose a collection { h kn : q C H n ( C , Σ) q C H n +1 ( C ′ , Σ ′ ) } k n of linearmaps, satisfying d in +1 ◦ h kn = h k − n − ◦ d in for i < k,d kn +1 ◦ h kn for i = k + 1 ,h kn − ◦ d i − n for i > k + 1 . (4.3) Then the alternating sums h n := P nk =0 ( − k h kn are components of a chain homotopy from f n := d n +1 ◦ h n to g n := d n +1 n +1 ◦ h nn . Given a morphism of endocategories (
F, ω ) : ( C , Σ) ( C ′ , Σ ′ ) define a chain map( F, ω ) ∗ : q C H • ( C , Σ) q C H • ( C ′ , Σ ′ ) by the formula( F, ω ) ∗ ( f ⊗ . . . ⊗ f n ) := ( ω ◦ Ff ) ⊗ Ff ⊗ . . . ⊗ Ff n , (4.4)where ω is used to fix the codomain of Ff . It is straightfoward to check that the as-signment ( F, ω ) (
F, ω ) ∗ is functorial and it can be shown that ( F, ω ) ∗ and ( F ′ , ω ′ ) ∗ are chain homotopic if the morphisms are naturally isomorphic (this also follows fromthe results of the following section).For a set of objects B ⊂ Ob( C ) define a subcomplex q C H • ( C | B, Σ) ⊂ q C H • ( C , Σ)by assuming that all x i in (4.1) are from B . Recall that B generates additively C in a graded sense if each object of C admits a homogeneous isomorphism, possibly ofnonzero degree, to a direct summand of some x ⊕ · · · ⊕ x r with x i ∈ B . Proposition 4.2.
The inclusion I : q C H • ( C | B, Σ) q C H • ( C , Σ) is a homotopyequivalence if B generates additively C in a graded sense. In particular, there are homo-topy equivalences q C H • ( C , Σ) ≃ q C H • ( C ⊕ , Σ) , and q C H • ( C , Σ) ≃ q C H • (Kar( C ) , Σ) induced by the canonical inclusions.Proof. We proof the statement in two steps. First assume each object x ∈ C admitsa decomposition x = x ⊕ · · · ⊕ x r with x i ∈ B , together with inclusions ι i : x i x andprojections π i : x x i , both homogeneous of degree 0. Pick the trivial decompositionif x is already in B . This leads to a chain map P : q C H • ( C , Σ) q C H • ( C | B, Σ) P n ( f ⊗ · · · ⊗ f n ) := X i ,...,i n (Σ π i n ◦ f ◦ ι i ) ⊗ ( π i ◦ f ◦ ι i ) ⊗ · · · ⊗ ( π i n − ◦ f n ◦ ι i n ) , which satisfies P ◦ I = id. To show that I ◦ P ≃ id, consider the family of maps42 kn ( f ⊗ · · · ⊗ f n ) := X i ,...,i k ( f ◦ ι i ) ⊗ ( π i ◦ f ◦ ι i ) ⊗ · · ·⊗ ( π i k − ◦ f k ◦ ι i k ) ⊗ π i k ⊗ f k +1 ⊗ · · · ⊗ f n and apply Lemma 4.1. Indeed, one checks directly that d n +1 ◦ h n = id, d nn +1 ◦ h nn = I n ◦ P n ,and the conditions (4.3) are satisfied.Hence, by extending B , we can assume that for each object x ∈ C there is b ( x ) ∈ B together with homogeneous morphisms ι x : x b ( x ) and π x : b ( x ) x , such that π x ◦ ι x = id x . Consider now the chain map P : q C H • ( C , Σ) q C H • ( C | B, Σ) P n ( f ⊗ · · · ⊗ f n ) := q | ι xn | (Σ ι x n ◦ f ◦ π x ) ⊗ ( ι x ◦ f ◦ π x ) ⊗ · · · ⊗ ( ι x n − ◦ f n ◦ π x n )where x i = dom( f i ). Again, P ◦ I = id, whereas the other composition is chain homotopicto the identity by Lemma 4.1 applied to the collection of linear maps h kn ( f ⊗ · · · ⊗ f n ) := ( f ◦ π x ) ⊗ ( ι x ◦ f ◦ π x ) ⊗ · · ·⊗ ( ι x i − ◦ f i ◦ π x i ) ⊗ ι x i ⊗ f i +1 ⊗ · · · ⊗ f n . Indeed, π x ◦ ι x = id x implies that d n +1 ◦ h n = id, and d n +1 n +1 ◦ h nn = I n ◦ P n is immediate.A graded ( A, A )–bimodule M can be understood as an endofunctor ( − ) ⊗ A M ong R ep ( A ) if it is finitely generated and projective as a right module. This leads to the iden-tification of the quantum Hochschild–Mitchell homology of the representation categoryof A with the quantum Hochschild homology of the algebra. Corollary 4.3.
Choose a graded algebra A and a graded ( A, A ) –bimodule M that isfinitely generated and projective as a right module. Then the chain complexes qCH • ( A, M ) and q C H • (g R ep ( A ) , M ) are homotopy equivalent.Proof. The category g R ep ( A ) is generated additively by B = { A } in a graded sense,where A is seen as a right A –module. Therefore, q C H • (g R ep ( A ) , M ) ≃ q C H • (g R ep ( A ) |{ A } , M ) ∼ = qCH • ( A, M ) , where the second isomorphism identifies f ∈ End A ( A ) with f (1) ∈ A and g ∈ Hom A ( A, M )with g (1) ∈ M . Remark . Considering R ep ( A ) as a trivially graded category, the above result can berephrased to say that C H • ( R ep ( A ) , M ) is homotopy equivalent to CH • ( A, M ). Let k - gCat be the 2-category of pregraded linear categories. It is locally pregradedwith a natural transformation η being homogeneous of degree d if each component η x is homogeneous of degree d . Objects of hTr q ( k - gCat ) are precisely pregraded endo-categories. Recall that a morphism in the horizontal trace from ( C , Σ) to ( C ′ , Σ ′ ) isrepresented by a pair ( F, ω ) consisting of a functor F : C C ′ and a natural transfor-mation ω : F Σ Σ ′ F . Hence, ( F, ω ) is actually a functor of endocategories. It is thennatural to ask, whether the chain map (
F, ω ) ∗ : q C H • ( C , Σ) q C H • ( C ′ , Σ ′ ) dependsonly on the image of the functor in the horizontal trace.43 heorem 4.5. The quantum Hochschild–Mitchell complex descends to a functor q C H : hTr q ( k - gCat ) Com b/h ( k ) . In particular, it is a quantum preshadow on k - gCat .Proof. Consider the following two diagrams in k - gCat C CC ′ C ′ Σ G Σ ′ FG σν q | ν | C CC ′ C ′ Σ F Σ ′ FGσ ν
Composing the 2–morphisms results in morphisms ( G, ( ν ◦ ) ∗ σ ) and ( F, q | ν | σ ∗ ( ◦ ν ))of endocategories. We check directly that the family of maps h kn ( f ⊗ . . . ⊗ f n ) := q | ν | ( σ x n ◦ Gf ) ⊗ Gf ⊗ . . . ⊗ Gf k ⊗ ν x k ⊗ F f k +1 ⊗ . . . ⊗ F f n where x i = dom( f i ), defines a chain homotopy between the induced chain maps. Indeed, d n +1 ◦ h n = q | ν | ( F, σ ∗ ( ◦ ν )) n by the naturality of ν , whereas d n +1 n +1 ◦ h nn = ( G, ( ν ◦ ) ∗ σ ) n is straightforward. Corollary 4.6. If ( F, ω ) and ( F ′ , ω ′ ) are isomorphic functors, then ( F, ω ) ∗ = ( F ′ , ω ′ ) ∗ .Proof. Suppose η : ( F, ω ) ( F ′ , ω ′ ) is an isomorphism. Then C CC ′ C ′ Σ F ′ F ′ Σ ′ ω ′ = CC ′ CC ′ ΣΣ ′ FF ′ η F ′ F η − ω = C CC ′ C ′ Σ F F Σ ′ ω in hTr q ( k - gCat ), where the second equality follows from the horizontal trace relation.It follows from Theorem 4.5 that the quantum Hochschild–Mitchell homology is a pre-shadow on k - gCat , which generalizes Example 3.8.1. In the view of Corollary 4.3 itinduces a preshadow on gRep with a priori a different cyclicity map. Proposition 4.7.
The homotopy equivalences q C H • (g R ep ( A ) , M ) ≃ qCH • ( A, M ) con-stitutes an isomorphism of preshadows.Proof. We have to check that the square qCH • ( B, N ′ ⊗ A N ) qCH • ( A, N ⊗ B N ′ ) q C H • (g R ep ( B ) , N ′ ⊗ A N ) q C H • (g R ep ( A ) , N ⊗ B N ′ ) θ N ′ ,N I PN ′∗ commutes up to a chain homotopy for bimodules N ∈ gRep ( A, B ) and N ′ ∈ gRep ( B, A ),where I and P are mutually inverse homotopy equivalences. Due to the naturality of both θ N ′ ,N and N ′∗ we can assume that N ′ is free as a right A –module with a graded basis44 n ′ , . . . , n ′ s } . Then the left action of B determines an algebra map B Mat s ( A ) thatassigns to b ∈ B the matrix ( a ij ( b )) i,j s determined uniquely by the formula bn ′ i = X j n ′ j a ij ( b ) . (4.5)One checks that a ij ( b ′ b ) = P k a kj ( b ′ ) a ik ( b ) for any b, b ′ ∈ B . Likewise, n ′ k ⊗ n ⊗ id N ′ isrepresented, as a right A –linear map, by the matrix ( δ jk n ⊗ n ′ j ) i,j s . It follows fromthe proof of Proposition 4.2 that the composition P ◦ N ′∗ ◦ I is the chain map( n ′ i ⊗ n ) ⊗ b r ⊗ · · · ⊗ b s X i ,...,i r =1 q | n ′ ir | ( n ⊗ n ′ i r ) ⊗ a i r − ,i r ( b r ) ⊗ · · · ⊗ a i ,i ( b ) . (4.6)To compute θ N ′ ,N , we need to lift the identity on N ′ ⊗ A N to a pair of graded chain maps p BA : R • ( B ) ⊗ B N ′ ⊗ A R • ( A ) ⊗ A N R • ( B ) ⊗ B N ′ ⊗ A N : s BA . By the uniqueness of a lift, p BA and s BA are mutually inverse homotopy equivalences.The map p BA is defined as the standard projection on R r ( B ) ⊗ B N ′ ⊗ A R ( A ) ⊗ A N = R r ( B ) ⊗ N N ′ ⊗ N and vanishes otherwise. The chain map s BA replaces each b ℓ , one-by-one, with entries of the matrix a ij ( b ℓ ): s BA ( b r ⊗ · · · ⊗ b ⊗ ( n ′ i ⊗ n )) := r X ℓ =0 s X i ,...,i ℓ =1 b r ⊗ · · · ⊗ b ℓ +1 ⊗ n ′ i ℓ ⊗ a i ℓ − ,i ℓ ( b ℓ ) ⊗ · · · ⊗ a i ,i ( b ) ⊗ n. (4.7)The equality (4.5) makes s BA commute with the differential.Up to the isomorphisms coInv q ( M ⊗ A R • ( A )) ∼ = coInv q ( R • ( A ) ⊗ A M ) we can express θ N ′ ,N as the composition p AB ◦ τ ◦ s BA , where τ it the graded twist from 3.8.5. Because | a ij ( b k ) | = | b k | and p AB vanishes unless none of b k appears, the composition coincideswith (4.6). K –theoretic invariance We say that a sequence of linear functors( F ′ , ω ′ ) ι ( F, ω ) π ( F ′′ , ω ′′ ) (4.8) semisplits if 0 F ′ x ι x F x π x F ′′ x x . Clearly, a split ex-act sequence semisplits, but not otherwise. Moreover, the sequence 4.8 remains semisplitwhen each of the three functors is pre- or post-composed with another functor, becauselinear functors preserve direct sums.Let gEndoCat ⊕ be the bicategory of small graded additive endocategories. We con-struct its semisplit Grothendieck category K ss ( gEndoCat ⊕ ) by replacing functor cate-gories with Z [ q ± ]–modules generated by isomorphism classes [ F, ω ] of functors modulothe relations [ F { } , ω { } ] = q [ F, ω ] and [
F, ω ] = [ F ′ , ω ′ ] + [ F ′′ , ω ′′ ] for every semisplitexact sequence (4.8). The composition is well-defined by the discussion above. A func-tor ( F, ω ) : ( C , Σ) ( C ′ , Σ ′ ) is a K –theoretic equivalence if it is an isomorphism in K ss ( gEndoCat ⊕ ). In other words, there is a functor ( F ′ , ω ′ ) : ( C ′ , Σ ′ ) ( C , Σ),called the K –theoretic inverse of ( F, ω ), such that the compositions ( F ′ , ω ′ ) ◦ ( F, ω ) and(
F, ω ) ◦ ( F ′ , ω ′ ) coincide in the corresponding Grothendieck groups with the images ofthe identity functors. 45 heorem 4.8 ( K –theoretic invariance) . The homotopy class of the chain map ( F, ω ) ∗ assigned to a functor of linear graded endocategories ( F, ω ) : ( C , Σ) ( C ′ , Σ ′ ) dependsonly on [ F, ω ] ∈ K ss ( gEndoCat ⊕ ) . In particular, ( F, ω ) ∗ is a homotopy equivalence ifthe functor is a K –theoretic equivalence.Proof. The relation ( F { } , ω { } ) ∗ = q ( F, ω ) ∗ follows from the proof of Proposition 4.2.Thence, we have to show that ( F, ω ) ∗ = ( F ′ , ω ′ ) ∗ + ( F ′′ , ω ′′ ) ∗ for a semisplit sequenceof functors (4.8). For that fix an isomorphism ϕ x : F x ∼ = F ′ x ⊕ F ′′ x for each x ∈ C that results from a splitting of the sequence 0 F ′ x F x F ′′ x
0. Thenevery morphism g ∈ C ( F x, F y ) can be identified with a 2 × g ij ). In particular,the proof of Proposition 4.2 implies that η ∗ is chain homotopic to f ⊗ . . . ⊗ f n X i ,...,i n ( ω ◦ F f ) i ,i n ⊗ ( F f ) i ,i ⊗ . . . ⊗ ( F f n ) i n ,i n − . (4.9)A simple diagram chasing reveals that for any f ∈ C ( x, y ) the matrix of Ff is upper trian-gular with F ′ f and F ′′ f on the diagonal, and likewise for ω x . Therefore, the only nontrivialsummands in (4.9) are the two with i = · · · = i n , which are precisely ( F ′ , ω ′ ) ∗ ( f ⊗ . . . ⊗ f n )and ( F ′′ , ω ′′ ) ∗ ( f ⊗ . . . ⊗ f n ). Let A be a graded algebra and ϕ ∈ Aut( A ) a graded automorphism. Following 2.4.6construct the bimodule ϕ A by redefining the left action as a · x := ϕ ( a ) x . The bimodules A ϕ and ϕ A ϕ are constructed likewise. The automorphism ϕ can be seen as an isomorphismof bimodules A ∼ = ϕ A ϕ ∼ = ϕ A ⊗ A A ϕ . Tensoring with ϕ A is an endofunctor on g M od ( A )denoted in 2.4.6 by ( − ) ϕ . A module V coincides with V ϕ , except that the action of A istwisted: v · a = vϕ − ( a ).Let B and ψ ∈ Aut( B ) be another graded algebra together with a graded automor-phism. Following 3.8.3 we put Σ M := A ϕ ⊗ A M ⊗ B ψ B for a graded ( A, B )–bimodule M .It coincides with M , except that ϕ and ψ are used to twist both actions. Suppose thereis a bimodule homomorphism ω : M Σ M . Then there is a B –linear map V ϕ ⊗ A M id ⊗ ω V ϕ ⊗ A ϕ M ψ ∼ = ( V ⊗ A M ) ψ for any right A –module V , where the second map is the isomorphism ϕ A ϕ ∼ = A . Hence,( M, ω ) determines a functor of endocategories.Let g B imod ( A, ϕ ; B, ψ ) be the category with objects graded pairs (
M, ω M ) as aboveand morphisms between ( M, ω M ) and ( N, ω N ) graded bimodule maps f : M N thatintertwine the structure, i.e. ω N ◦ f = f ◦ ω M . One checks directly that ω M (ker f ) ⊂ ker f and ω N (im f ) ⊂ im f , so that g B imod ( A, ϕ ; B, φ ) has kernels and cokernels. Thence,it is an abelian category. Restricting bimodules to those that are finitely generated andprojective as left modules picks the subcategory g R ep ( A, ϕ ; B, ψ ). It is additive, butnot abelian. Note that the categories are slightly different from morphism categories of f gBimod and f gRep from Section 3.8.3.Following the usual convention we shall write G ( A, ϕ ; B, ψ ) and K ( A, ϕ ; B, ψ ) forthe (exact) Grothendieck groups of g B imod ( A, ϕ ; B, ψ ) and g R ep ( A, ϕ ; B, ψ ) respec-tively. The inclusion of categories induces a linear map K ( A, ϕ ; B, ψ ) G ( A, ϕ ; B, ψ ), That is the isomorphism class of a bimodule [ M ] is identified with [ M ′ ] + [ M ′′ ] whenever there isan exact sequence 0 M ′ M M ′′ R ep ( A, ϕ ; B, ψ ) splits, K ( A, ϕ ; B, ψ ) is usually a proper quotient of the split Grothen-dieck group.The tensor product ( M ⊗ B M ′ , ω ⊗ B ω ′ ) of bimodules ( M, ω ) ∈ g R ep ( A, ϕ ; B, ψ )and ( M ′ , ω ′ ) ∈ g R ep ( B, ψ ; C, ρ ), belongs to g R ep ( A, ϕ ; C, ρ ). Hence, the categoriesg R ep ( A, ϕ ; B, ψ ) constitute a bicategory. Moreover, tensoring with (
M, ω ) is exact. In-deed, exact sequences in g R ep ( B, ϕ ; C, ρ ) split when considered as sequence of left mod-ules, which is enough to prove the exactness of (
M, ω ) ⊗ B ( − ), whereas the exactness of( − ) ⊗ A ( M, ω ) follows from M being projective as a left module. Hence, the Grothendieckgroups K ( A, ϕ ; B, ψ ) form a category as well. Following [Ke98] we say that (
M, ω ) isa K –theoretic equivalence if [ M, ω ] is invertible in this category. The following is an im-mediate translation of Theorem 4.8.
Proposition 4.9 (Algebraic K –theoretic invariance) . The homotopy class of the chainmap ( M, ω ) ∗ : qCH ψ • ( B ) qCH ϕ • ( A ) induced by ( M, ω ) ∈ g R ep ( A, ϕ ; B, ψ ) dependsonly on [ M, ω ] ∈ K ( A, ϕ ; B, ψ ) . In particular, ( M, ω ) ∗ is a homotopy equivalence if ( M, ω ) is a K –theoretic equivalence. Choose now a positively graded algebra A = L d ∈ N A d . The degree zero subalgebra A ⊂ A is preserved by any graded automorphism ϕ ∈ Aut( A ). Hence, the inclusion andprojection induce chain maps qCH ϕ • ( A ) qCH ϕ • ( A ) and qCH ϕ • ( A ) qCH ϕ • ( A )respectively, where ϕ ∈ Aut( A ) is the restriction of ϕ . One of the compositions isclearly the identity map, but not the other.These chain maps may not be homotopy equivalences. For instance, it is known thatthe algebra of dual numbers k [ x ] / ( x ), where deg x = 2, has unbounded Hochschild ho-mology, whereas the homology of k is one-dimensional. The situation changes drasticallywhen A has finite global dimension: the inclusion A A induces an isomorphism onHochschild homology under some other technical conditions [Ke98]. Here we reprove thisresult for the twisted homology. Theorem 4.10.
Let k be a field and A = L d ∈ N A d a positively graded k –algebra of finitedimension. If each simple A –module is one dimensional and A has finite global dimen-sion, then the inclusion A ⊂ A induces a homotopy equivalence qCH ϕ • ( A ) qCH ϕ • ( A ) for any graded automorphism ϕ ∈ Aut( A ) .Proof. The chain map qCH ϕ • ( A ) qCH ϕ • ( A ) is induced by ( A, ϕ ), seen as an ob-ject of g R ep ( A, ϕ ; A , ϕ ). Likewise, the other chain map is induced by ( A , ϕ ) ∈ g R ep ( A , ϕ ; A, ϕ ), where the right action of A on A is given by the projection. A quickcomputation shows that ( A , ϕ ) ⊗ A ( A, ϕ ) ∼ = ( A , ϕ ) in g R ep ( A , ϕ ; A , ϕ ). Hence,the composition qCH ϕ • ( A ) qCH ϕ • ( A ) qCH ϕ • ( A ) is homotopic to the identity.In the view of Proposition 4.9 it is now enough to show that ( A, ϕ ) and (
A, ϕ ) ⊗ A ( A , ϕ )coincide in K ( A, ϕ ; A, ϕ ).We check first that the bimodules have the same image in G ( A, ϕ ; A, ϕ ). Indeed, A > d /A >d ∼ = ( A > d ⊗ A A ) /A >d ⊗ A A ) , which shows that the graded associate of the bimodules coincide, and so their compositionseries (which are finite due to A being of finite dimension). It remains to show that every( A, A )–bimodule of finite length can be represented in K ( A, ϕ ; A, ϕ ).47e first notice that A e := A ⊗ A op has finite global dimension: each simple A e –moduleis of the form L ′ ⊗ L ∗ for certain simple A –modules L ′ and L , and as such it has a uniformlybounded projective resolution. Thence, every ( A, A )–bimodule M of finite length hasa finite projective resolution P • ( M ). Because A ϕ and ϕ A are projective as left and rightmodules, Σ is exact and preserves projective bimodules. Hence, P i (Σ M ) := Σ( P i ( M )) isa projective resolution of Σ M . Lifting ω to a chain map P • ( ω ) between these complexes,we construct a resolution of ( M, ω ) in g R ep ( A, ϕ ; A, ϕ ). The assignment[
M, ω ] X i ( − i [ P i ( M ) , P i ( ω )] (4.10)is then a two-sided inverse of K ( A, ϕ ; A, ϕ ) G ( A, ϕ ; A, ϕ ). Hence, the bimodules(
A, ϕ ) and (
A, ϕ ) ⊗ A ( A , ϕ ) coincide in K ( A, ϕ ; A, ϕ ) as desired.
With this section we move from the algebraic to the topological part of the paper. Itstarts with a brief description of the formal bracket, a very generic approach to Khovanov-type homology due to Bar-Natan [BN05], which is followed by a list of TQFT functorsproducing link homologies used in this paper. The main purpose of this section is to fixnotation and conventions. In particular, the reader should be aware that our conventionfor the quantum grading makes Chen–Khovanov algebras negatively graded.
Let us start with a brief reminder of the formal Khovanov bracket following [BN05]. Inthis section F stands for a surface, possibly with boundary.Choose a diagram D ⊂ F of an oriented tangle T ⊂ F × R and let n stand forthe number of crossings in D . To compute the formal Khovanov bracket of T one beginswith creating the n -dimensional cube of resolutions I ( D ) of D defined as follows: • each vertex ξ of I ( D ) is decorated with the resolution D ξ of D , i.e. the collectionof circles and proper intervals in F obtained from D by forgetting the orientationand replacing each i –th crossing by its horizontal or vertical smoothing for ξ i = 0 and ξ i = 1 respectively, and • each edge is decorated by a cobordism with a unique saddle point over the smoothingbeing changed and directed from the resolution with less vertical smoothings tothe one with more of them.We can view I ( D ) as a commutative cubical diagram in C ob ( F ), the category withobjects (non-oriented) flat tangles in F and morphisms the isotopy classes of surfaces in F × I . The degree of a surface S ⊂ F × I is given by the formuladeg S := χ ( S ) − B B is the set of corners and χ ( S ) the Euler characteristic of S . Alternatively, deg S counts the critical points of the natural height function h : S I with signs: a point ofindex µ contributes ( − µ towards deg. Thence, to ensure that each morphism in I ( D )has degree 0, we introduce formal degree shifts and place at each vertex ξ the shiftedresolution D ξ {| ξ |} for | ξ | := ξ + · · · + ξ n . Formally, objects in C ob ( F ) are now symbols T { i } formed by a flat tangle T ⊂ F and i ∈ Z . C ob ( F ) ⊕ be the additive closure of C ob ( F ). The formal Khovanov bracket J D K ofthe tangle diagram D is the complex in C ob ( F ) ⊕ obtained from the cube by distributingsigns over some edges to make every square anticommute, then taking direct sums alongdiagonal sections of the cube, and finally applying suitable degree shifts: J D K i := M | ξ | = i + n − D ξ { i + n + − n − } (5.2)where n ± stand for the amount of positive or negative crossings in D . Theorem 5.1 (cf. [BN05]) . The homotopy type of the complex J D K is a tangle invariantafter imposing on C ob ( F ) ⊕ the following local relations ( S ) = 0 ( T ) = 2( ) + = + introduced first by Bar-Natan [BN05]. Let , , , and be tangle diagrams that coincide everywhere except a small disk,in which they look as indicated by the pictures. It follows from the construction thatthe formal brackets of the first two diagrams are mapping cones of chain maps betweenthe formal complexes of the other two diagrams.
Proposition 5.2 (cp. [Kh99, BN05]) . Let , , , be four link diagrams as above,where we choose any orientation for , and write e = n − ( ) − n − ( ) . Then there areisomorphisms of formal complexes J K ∼ = cone (cid:0) J K { } [ − J K { e + 2 } [ − − e ] (cid:1) , and (5.3) J K ∼ = cone (cid:0) J K { e − } [ − e ] J K {− } (cid:1) , J K J K (5.4) where the chain maps are induced by saddle cobordisms.
The isomorphisms (5.3) and (5.4) were first observed in [Kh99, Section 4.2] for knotsin R and then by Bar-Natan in the framework of the formal bracket [BN05, Lemma 4.4].They lead to distinguished triangles in the homotopy category of C ob ( F ) ⊕ , which can beseen as categorified versions of the Kauffman skein relation.To retrieve homology groups from J D K one has to replace C ob ( F ) ⊕ with an abeliancategory. This is done by applying to the above construction a graded TQFT functor thatpreserves the relations S , T , and . Most known functors of this type factorize throughthe universal Bar-Natan skein category g B N ( F ) [BN05], the additive linear category gen-erated by circles and proper intervals in F as objects, and formal linear combinations ofcobordisms in F × I decorated by dots (each dot decreases the degree of a cobordism by2) as morphisms, subject to the local relations( S ) = 0 , ( D ) b = 1 , Locality means that each picture represents a part of a cobordism inside a ball in F × I . N ) = b + b − bb . The last relation is commonly referred as the neck cutting relation and it allows to reduceany surface to a linear combination of surfaces of genus zero. Both T and follows,and a contractible circle can be replaced with a pair of shifted copies of an empty set. Inparticular, the empty set generates g B N ( F ) when F is either R or S . Proposition 5.3 (Delooping, cf. [BN07]) . There is a pair of mutually inverse isomor-phisms in g B N ( F ) ∅{− }∅{ +1 }⊕ − bb + b b (5.5) for every circle bounding a disk in F . The TQFT functors we consider in this paper factor through certain quotients of g B N ( F ). Consider the following two relations:( TD ) b b = 0( B ) γ b = 0 if [ γ ] = 0 in π ( F × I ) . The first one asserts that two dots annihilate a cobordism when placed on the sameconnected component. Together with the neck cutting relation it implies further thata cobordism is annihilated when a dot is placed on its component of positive genus.The second relation prohibits a component of a cobordism to carry a dot if the componentcontains a closed curve that is nontrivial in π ( F × I ). In particular, a component ofa cobordism cannot carry a dot if its boundary curves are not contractible in F .We write B N ( F ) when only TD is imposed and B B N ( F ) when both. B N ( F ) iscommonly called the Bar-Natan skein category and it was first defined in [BN05]. The caseof the annulus was extensively studied by Russell [Rus09]. The relation B was introducedby Boerner [Boe08] for any surface F and we call B B N ( F ) the Boerner–Bar-Natan skeincategory . Functoriality
The construction of the formal bracket J T K is functorial up to signs: given a cobordism S between tangle diagrams T and T ′ there is a chain map J S K : J T K J T ′ K defined upto the factor ±
1, such that J SS ′ K = ± J S K ◦ J S ′ K . Hence, the same type of functorialityholds for any TQFT functor F : C ob ( F ) A satisfying the relations S , T , and .The chain map J S K is computed from a movie presentation of S , a sequence of genericsections S t = S ∩ ( F × { t } ) called movie clips , such that the part S | [ t,t ′ ] of S between twoconsecutive clips S t and S t ′ is either one of the Reidemeister moves, a saddle cobordism,a cap, or a cup [CS98]. There is a well-defined chain map for each of the parts and J S K is defined as the composition of these pieces. A cobordism S admits many moviepresentations and it is proven that up to sign J S K does not depend on the presentation50hosen [BN05]. Unfortunately, direct computation shows that the sign of J S K does dependon the presentation [Jac04].There are a few approaches to attack the sign issue. In case F = R one can use the Leedeformation of the Khovanov homology [Le05] to define canonical generators , which arepreserved by J S K up to sign [Ras05]. We can then redefine J S K so that the generators areactually preserved. This approach was used in [GLW15] to fix signs in certain cases.A different idea is to replace C ob ( F ) with another category. This was done successfullyfor F = R by Clark, Morrison and Walker using cobordisms with seams and coefficientsin the ring of Gaussian integers Z [ i ] [CMW09], then by Blanchet using nodal foams [Bla10]and by Vogel using mixed cobordisms : locally oriented cobordisms with certain disorien-tation curves [Vo15]. The first strictly functorial construction for (2 m, n )–tangles is dueto Caprau [Ca07], and is defined over Gaussian integers. It assigns to a tangle a directsummand of the corresponding invariant due to Chen and Khovanov, which is used inthis paper. In a forthcoming paper [BHPW18] we address the functoriality of the Chen–Khovanov invariant by rephrasing it in terms of gl foams and constructing an explicitisomorphism between the new and the original invariant. Finally, the functoriality ofthe bracket for links in thickened surfaces has been fixed in a recent work of Queffelecand Wedrich [QW18], who constructed certain foam categories that extend the nodalfoams introduced by Blanchet. R Let R be a commutative algebra and A a Frobenius algebra over R of rank 2. Thisdatum determines a TQFT functor with F ( ∅ ) := R and F ( ) := A , and it producesan invariant chain complex for link diagrams on a plane [Kh06]. For instance, Khovanov’sfunctor F Kh : C ob ( R ) M od ( k ) is defined this way by taking R := k and equipping A := Rw + ⊕ Rw − with the structure maps m : A ⊗ A A w + ⊗ w + w + ,w ± ⊗ w ∓ w − ,w − ⊗ w − , (5.6)∆ : A A ⊗ A (cid:26) w + w + ⊗ w − + w − ⊗ w + ,w − w − ⊗ w − , (5.7) η : k A (cid:8) w + , (5.8) ǫ : A k (cid:26) w + ,w − . (5.9)The functor is graded if we set deg w ± := ±
1. We shall write Kh ( D ) for the homology of CKh ( D ) := F Kh J D K , where D is a diagram of a link L ; it is called the Khovanov homology of the link L .Khovanov’s functor factorizes through B N ( R ), where a dot is understood as multi-plication by w − . In particular, both generators are images of 1 ∈ k under cup cobordisms F Kh (cid:16) (cid:17) : 1 w + F Kh (cid:16) b (cid:17) : 1 w − (5.10)which motivates the following graphical description of F Kh . Given a collection of curves Γ ⊂ R we identify F Kh (Γ) with the module generated freely by all diagrams obtained51rom Γ by decorating some curves with dots and imposing the relation that two dots ona single curve annihilate the diagram. For example, F Kh (cid:18) (cid:19) := span k (cid:26) , b , bb b , . . . (cid:27), b b = 0The generators w + and w − of the algebra A are represented by the circle without andwith a dot respectively. To redefine F Kh on a cobordism S we use the following rules: • if S creates a circle, then F Kh ( S ) modifies a diagram by inserting the new circlewith no dot on it, • if S contracts a circle, then F Kh ( S ) removes the circle from a diagram if it wasdecorated by a dot, or takes the diagram to 0 otherwise, and • we use the following local surgery formulas to define F Kh ( S ) if S is a merge or a split(5.11) b + b (5.12)where the blue thick arcs visualize the saddle of S .Notice that a merge of two curves is zero, when each curve carries a dot, as the surgery(5.11) produces a curve with two dots. Likewise, a split of a curve with a dot results inone diagram, as one of the two terms at the right hand side of (5.12) vanishes. There are two types of closed curves in the annulus: trivial curves bounding disks in A ,and essential curves , parallel to the core of A . The value of an annular TQFT functor F : B N ( A ) M od ( k ) on trivial curves is determined by Bar-Natan’s relations, but notthe value on essential curves.The first construction of an annular TQFT functor is due to Asaeda, Przytycki, andSikora [APS04]. The APS functor F A : B N ( A ) M od ( k ) assigns to a trivial andan essential curve the free modules W := span k { w + , w − } , V := span k { v + , v − } , (5.13)respectively, with the degree defined on generators asdeg w ± = ± , deg v ± = 0 . (5.14)This degree is denoted by j ′ in [GLW15] and differs from the one used in [Rob13]. Inaddition, the modules admit the annular grading , denoted adeg and defined asadeg w ± = 0 , adeg v ± = ± . (5.15)One can define F A by comparing it to F Kh . Indeed, V and W are isomorphic as ungradedmodules, but (5.14) induces a filtration on F Kh and the functor F A can be constructed asthe graded associate [Rob13]. For completeness we write down the maps correspondingto the elementary saddle moves. A merge is assigned one of the maps Clearly, those diagrams in which each circle carries at most one dot form a free basis for F Kh (Γ). ⊗ W Ww + ⊗ w + w + w ± ⊗ w ∓ w − w − ⊗ w − V ⊗ W Vv ± ⊗ w + v ± v ± ⊗ w − V ⊗ V Wv ± ⊗ v ± v ± ⊗ v ∓ w − depending on the curves involved, whereas for splits we choose one of W W ⊗ Ww − w − ⊗ w − w + w + ⊗ w − + w − ⊗ w + V V ⊗ Wv ± v ± ⊗ w − W V ⊗ Vw − w + v + ⊗ v − + v − ⊗ v + The value of F A on caps and cups is unchanged.The graphical description of F Kh can be extended to the annular case. Trivial curvescan again carry dots, but the essential ones cannot, because the merge cobordism takes v ± ⊗ w − to zero. Therefore, we shall visualize the two generators of V by choosingan orientation of the essential curve, anticlockwise for v + and clockwise for v − : v + v − (5.16)We use the usual surgery formulas for merging a trivial curve to an essential one orsplitting it off, keeping in mind that an essential curve cannot carry dots: b (5.17)Two essential curves can be merged together only if they have opposite orientations, inwhich case we decorate the resulting trivial curve with a dot, and otherwise we have zero: b Action of sl It has been recently observed in [GLW15] that the annular link homology admits an actionof sl if we consider W as a trivial representation and V is identified with the fundamentalone V = span k { v + , v − } or its dual V ∗ = span k { v ∗ + , v ∗− } , depending on the nestedness of53he associated essential curve, i.e. V and V ∗ are assigned alternatively. The followingtables describe the action of sl .The action on V The action on V ∗ Ev + = 0 Ev − = v + Ev ∗− = 0 Ev ∗ + = − v ∗− F v + = v − F v − = 0 F v ∗− = − v ∗ + F v ∗ + = 0 (5.20)There is an obvious isomorphism of sl –modules V ∼ = V ∗ , which identifies v ± with ± v ∗∓ .However, the action on the annular chain complex is defined using instead the linearisomorphism V ∼ = V ∗ that sends v ± to v ∗∓ , so that the action depends on the position of V in the tensor product. For instance, two essential curves are assigned V ⊗ V ∼ = V ∗ ⊗ V with sl acting in the following way: E ( v + ⊗ v + ) = 0 F ( v + ⊗ v + ) = v − ⊗ v + − v + ⊗ v − E ( v + ⊗ v − ) = − v + ⊗ v + F ( v + ⊗ v − ) = v − ⊗ v − E ( v − ⊗ v + ) = v + ⊗ v + F ( v − ⊗ v + ) = − v − ⊗ v − E ( v − ⊗ v − ) = v + ⊗ v − − v − ⊗ v + F ( v − ⊗ v − ) = 0It follows that the maps V ⊗ V W and
W V ⊗ V intertwine the action andthe annular TQFT functor is upgraded to F A : B N ( A ) g R ep ( sl ). In particular, sl acts on the triply graded annular homology. Remark.
The action admits the following graphical description: each clockwise orientedcurve in a diagram w contributes to Ew a diagram obtained from w by reversing the curve,and scaling it by ( −
1) if it is separated from the outer boundary by an odd number ofcurves. Likewise for F we reverse orientations of anticlockwise oriented curves. Beyond links in a thickened annulus, we also consider links in the twisted line bundle overthe M¨obius band (“twisted” means that the monodromy along the orientation reversingcurve is − id , so that the bundle is an orientable 3-manifold). Let us recall the constructionof the APS functor in this case.A M¨obius band M admits three types of curves: trivial curves bounding disks, sep-arating curves cutting an annulus out of M , and nonseparating ones . The APS functor F M : B N ( M ) M od ( k ) assigns to them the following free modules W := span k { w + , w − } , V := span k { v + , v − } , and U := span k { u + , u − } , (5.21)with the degree function vanishing on both V and U , and deg w ± = ± M × I than in A × I , and F M vanishes on thosewithout trivial circles in the boundary. Otherwise, it is defined as in the annular case formerges and splits (where both V and U can play the role of the “annular” V ) and one ofthe maps W Vw + v + + v − w − V Wv ± w − , b (5.23)We shall write CKh M ( D ) := F M J D K for the chain complex for a link diagram D on M , and Kh M ( D ) for its homology. A resolution of D can have at most one non-separating curve—such a curve cuts the band into an annulus. In particular, U onlyappears in CKh M ( D ) when D meets any cross section of M in an odd number points. Ifso, CKh M ( D ) ∼ = CKh M ( D ) ⊗ U , where we write CKh M ( D ) for the chain complex of D computed with k assigned to non-separating curves instead of U . Tangle diagrams in a thickened stripe R × I form a category. Thence, the formal Khovanovbracket of a tangle is a chain complex built over the bicategory Cob = C ob ( R × I )of points on a line, flat tangles in a stripe R × I , and surfaces in ( R × I ) × I . Thisbicategory is graded with the degree of a surface S defined in (5.1). To preserve thisricher structure, the homology for tangles is constructed by Chen and Khovanov [CK14]using a 2–functor F CK : Cob gBirep valued in the bicategory of graded bimodulesthat are sweet , i.e. finitely generated and projective as left and as right modules, but notnecessarily as bimodules. In fact, F CK factors through the Bar-Natan skein bicategory BN = B N ( R × I ). We begin with describing the modules assigned to tangles, thenthe algebras assigned to points, and finally reconstructing the bimodule structure. Cup diagrams with platforms A crossingless matching between 2 n points in a line is a collection of n disjoint arcsattached to the points. We shall draw the arcs in the lower half-plane R × R − and referto them as a cup diagram . Following [CK14] we generalize cup diagrams to allow semi-infinite arcs, each attached to one point only and going left or right towards infinity. Thiscan be visualized by drawing two vertical platforms going out of the horizontal line, oneto the left and one to the right of all the points, and attaching semi-infinite arcs to them.In particular, odd number of points are allowed. We shall call the points on the line termini to distinguish them from the endpoints on the platforms. Figure 7 presents allcup diagrams with three termini.Let G M n be the set of such diagrams with n termini. We define the weight of a diagram a ∈ G M n as wt( a ) := r − ℓ , where r and ℓ count respectively the arcs terminating onthe right and on the left platform. In what follows we shall write G M n ( λ ) ⊂ G M n for55igure 7: The generalized cup diagrams with three termini.the subset of diagrams of weight λ . Notice that G M n ( λ ) is empty unless λ has the sameparity as n .Dually we define the set G M n of cap diagrams with platforms with arcs drawn inthe upper half-plane. The reflection along the horizontal line induces a bijection of sets G M n ( λ ) ∋ a a ! ∈ G M n ( λ ) (5.24)for every n and λ . An extension of F Kh A pair of cup diagrams a ∈ G M m ( λ ) and b ∈ G M n ( λ ) can be used to produce a planarclosure b ! T a of any flat tangle T ∈ Cob ( m, n ). The closure is constructed by gluing a to the bottom of T and b ! to the top, then turning the platforms towards themselvesand identifying the endpoints of the arcs from inside out. In case m = n there will beunmatched endpoints, the same number at each side, because a and b have equal weights.We connect them with half-circles, see Figure 8.Figure 8: The construction of a planar closure of a (1 , F Kh : C ob ( R ) M od ( k ) is extended to collections of curves with plat-forms by assigning to such a collection a module generated by all possible decorations ofthe curves with dots as before, but with more restrictions:1) a diagram vanishes when it contains a curve intersecting any of the platforms twice,2) a dot annihilates a diagram when placed on a curve that intersects a platform, and3) as before, two dots on one curve annihilate the diagram.A diagram is nonadmissible if one of the above situations happens, see Figure 9. Surgeries(5.11) and (5.12) on nonadmissible diagrams produce nonadmissible ones, so that F Kh ( S )is well-defined for any surface S , see also [CK14]. The Chen–Khovanov functor assignsto a flat tangle T ∈ Cob ( m, n ) the module F CK ( T ) := M λ F CK ( T ; λ ) , with F CK ( T ; λ ) := M a ∈ G M m ( λ ) b ∈ G M n ( λ ) F Kh ( b ! T a ) , (5.25)56nd to a cobordism S between flat tangles T and T the linear map F CK ( S ) := M λ F CK ( S ; λ ) , with F CK ( S ; λ ) := M a ∈ G M m ( λ ) b ∈ G M n ( λ ) F Kh ( b ! Sa ) , (5.26)where b ! Sa stands for the surface ( b ! × I ) ∪ S ∪ ( a × I ). b Figure 9: Examples of generators of F CK (cid:0) (cid:1) . The first two diagrams are nonadmissi-ble, because they contain either a turnback or a dot on an open arc. Remark.
The closures of in Figure 9 are drawn without identifying the platformsof cup diagrams. Not only makes this smaller diagrams, but also easier to describethe module structure on F CK ( T ) once the Chen–Khovanov algebras are introduced. Example 5.4.
Consider the saddle cobordism S := : between the iden-tity (2 , F CK ( ) has seven generators, on which F CK ( S ) takes the following values:0 b b bb b b + b For example, the top right component of F CK ( S ) is a merge when the platforms areidentified: : Example 5.5.
Consider now the cobordism S := : going in the otherdirection. The module F CK ( ) has eight generators, on which F CK ( S ) is defined asbelow: bb , b , b b b , b S in this case is a split with each circle in its output touching a platform:: b + b = 0 . (5.27) Arc algebras and diagrammatic bimodules
Let c ∈ G M n be a generalized cup diagram and write S c for the cobordism from c ⊔ c ! to2 n vertical lines obtained by a sequence of n surgeries, one per arc in c , see Figure 10.The collection of such cobordisms defines linear maps µ T ′ ,T : F CK ( T ) ⊗ F CK ( T ′ ) F CK ( T ′ T ) , (5.28)one per a pair of tangles T ∈ Cob ( m, n ) and T ′ ∈ Cob ( n, k ). Explicitly, µ T ′ ,T ( x ⊗ y ) = 0for x ∈ F CK ( b ! T a ) and y ∈ F CK ( d ! T ′ c ) unless b = c , in which case µ T ′ ,T = F Kh ( d ! T ! S c T a ).Figure 10: A sequence of surgeries replacing a disjoint union of a cup diagram and itsvertical flip with vertical lines.The
Chen–Khovanov algebra A n is the module assigned to the tangle formed by n vertical lines, with x · y := µ ( x ⊗ y ). It admits a weight decomposition A n = M λ A n ( λ ) , (5.29)which is related to that from [CK14] by setting A n − k,k = A n ( n − k ). There is a uniqueprimitive idempotent e c ∈ A n for each closure c ∈ G M n given by the diagram c ! c with nodots. The idempotents are mutually orthogonal, and their sum is a unit in A n . Example 5.6.
The algebra A has generators in weights −
2, 0, and 2. Both A ( −
2) and A (2) are one dimensional, whereas A (0) has five generators: b (5.30)of which the first two are idempotents and the other square to zero. Furthermore, · = b and · = 0 . (5.31)The product in A n can be described explicitly using generalized surgeries as in [BS11].Because we do not identify platforms when drawing diagrams, each diagram has fourplatforms drawn vertically. The product x · y , when nonzero, can be then computedgraphically by placing y on x , connecting the platforms in between, and following the twosteps below. 58 tep I: surgeries at platforms. Replace two opposite arcs touching one of the platformswith a vertical line and decorate with a dot each closed loop created that way: b (5.32) Step II: surgeries on half-circles.
When no arc at inner platforms is left, perform surgerieson the remaining arcs using the usual surgery formulas (5.11) and (5.12), except thata merge of two open arcs is zero and a diagram with a dot on an open arc vanishes (inparticular, the first term in the result of the second surgery may vanish): b + b . (5.33)The merge of two arcs vanishes in the second step, because the two arcs have endpoints onthe outer platforms and they belong to the same circle when the platforms are identified(compare with (5.27)).It follows from the construction that the maps µ T ′ ,T are natural with respect totangle cobordisms. Furthermore, (5.26) can be extended to dotted surfaces, because F CK is defined by F Kh , in which case F CK ( S ) = 0 if any component of S carries two dots.Thence, the following result holds, see also [CK14]. Proposition 5.7. F CK ( T ) is a sweet ( A n , A m ) –bimodule for any flat tangle ( m, n ) –tangle T , where the actions of the algebras are given by µ . Moreover, (5.28) descend to naturalisomorphisms of bimodules F CK ( T ) ⊗ A n F CK ( T ′ ) ∼ = F CK ( T ′ T ) , (5.34) so that there is a strong bifunctor F CK : BN Birep . Throughout the paper we call F CK ( T ) a diagrammatic bimodule . They are called geometric in [CK14]. Each F CK ( T ) has a two sided dual F CK ( T ! ), where T ! is the verticalflip of T . It is also known that each weight component F CK ( T ; λ ) is indecomposable when T contains no loops [BS10, Theorem 4.14], and otherwise F CK ( T ; λ ) ∼ = F CK ( e T ; λ ) ⊗ k ℓ ,where e T is the tangle T with ℓ loops removed. Therefore, the category of diagrammaticbimodules has duals and is closed under direct summands. Grading
The grading on A n is defined in [CK14] by shifting by n the grading induced by the functor F Kh . This does not work well for bimodules assigned to tangles, though. For instance, re-garding a cup diagram c ∈ G M n as a flat (0 , n )–tangle, there is an isomorphism of gradedbimodules F CK ( c ) ⊗ F CK ( c ! ) ∼ = e c A n e c , but e c A n e c is graded differently from F CK ( c ! c ).In [BS11, BS10] a grading is computed differently. It agrees with that from [CK14] forarc algebras, and it is coherent with tensor products of tangle bimodules. On the other Recall that M ◦ N := N ⊗ M in Birep . T be a flat tangle with ℓ loops and c arcs connecting bottom endpoints. Givena diagram x ∈ F CK ( T ) orient all its curves counterclockwise. The platforms and boundarylines of T split some curves of x into vertical lines, caps, and cups. Let a be the numberof cups and caps with clockwise orientation. Then the degree of x is given by the formula,deg x := ℓ + c − a − d, (5.35)where d is the number of dots. For example,deg b = − , and deg = 1 , (5.36)where the arcs with clockwise orientation are thickened. A quick look on the surgeryformulas (5.11) and (5.12) reveals that for any tangle cobordism S the map F CK ( S )is homogeneous of degree deg S . Furthermore, µ T ′ ,T : F CK ( T ) ⊗ F CK ( T ′ ) F CK ( T ′ T )preserves the degree, see [BS10, Theorem 3.5 (iii)], so that (5.34) is an isomorphism ofgraded bimodules. However, F CK ( T ! ) is dual to F CK ( T ) only up to a degree shift. Chain complex and homology
Assume now that T is an oriented tangle with m points at the bottom and n at the top.The Chen–Khovanov complex C CK ( T ) := F CK J T K is a chain complex of graded sweet( A m , A n )–bimodules, obtained from the formal bracket by applying the bifunctor F CK component-wise. We refer to the homology Kh • ( T ) := H • ( C CK ( T )) as the Chen–Khovanovhomology . It is a triply graded theory: beyond the homological and quantum grading Kh • ( T ) admits a weight decomposition. Decategorification
Let V be the fundamental representation of U q ( sl ). It is known that flat tangles can beintrepreted as intertwiners between tensor powers of V , see Appendix A.1. More precisely,there is a linear Temperly–Lieb category TL , the morphisms of which are generated byflat tangles, and a functor F T L : TL R ep ( U q ( sl )) that takes a collection of n pointsto V ⊗ n , whereas a cap and a cup to the evaluation and coevalution map (A.6) respectively.The Chen–Khovanov construction categorifies this functor. Theorem 5.8 (cf. [CK14]) . There are isomorphisms γ n : K ( A n ) ⊗ Z [ q ± ] k ∼ = V ⊗ n , suchthat γ n ◦ [ F CK ( T )] ◦ γ − m = F T L ( T ) for any flat ( m, n ) –tangle T . Let us briefly recall from [CK14] how γ n is constructed. The group K ( A n ) is generatedfreely by indecomposable projectives P a := e a A n , one projective for each generalized cupdiagram a ∈ G M n . On the other hand, cup diagrams are flat tangle. The isomorphism γ n is defined by γ n ([ P a ]) := F T L ( a )( v ⊗ ℓ − ⊗ v ⊗ r + ), where ℓ and r count points of a on the leftand right platform respectively. 60 Quantization of the annular link homology
This section discusses the construction and properties of the quantum annular link ho-mology. We start with redefining the action of sl in a way that motivates the search forthe quantization. Section 6.2 contains a detailed construction of the quantum annularTQFT, which is then applied to (a quantization of) the formal bracket to obtain the newinvariant. sl revisited The action of sl on the annular link homology can be understood already at the level ofthe skein category. Consider now an operation that takes a flat tangle T into the surface S × T ⊂ A × I . A closed loop in T corresponds to a toroidal component of the sur-face, which is evaluated to 2 in B N ( A ) due to the neck cutting relation. Hence, there isa well-defined functor S × ( − ) : TL | q =1 B N ( A ), where TL | q =1 is the Temperly–Liebcategory specialized at q = 1, see Appendix A.1. In fact, it takes values in B B N ( A ),the quotient of B N ( A ) by the Boerner’s relation that forces a dot to annihilate a con-nected surface with an essential circle in its boundary. Remark.
Because
B B N ( A ) is both graded and additive, from now on we make twomodification to TL : we introduce a formal degree shift despite all morphisms in TL having degree 0, and formal direct sums, so that TL becomes a graded additive category.The functor F TL extends naturally to a faithful functor F TL : TL g R ep ( U q ( sl )) and S × ( − ) is still well-defined when q = 1. Proposition 6.1.
The functor S × ( − ) : TL | q =1 B B N ( A ) is an equivalence ofcategories.Proof. By the Delooping Lemma, each object in
B B N ( A ) is isomorphic to a collectionof essential curves with shifted degree, whereas the neck cutting relation implies thatmorphisms between such collections are generated by incompressible surfaces, i.e. annuli[AF07]. These are graded morphisms only when the degree shifts of the collection ofcurves at the bottom and top agree. Hence, the functor S × ( − ) is full and essentiallysurjective. Faithfulness follows, because the annuli are linearly independent in B B N ( A ),see [Rus09].Specializing q = 1 makes F TL valued in g R ep ( sl ). We check directly that the triangleof functors TL | q =1 B B N ( A )g R ep ( sl ) S × ( − ) F TL F A (6.1)commutes. This equips the annular TQFT with an action of sl that coincides withthe one from [GLW15]. Our goal is to quantize the annular skein category, so that Proposition 6.1 holds for all val-ues of q . We achieve this by identifying B N ( A ) with the additive closure of the horizontaltrace of the bicategory BN (taking the additive closure is necessary, because hTr( BN )61s only linear), then deforming the trace relation. Because the horizontal traces are com-pared with Bar-Natan skein categories, it is understood that they are made graded byintroducing formal degree shifts. Proposition 6.2.
There is an equivalences of categories hTr ⊕ ( BN ) ≃ B N ( A ) .Proof. The proof follows the same argument as the one of Theorem A.The bicategory BN is locally pregraded and each cobordism S is a homogeneousmorphism of degree deg S = χ ( S ) − B − d, (6.2)where B is the set of corners of S and d the number of dots. Thence, we can deform B N ( A ) by taking quantum horizontal traces. Namely, we define the quantum Bar–Natanskein category of the annulus as the additive closure of the quantum horizontal trace B N q ( A ) := hTr ⊕ q ( BN ) . (6.3)This category admits the following graphical description. The identified boundaries of( R × I ) × I form a membrane in the resulting solid torus, and the orientation of the core of A equips the membrane with a coorientation. Isotopic cobordisms are identified wheneverthe isotopy fixes the membrane. Otherwise, we scale the target cobordism according tothe following rules: = q = q − = q b = q − b (6.4)where, in each equality, the tangles draw on the membrane by the cobordism havethe same formal shift. For instance, a torus wrapped once around the annulus evalu-ates to q + q − : = b + b = q − b + q b . (6.5)It follows also that (1 − q ) annihilates a surface S that has a connected componentwith both an essential boundary and a dot. Indeed, moving the dot along the essentialboundary curve does not change the isotopy class of S , yet it requires to pass the dotthrough a membrane. Therefore, from now on we impose the Boerner relation, writing B B N q ( A ) := B N q ( A ) . B = hTr ⊕ q ( BN ) . B (6.6)for the quotient category. According to (6.5), the Cartesian product with a circle isa well-defined functor S × ( − ) : TL B B N q ( A ) (6.7)for any value of q . The rest of this section is devoted to prove the following statement.62 heorem 6.3. There is a commuting diagram
TL B B N q ( A )g R ep ( U q ( sl )) S × ( − ) F TL F A q (6.8) with the horizontal functor an equivalence of categories. We first show the surjectivity of S × ( − ). Hereafter we write µ = { } × R ⊂ S × R for the seam , the arc formed by identifying the boundaries of R × I . Lemma 6.4.
The canonical embedding i : vTr ⊕ q ( BN ) hTr ⊕ q ( BN ) is an equivalenceof categories.Proof. The functor i is full and faithful, see Section 3.4. Therefore, we need only to showthat every object of hTr ⊕ q ( BN ) = B N q ( A ) is isomorphic to one from the image of i .Let Γ be a collection of curves in A . If α ⊂ Γ is an embedded arc such that ∂α = µ ∩ α ,then we shall say that Γ is retractible if there is an embedded disk D ⊂ A with interiordisjoint from Γ, and which boundary is formed by α and the subarc of µ that lies betweenthe two endpoints of α , see Figure 11.Assume first that Γ has no retractible arcs, so that it is a collection of essential andtrivial circles. Essential circles are in the image of i , and the trivial ones can be removedusing the delooping isomorphism from Proposition 5.3.It remains to show that collections of curves with no retractible arcs generate B N q ( A ).For that pick a retractible arc α in Γ with a corresponding disk D and apply an isotopythat pushes α across D , taking it off µ . This procedure reduces the geometric intersectionnumber of Γ with µ by 2. Hence, applying this step several times, we end up with Γ ′ thatcontains no retractible arcs, hence from the image of i . µ ΓFigure 11: An example of a positive (to the left) and a negative (to the right) retractiblearc, each with the retracting disk. The top arc is not retractible, but it will be afterthe positive arc is retracted.
Corollary 6.5.
The functor S × ( − ) : TL B B N q ( A ) is full and essentially surjective.Proof. In the view of (6.6) and Lemma 6.4 we only need to show that every cobordism in
B B N q ( A ) is a linear combination of those of the form S × T , where T is a Temperly–Liebdiagram. We achieve that by using the Bar-Natan and the trace relations.Using the neck-cutting relation we reduce first a surface S to a linear sum of surfaces S i of genus 0. Because all closed components evaluate to scalars, we may assume each S i has a boundary component, which is an essential curve in A intersecting the seam63nce. Hence, S i is an annulus that intersects the membrane in an arc and, perhaps,in a collection of circles. The latter can be removed at the cost of some power of q with the left relations in (6.4). The resulting surface is isotopic to S × T , where T isthe intersection of S i with the membrane. Furthermore, the Boerner relation prohibitsthe surface from carrying dots.To show faithfulness of S × ( − ), so that B B N q ( A ) does not collapse, we constructthe functor F A q : B B N q ( A ) M od ( k ) using the diagramhTr q ( BN ) hTr q ( gBirep ) B N q ( A ) M od ( k ) hTr q ( F CK ) qHH F A q (6.9)where hTr q ( F CK ) is the functor induced by the Chen–Khovanov TQFT. Here gBirep isthe bimodule bicategory restricted to sweet bimodules, i.e. those that are finitely gen-erated and projective as both left and right modules. This bicategory has duals, sothat the shadow qHH factorizes through the horizontal trace by Theorem 3.5, providingthe right vertical map. Explicitly, F A q ( b T ) := qHH ( A n , F CK ( T )) (6.10)for a flat ( n, n )–tangle T . Equivalently, F A q can be constructed by pulling back qHH to BN along F CK and factorizing it through the horizontal trace. The following justifiestaking only the 0th Hochschild homology. Proposition 6.6.
Suppose k is flat over Z [ q ± ] . Then the inclusion A n ⊂ A n inducesan isomorphism of quantum Hochschild homology. In particular, the Chern character h : K ( A n ) ⊗ Z [ q ± ] k qHH ( A n ) is an isomorphism and qHH i ( A n ) = 0 for i > .Proof. It is enough to check the case k = Z [ q ± ]. Consider quantum Hochschild homologyas abelian groups and let R be any ring. The Universal Coefficient Theorem providesa commuting diagram0 Tor( qHH i − ( A n ) , R ) qHH i ( A n ⊗ R ) qHH i ( A n ) ⊗ R
00 Tor( qHH i − ( A n ) , R ) qHH i ( A n ⊗ R ) qHH i ( A n ) ⊗ R ∼ = α i β i γ i with exact rows, where the vertical homomorphisms are induced by the inclusion ofalgebras. Theorem 4.10 implies that β i is an isomorphism if R = Z p for prime p , becausethe global dimension of A n ⊗ Z p is finite [BS11] and simple modules are one-dimensional.Thus γ is an isomorphism, because the left groups vanish when i = 0. Hence, it isan isomorphism for R = Z , and so must be α . Using 5-Lemma we can now prove byinduction that β i is an isomorphism for R = Z and any i , which shows the first claim.The second claim follows from a direct computation. The algebra A n ∼ = k n is gen-erated by 2 n orthogonal idempotents, so that qHH ( A n ) ∼ = k n ∼ = K ( A n ) ⊗ Z [ q ± ] k andhigher homology vanishes. Corollary 6.7.
We have qHH i ( A n , F CK ( T )) = 0 for a flat ( n, n ) –tangle T and i > . roof. In the view of Lemma 6.4 every flat ( n, n )–tangle T is isomorphic in B N q ( A ) tothe disjoint union of the identity tangle on m points and ℓ trivial loops for some m, ℓ > qHH i ( A n , F CK ( T )) ∼ = qHH i ( A m ) ⊗ W ⊗ ℓ , where W = span k { w + , w − } is the moduleassigned by F CK to a circle. The thesis follows from Proposition 6.6.We are now ready to show the main result of this section. Proof of Theorem 6.3.
The naturality of the Chern character provides a commuting square K ( A n ) K ( A m ) qHH ( A n ) qHH ( A m ) [ F CK ( T )] h h F CK ( T ) ∗ (6.11)in which both compositions take the class [ P ] of a projective to the quantum Hattoring–Stallings trace of the identity morphism on P ⊗ A n F CK ( T ), see 3.8.2. The vertical arrowsare isomorphisms by Proposition 6.6 and the top horizontal map can be identified with F T L ( T ) due to Theorem 5.8. Finally, the bottom horizontal map coincides with F A q ,because it is the result of applying the pullback shadow ( F CK ) ∗ qHH to the morphism invTr q ( BN ) ⊂ hTr q ( BN ) represented by the square [ T, I × T ]. We use the deformed skein category together with the functor F A q to produce a quanti-zation of the annular link homology. The construction follows the usual pattern.1) Choose an annular link diagram D that is transverse to the seam µ ⊂ A . Inparticular, all crossings are assumed to be away from µ .2) Construct the formal bracket J D K q in Com b/h ( B B N q ( A )) as explained in Section 5.1,where we use the subscript q to emphasize that the resolutions of D are taken inthe quantized skein category3) Apply the functor F A q component-wise to J D K q to get the quantum annular chaincomplex CKh A q ( D ) := F A q J D K q .The quantum annular homology Kh A q ( L ) of an annular link L is defined as the homologyof CKh A q ( D ). It is a triply graded k –module with a homological grading, a quantumgrading coming from the grading on F CK , and an annular grading arising from the sl weight decomposition. In what follows we prove that it is well-defined and examinefunctoriality with respect to annular link cobordisms. We begin with a detailed lookon the quantized formal bracket. In what follows, we say that a functor or a shadow is projective or q -projective if it is defined on 2-morphisms up to scaling by ± ± q ± k forsome k ∈ Z respectively. Proposition 6.8.
The quantized formal bracket J − K q : L inks ( A ) Com b/h ( B N q ( A )) isa q –projective functor. In other words, the homotopy class of the formal chain complex J D K q is an invariant of annular links and an annular link cobordism W : L L ′ inducesa formal chain map J W K q : J D K q J D ′ K q , defined up to an overall sign and power of q ,where D and D ′ are diagrams of L and L ′ respectively.Proof. Let k Tan be the linear extension of the tangle bicategory, 2-morphisms of whichare finite sums of tangle cobordisms with coefficients in k . Motivated by Theorem Awe define L inks q ( A ) := hTr q ( k Tan ). It is a deformed linear extension of the category65f annular links, in which an annular link cobordism gets scaled by a power of q whenisotoped through the membrane. Clearly, the formal bracket on Tan extends to k Tan .The universal shadow d ( − ) : BN B N q ( A ), defined as the annular closure of flat tan-gles, induces a Lefschetz shadow on Com b/h ( BN ), see Lemma 3.16. Its pullback to k Tan along J − K is thence a projective shadow. The universality of the horizontal trace givesthen a projective functor J − K ′ q : L inks q ( A ) Com b/h ( B N q ( A )). One can easily checkthat J W K ′ q = J W K q for a fixed presentation of a link cobordism W , and it remains to checkhow dropping the membrane affects the bracket. For that choose isotopic link cobordisms W and W ′ that are related by a trace move. Regarded as morphisms in L inks q , theysatisfy a relation W = q k W ′ for some k . Hence, J W K q = J W K ′ q = ± q k J W ′ K ′ q = ± q k J W ′ K q ,which ends the proof. Theorem B.
The quantum annular homology Kh A q ( L ) is a triply graded invariant ofan annular link L , which is q –projectively functorial with respect to annular link cobor-disms. Moreover, it admits an action of the quantum group U q ( sl ) that commutes withthe differential and the maps induced by annular link cobordisms intertwine this action.Proof. It follows from Proposition 6.8 that the functor
CKh A q ( − ) := F A q J − K q is q –projective functor and the action of U q ( sl ) follows from Theorem 6.3.It was conjectured in [AGW15] that the Hochschild homology of Chen–Khovanovcomplexes recovers the annular chain complexes, with a proof for the algebras A ,n − ,the next–to–highest weight subalgebra of A n . The conjecture follows from Theorem 6.3and the computation of Hochschild homology in Corollary 6.7. Theorem C.
Let b T be the annular closure of an ( n, n ) –tangle T . Then there is an iso-morphism Kh A q ( b T ) ∼ = qHH • ( A n , C CK ( T )) , (6.12) natural with respect to chain maps associated to tangle cobordisms. The annular gradingin Kh A ( b T ) corresponds to the weight decomposition of C CK ( T ) .Proof. The right hand side of (6.12) is the total homology of the bicomplexcoInv q ( C • CK ( T ) ⊗ A n R • ( A n )) , where R • ( A n ) is the bar resolution of A n . Consider the associated spectral sequence E r that starts with the Hochschild differential. The first page E ij = qHH j ( A n , C iCK ( T )) hasvanishing rows except the one with j = 0 due to Corollary 6.7. Hence, the sequencecollapses at the second page with E i = Kh i A q ( b T ) and E ij = 0 for r = 0, which endsthe proof. j i − Z Z b bbb b Z b bbb bbb E ij = HH i ( Kh jCK ( ))There is another spectral sequence { E r } associated to coInv q ( C • CK ( T ) ⊗ A n R • ( A n )),where the Chen–Khovanov differential is computed first. Its second page E rij = qHH i ( A n , Kh jCK ( T )) (6.13)is already an invariant of the annular link b T . Contrary tothe previous case, this sequence may not collapse imme-diately. The second page for T = has three non-trivialentries and there is a non-trivial differential that killstwo generators, see the diagram to the left for the case k = Z and q = 1. It is the third page that agrees with the annular homology of b T , whichis the homology of ( W V ⊗ ) with W in homological degree − .4 Homology for (2 , n ) torus links Consider the subcategory E ⊂ B B N q ( A ) generated by objects intersecting the seam µ in exactly two points. Note that every such object is of the form Γ I ∪ Γ N , where Γ N isa (possibly empty) union of trivial circles not intersecting µ , and Γ I is either a trivialcircle intersecting µ in two points, or a pair of essential circles, each intersecting µ ina single point.In what follows we shall write W or W I for the module assigned to a trivial circledepending on whether it is disjoint from µ or not. W is freely generated by w + and w − , the images of 1 ∈ k under the maps induced by a cup cobordism disjoint fromthe membrane M = µ × I , without and with a dot respectively. To pick generators for W I consider a cup cobordism that intersects M in a single arc α and define w + , w −− and w + − as the images of 1 ∈ k under the maps induced by the cobordism respectively withoutany dot, a dot on the negative side of M , and a dot on the positive side of M . All threegenerate W I , but they are not linearly independent: w −− = q w + − .We can represent elements of the modules graphically as in Section 5.3: the generatorsof V are visualized by orienting the essential circles, and those of W and W I are given asthe trivial circle without or with a dot. The relation in W I between w −− and w + − can bethen written diagrammatically as b = q b (6.14)Here we choose the counter-clockwise orientation of the core of the annulus, so thatthe seam µ is cooriented upwards. Because the intertwners (A.6) are not symmetric,the essential circles must be ordered. We choose the left–to–right ordering read fromthe seam.Capping off a trivial circle touching the seam vanishes on w + and takes w ±− to q ∓ ,as the result is a sphere without or with a dot respectively, but intersecting the mem-brane, and isotoping it off the membrane using (6.4) introduces a power of q . It is nowstraightforward to compute the saddle cobordisms in E using the comparison with F TL .A merge of two essential circles followed by capping off the trivial circle is the evaluationmap, which implies the following surgery rules : b q b (6.15)The other saddle cobordism is even easier to find out. For degree reasons it must vanishif the trivial circle carries a dot, and otherwise it is the coevaluation map:+ q − (6.16)Fix n > T ,n denote the annular (2 , n ) torus link: the annular closure of thebraid σ − n , where σ is the positive generator of the 2–strand braid group.67 roposition 6.9. The quantum annular Khovanov homology of the annular (2 , n ) toruslink is given byKh i,j A q ( T ,n ) = V if i = 0 and j = − n,V / ( q + ( − i ) if − n + 1 i − and j = 2 i − n,K ( q − ( − i ) if − n i − and j = 2 i − n + 2 ,V if i = − n and j = − n, else , (6.17) where V := ( V ⊗ V ) / span k { v + ⊗ v − + q − v − ⊗ v + } is the simple representation of U q ( sl ) of dimension 3, and K ( a ) := { v ∈ V | av = 0 } for any a ∈ k . Here i and j represent respectively the homological and the quantum degree as definedin (5.14). The latter is denoted by j ′ in [GLW15]. Proof.
Let D ⊂ A be a standard diagram for T ,n such that cutting D along µ results ina diagram for σ − n . Then each resolution of D belongs to E . We introduce the notations u q , l q : W I W I for the maps that put a dot on the positive and negative side of the circlerespectively, and w q : W I V ⊗ V for the split map. Explicitly, u q ( w + ) = w + − , u q ( w − ) = 0 , (6.18) l q ( w + ) = w −− = q w + − , l q ( w − ) = 0 , (6.19) w q ( w + ) = v + ⊗ v − + q − v − ⊗ v + , w q ( w ±− ) = 0 . (6.20)Let { m } denote the grading shift functor which raises the j –degree by m . Arguing asin [Kh99, Proposition 26], one can show that CKh A q ( D ) is quasi-isomorphic to the chaincomplex0 W I {− n + 1 } ∂ − n W I {− n + 3 } ∂ − n +1 . . . ∂ − W I {− n − } ∂ − W I {− n − } ∂ − V ⊗ V ∗ {− n } ∂ − = w q and ∂ i = u q − ( − i l q for − n ≤ i ≤ −
2. One can write the abovecomplex more explicitly by writing each W I as a direct sum W I = span k { w + , w − } = V { +1 } ⊕ V {− } , and by noting that the map u q − ( − i l q is given by u q − ( − i l q = (cid:18) − ( − i q (cid:19) with respect to this direct sum decomposition. It then follows that the above complex isisomorphic to a direct sum of complexes (cid:16) V w q V ⊗ V (cid:17) {− n } (cid:16) V − i q V (cid:17) { i − n } for − n + 1 ≤ i ≤ − (cid:16) V (cid:17) {− n } where, in each of these complexes, the bidegree of the rightmost nonzero term is supportedon the diagonal j = 2 i − n . The proposition now follows by passing to homology.68 ❍❍❍❍❍ j i − − − − − − V − V † V † − V ‡ V ‡ − V † V † − V ‡ V ‡ − V Table 1: The quantum annular homology for the torus knot T , . The representationsmarked by a dagger ( † ) only occur if q = 1, and the ones marked by a double dagger ( ‡ )only occur if q = −
1. The unmarked representations are always there.If k is a field and q = ±
1, then q + ( − i is invertible. It follows from the aboveproof that in such a case Kh i,j A q ( D ) = V if ( i, j ) = (0 , − n ) ,V if ( i, j ) = ( − n, − n ) , . (6.21)On the other hand, the quantum homology contains additional copies of V if q = ± q = 1. Here we discuss functorial properties of the quantum annular link homology, as well asgeneralizations, such as the quantized APS homology for links in a thickened M¨obius orthe twisted annular homology (which provides a way to describe homology of satelliteknots).
It follows from Theorem B that the quantum annular homology assigns a chain map toannular link cobordisms. Let us now recall how this map is computed.Let L and L ′ by annular closures of an ( m, m )–tangle T and an ( n, n )–tangle T ′ respectively. A link cobordism W : L L ′ intersects the membrane in an ( n, m )–tangle P , so that it can be represented by a tangle cobordism f W : P T T ′ P . Applyingthe Chen–Khovanov functor results in a square A m A m A n A nC CK ( T ) C CK ( P ) C CK ( P ) C CK ( T ′ ) f W ∗ (7.1)The quantum annular complex is computed by replacing each component of the Chen–Khovanov complex with its Hochschild homology. This is an example of a Lefschetz69hadow discussed in Section 3.6. Hence, the component of the chain map W ∗ at homo-logical degree i and quantum degree j is given by summing up the compositions qHH ( A m , C i,jCK ( T )) coev ∗ qHH ( A m , C i,jCK ( T ) ⊗ A m C i ′ ,jCK ( P ) ⊗ A n C i ′ ,jCK ( P ) ∗ ) f W ∗ qHH ( A m , C i ′ ,jCK ( P ) ⊗ A n C i,jCK ( T ′ ) ⊗ A n C i ′ ,jCK ( P ) ∗ ) ( − i ′ ( i + i ′ ) θ qHH ( A n , C i,jCK ( T ′ ) ⊗ A n C i ′ ,jCK ( P ) ∗ ⊗ A m C i ′ ,jCK ( P )) ev ∗ qHH ( A n , C i,jCK ( T ′ )) (7.2)over all indices i ′ . Notice that the third map takes the class of x ⊗ y ⊗ α to the class of( − i ′ ( i + i ′ ) q qdeg( x ) y ⊗ α ⊗ x . Proposition 7.1.
Let the cobordism W : L L trace the rotation of L along the an-nulus. The induced map Kh A q ( W ) scales x ∈ Kh A q ( L ) by q qdeg( x ) .Proof. Write L as the annular closure of T . Then W is represented by the identitycobordism from T T to itself, and the only components (7.2) that contribute are thosewith i = i ′ . In particular, the second map scales the argument by q j , but does not changethe sign.It follows that quantum annular homology detects twists: the chain map W ∗ inthe above proposition is not scaled identity. Even more, if a resolution T ζ of T is a col-lection of n vertical lines, the restriction of the twist to T ζ is q d , where d is the degreeshift of T ζ in the complex. Indeed, the component of the twist is represented by A n A n A n A n F CK ( T ζ ) { d } F CK ( T ζ ) { d } F CK ( T ζ ) { d } F CK ( T ζ ) { d } id (7.3)whereas the identity map has F CK ( T ζ ) with no degree shifts as vertical maps.Because Kh A q is defined using a Lefschetz shadow, we can use Corollary 3.22 to com-pute the invariant for closed surfaces. Theorem D.
Let c W ⊂ S × R be a closed surface obtained as an annular closure ofa link cobordism W : L L with L ⊂ R . Then Kh A q ( c W ) = Λ( W ∗ ) is the gradedLefschetz trace of W ∗ : Kh ( L ) Kh ( L ) , the endomorphism of the Khovanov homologyof L . In particular, Kh A q ( S × L ) coincides with the Jones polynomial J ( L ) .Proof. The first statement is an immediate consequence of Corollary 3.22. Applying it to W = L × I results in computing the graded Euler characteristic of Kh ( L ), the Khovanovhomology of L .The APS homology does not distinguish closed surfaces. Hence, Kh A q is a nontrivialdeformation of the annular sl –homology. 70 .2 An action of tangles on cablings The action of the braid group on the annular homology of a cabling of a framed annularknot K was studied in [GLW15]. In what follows we compute the action of the entirecategory of oriented tangles on the quantum annular homology of cablings of K . Wework in this section in characteristic 2, so that the Chen–Khovanov functor is strictlyfunctorial.To a framed knot K ⊂ A × I we can associate an embedding ν K : A × I A × I withthe tubular neighborhood as its image. It determines a functor K ∗ : T an L inks ( A )such that • a collection of oriented points B ⊂ R is taken to K B := ν K ( S × B ), an orientedcabling of K , and • an oriented tangle T ⊂ R × I is mapped to the oriented cobordism K T := ν K ( S × T )between the cablings.Applying the quantum annular homology produces a map of homology Kh A q ( K T ) : Kh A q ( K B ) Kh A q ( K B ′ )for any oriented tangle T ∈ T an ( B, B ′ ), defined up to an overall power of q . These givea projectively functorial action of T an , i.e. Kh A q ( K T ′ T ) = q k (cid:16) Kh A q ( K T ′ ) ◦ Kh A q ( K T ) (cid:17) for composable tangles T , T ′ , and some k ∈ Z .This action has an interpretation in the framework of horizontal traces when K isthe annular closure of a framed (1 , τ . Denote by ν τ : R × I R × I the associated parametrization of the tubular neighborhood of τ and choose a tangle T with collections of points B and B ′ as its bottom and top boundary respectively. Thenthe cobordism K T intersects the membrane in T and is represented by the square B BB ′ B ′ τ B T Tτ B ′ τ T (7.4)where τ B := ν τ ( B × I ) and τ ( B ′ ) := ν τ ( B ′ × I ) are cablings of τ , and τ T := µ τ ( T × I )traces the isotopy that slides T along τ . We use this to prove that the action satisfiesthe Jones relation. Theorem E.
Let K be a framed annular link, considered as an object in L inks q ( A ) .There is a functorial action of T an on the quantum annular homology of oriented cablingsof K , that takes a tangle T to the chain map Kh A q ( K T ) , and which intertwines the actionof U q ( sl ) . It factors through the Jones skein relation q Kh A q ( K ) − q − Kh A q ( K ) = ( q − q − ) Kh A q ( K ) (7.5) if K intersects the membrane in one point.Proof. Functoriality has been discussed before and compatibility with the action of U q ( sl )follows from the construction—the action of T an is already defined at the level of the quan-tized formal bracket. It remains to show that the Jones relation holds. We prove it forthe formal bracket. 71et K be the annular closure of a (1 , τ .Given a finite sequence ǫ = ( ǫ , . . . , ǫ k ) with ǫ i ∈ {− , +1 } we write τ ǫ for the oriented k –cabling of τ , in which each i -th cable is oriented parallel to τ when ǫ i = +1 and oppositeto τ otherwise. We write τ k when each ǫ i = 1. Up to degree shifts, the formal bracketdepends only on k . Explicitly, J τ ǫ K = J τ k K h w ( τ )2 ( k − k ǫ k ) i n w ( τ )2 ( k − k ǫ k ) o , (7.6)where we write w ( T ) = n + ( T ) − n − ( T ) for the writhe of T and k ǫ k := P ǫ i . Notice that k ǫ k ≡ k mod 2, so that both numbers are integers. To derive this formula it is enoughto compare degree shifts of a single resolution, which is left as an exercise.It follows from functoriality of the action and (7.6) that the Jones relation has to bechecked only for the first three tangles shown below: · · · · · · i i +1 k · · · · · · i i +1 k · · · · · · i i +1 k · · · · · · i i +1 k T + T − T T h We use the forth one to express the formal brackets of T + and T − as mapping conesfollowing Proposition 5.2, obtaining distinguished triangles J T K [ − { } J T h K [ − { } J T + K J T K { } J T h K {− } J T K {− } J T − K J T h K [1] {− } sd in prsd in pr in which the left morphisms are saddle cobordisms in BN . Consider now the diagram J T K [ − ⊗ J τ k K J T h K [ − ⊗ J τ k K J T + K ⊗ J τ k K J T K ⊗ J τ k KJ τ k K ⊗ J T K [ − J τ k K ⊗ J T h K [ − J τ k K ⊗ J T + K J τ k K ⊗ J T K sd in prsd in pr J τ T K [ − J τ Th K [ − J τ T + K J τ T K (7.7)where the quantum degree shifts are dropped for clarity. The left and right vertical mapsare identities, whereas those in the middle are induced by a sequence of Reidemeistermoves. The left square commutes up to a formal chain homotopy H by the functorialityof J − K . In particular, J τ T K and J τ T h K together with H induce a chain map α = (cid:18) J τ T K − H J τ T h K [ − (cid:19) : J τ k K ⊗ cone( sd ) cone( sd ) ⊗ J τ k K that would make the other squares in (7.7) commute when placed as the third verticalmap. Hence, it is enough to show that α and J τ T + K are chain homotopic.It is shown in [BN05, Section 8] that every degree 0 automorphism of J T K is homotopicto ± id if T is obtained from a crossingless tangle by twisting its endpoints. Therefore,there exists at most one homotopy equivalence J T K ≃ J T ′ K when T and T ′ are two suchtangles (as we work over Z ). In particular, there are unique homotopy equivalences uwv T ∗ · · ·· · · }(cid:127)~ ≃ uwv T ∗ · · ·· · · }(cid:127)~ and uwv T ∗ · · ·· · · }(cid:127)~ ≃ uwv T ∗ · · ·· · · }(cid:127)~ T ∗ is any of the four tangles T + , T − , T , or T h . On the other hand, each map J τ T ∗ K is a composition of the above equivalences, and so is the chain map α . Hence, α and J τ T + K agree up to a chain homotopy, which makes the middle and right squares in (7.7)commute in the homotopy category.The quantum annular bracket is a quantum Lefschetz shadow on Com b/h ( BN ). Hence,taking the degree shifts into acount, we have from Proposition 3.17 that J K T + K q = q J K T K q − q J K T h K q . Likewise, using the other distinguished triangle we obtain J K T − K q = q − J K T h K q − q − J K T K q . These two equalities imply the Jones skein relation.The commutativity of (7.7) can be also checked directly, as there are explicit formulasfor all the chain maps, see [BN05]. Although it requires more work, with this approachone can prove Theorem E for the homology with integral coefficients, once a strictlyfunctorial version of the construction due to Chen and Khovanov is used. This approachis used in a following paper. Alternatively, one can fix the signs explicitly as it was donesuccessfully in [GLW15].One can forget the membrane and work with true annular links, instead of L inks q ( A ).Although the map Kh A q ( K T ) is a priori defined only up to an overall power of q , makingthe relation (7.5) problematic, our definition of the map uses a special presentation ofthe cobordism: as the isotopy sliding T along τ , the opening of K . For this particularpresentation the power of q is well-defined, so that the Jones relation holds. However,one must be careful when starting to isotope the cobordism. Let ρ ∈ Diff ( R × I ) be the flip along the interval, i.e. ρ ( x, t ) = ( − x, t ). It inducesan endobifunctor ρ ∗ on BN . Identifying ( x,
1) with ( − x,
0) produces a M¨obius band M ,and, by the argument from the proof of Theorem 2.19, hTr ⊕ ( BN , ρ ∗ ) and B N ( M ) areequivalent categories. In an analogy to the case of annulus we define B N q ( M ) := hTr ⊕ q ( BN ( R × I ) , ρ ∗ ) (7.8)The reflection ρ induces also an automorphism of A n , which we denote with the samesymbol. The following result is immediate from the definition of F CK . Lemma 7.2.
Given an ( m, n ) –tangle T write T flip for its reflection along the verticalaxis. Then ρ F CK ( T ) ρ ∼ = F CK ( T flip ) (7.9) as ( A m , A n ) –bimodules. Thence, we can define a TQFT functor F M q : B N q ( M ) g M od ( k ) with the help ofthe pullback shadow ( F CK ) ∗ qHH ρ . The argument from Proposition 6.6 applied to qHH ρ implies that qHH ρ> ( A n ) = 0 and qHH ρ ( A n ) ∼ = coInv ρ ( A n ) , (7.10)73here the latter is generated by idempotents corresponding to symmetric cup diagrams(i.e. those fixed by ρ ). In particular, if we write c n for the collection of n parallel separatingcurves, each wrapping M twice, and γ for the nonseparating curve, then F M q ( c n ) ∼ = qHH ρ ( A n ) ∼ = A n ∼ = V ⊗ n and F M q ( γ ∪ c n ) = 0 (7.11)where A n ∼ = qHH ρ ( A n ) takes an idempotent e ∈ A n to e ⊗ ρ ( e ) ∈ A n . In particu-lar, we are interested only in the subcategory B N ev q ( M ) generated by those collectionsthat do not contain γ . These collections are characterized by the following property:if µ ⊂ M cuts the band into a square, then each object from B N ev q ( M ) intersects µ inan even number of points. The argument from Lemma 6.4 adapted to this case showsthat vTr ⊕ q ( BN ev , ρ ∗ ) hTr ⊕ q ( BN eb , ρ ∗ ) = B N ev q ( M ) is an equivalence of categories.The following result is straightforward. Lemma 7.3.
Let γ ⊂ M be a nonseparating curve. Then there is a commutative diagramof functors B N q ( A ) B N ev q ( M ) M od ( k ) Φ F A q F M q (7.12) where Φ :
B N q ( A ) B N q ( M ) is induced by the diffeomorphism A ≈ M − γ .Proof. The functor Φ, seen as hTr ⊕ q ( BN ) hTr ⊕ q ( BN , ρ ∗ ), adds n vertical lines next toan ( n, n )–tangle and sends a morphism [ T, W ] to [ T ∪ T flip , W ∪ ( I × T flip )]. The differencein powers of q appears, because going once through a membrane in hTr ⊕ q ( BN ) ∼ = B N q ( A )corresponds to going twice through the membrane in hTr ⊕ q ( BN , ρ ∗ ) ∼ = B N q ( M ).Each cobordism in vTr ⊕ q ( BN ev , ρ ∗ ) can be decomposed into a composition of cobor-disms from the image of Φ and projective planes with a disk removed, each with a non-trivial curve in M as its boundary. These can be seen in turn as the saddle cobordismsbetween a trivial and a nontrivial curve, with the trivial curve capped off. They corre-spond under the quotient map ( R × I ) × I M e × I to surfaces × I and × I , andwe denote them by S and S respectively. Lemma 7.4. F M q ( S ) : V k evaluates v + and v − to q and respectively, whereas F M q ( S ) : k V takes to v + + q − v − .Proof. The module V := qHH ρ ( A ) assigned by F M q to a nontrivial curve has canonicalgenerators b := and b := (7.13)which corresponds under (7.11) to v + + q − v − and v − respectively. The thesis now followsfrom direct computations. Indeed, the map F M q ( S ) is given by the sequence Z [ q ± ] = qHH ρ ( Z ) qHH ρ ( Z , F CK ( ) ⊗ A ∗ F CK ( )) θ qHH ρ ( A , ∗ F CK ( ) ⊗ F CK ( )) qHH ρ ( A )which takes 1 ∈ k into b . Dually, S is assigned the sequence qHH ρ ( A ) qHH ρ ( A , F CK ( ) ⊗ ∗ F CK ( ))74 qHH ρ ( Z , ∗ F CK ( ) ⊗ A F CK ( )) qHH ρ ( Z ) = Z [ q ± ] , which takes b and b to q + q − and 1 respectively.The above is enough to compute F M q on all morphisms in B N q ( M ). A comparison withthe formulas from Section 5.4 shows that F M q is a deformation of the APS construction. Theorem 7.5.
Let b T be the closure in M of a (2 n, n ) –tangle T . Then there is an iso-morphism Kh M ( b T ) ∼ = HH ρ • ( A n , C CK ( T )) (7.14) natural with respect to the chain maps associated to link cobordisms. A family of homotopy equivalences C CK ( τ ǫ ′ ) ⊗ A n C CK ( T ) ≃ C CK ( T ) ⊗ A m C CK ( τ ǫ ) wasused in Section 7.2 to construct the action of oriented tangles, where T is placed verticallyon the membrane. However, one can also place T horizontal and understand the homotopyequivalences as components of a natural endotransformation τ ∗ of the identity functoron Com b/h ( DB ), where DB := F CK ( BN ) is the bicategory of diagrammatic bimodules .Therefore, the pair (Id , τ ∗ ) is a functor from the endocategory ( Com b/h ( DB ) , Id) to itself,and we can obtain a new homology theory by pulling back the shadow ( F CK ) ∗ qHH . Thismotivates the following definition. Definition 7.6. An annular twistor is a family M := { M ǫ } of chain complexes of( A n , A n )–bimodules, parametrized by finite sequences ǫ = ( ǫ , . . . , ǫ n ) with ǫ i ∈ {− , +1 } ,together with natural chain maps φ T : M ǫ ⊗ A m C CK ( T ) C CK ( T ) ⊗ A n M ǫ ′ , (7.15)one per oriented ( m, n )–tangle T with orientation of input and output encoded by ǫ and ǫ ′ respectively.Although not stated this way, the chain map (7.15) is a homotopy equivalence. Up todegree shifts, its inverses can be built from φ T ! together with evaluation and coevalutionmaps.Let L be the annular closure of an ( n, n )–tangle T with orientation of endpointsencoded by a sequence ǫ . It follow from Proposition 3.8 that Kh A q ( L, M ) := qHH • ( A n , M ǫ ⊗ A n C CK ( T ))does not depends on the choice of T . We call it the annular homology twisted by M .When M ǫ = C CK ( τ ǫ ) is the twistor described above, Kh A q ( L, M ) is the quantum annularhomology of the satellite of L with companion b T . There is another generalization of the annular homology, which is very close to twisting.In this section we work in characteristic two, so that the Chen–Khovanov construction isstrictly functorial. 75et us fix a (1 , T and denote by T ! its mirror image. Write V T := CKh A q ( b T )and V ∗ T := CKh A q ( b T ! ) for the quantized annular chain complexes of the annular closuresof T and T ! . They form a dual pair, with evaluation and coevalution maps induced bycobordisms T × and T × in ( A × I ) × I respectively, see Appendix A.4 and Figure 12for a picture of the evaluation cobordism.We generalize F A q : B N q ( A ) g R ep ( U q ( sl )) by assigning to essential circles al-ternatively V T and V ∗ T . An annulus with essential boundary is then assigned one ofthe evaluation of coevaluation maps, and merging a trivial circle to an essential one isdetermined by the cobordisms merging an unknot to b T or b T ′ . It follows from the functo-riality that this produces a well-defined TQFT functor F T A q : B N ( A ) Com b/h ( U q ( sl )),valued in the homotopy category of representations of U q ( sl ). Definition 7.7.
Let T be a (1 , L with diagram D .The T –annular homology Kh T A q ( L ) of L is the homology of the chain complex F T A q J D K .It is tempting to express the above construction using annular twistors. Indeed, onecan define M ǫ , where ǫ has length n , as the alternating tensor product V T ⊗ V ∗ T ⊗ V T ⊗ . . . of n factors. The duality between V T and V ∗ T can be used to construct a chain map ν T : M ǫ ⊗ A m C CK ( T ) C CK ( T ) ⊗ A n M ǫ ′ for every ( m, n )–tangle T . However, it is not natural. For instance, seeing T = asa composition of a cup and a cap, the composition M ⊗ C CK ( ) ⊗ A C CK ( ) ν C CK ( ) ⊗ A M ⊗ A C CK ( ) ν C CK ( ) ⊗ A C CK ( ) ⊗ M is the multiplication by the quantum dimension of V T ⊗ V ∗ T , the Jones polynomial of b T b T ′ . On the other hand, C CK ( ) = k {− } ⊕ k { +1 } , suggesting that ν = id. Thisseams to be the only issue, because for the decomposition of a zigzagthe induced map is a homotopy equivalence due to the relation between the evaluationand coevaluation morphisms. A Background survey
The material presented here is widely known, and the main goal of this section is to fixthe notation. Bicategories are treated in the excellent paper [Be67], whereas [Lei98] isa brief list of basic definitions. The reader is also referred to [EGNO09], because manyresults about monoidal categories immediately translates to bicategories.
A.1 Representations of U q ( sl ) As usual we fix a commutative unital ring k together with an invertible element q . Bydefinition, U q ( sl ) is the unital associative k –algebra with generators E , F , K , K − andrelations KE = q EK, KK − = 1 = K − K, F = q − F K, K − K − = ( q − q − )( EF − F E ) . It is a Hopf algebra with the comultiplication ∆ : U q ( sl ) U q ( sl ) ⊗ U q ( sl ), the counit ǫ : U q ( sl ) k , and the antipode S : U q ( sl ) U q ( sl ) defined by∆( E ) = E ⊗ K + 1 ⊗ E, ǫ ( E ) = 0 , S ( E ) = − EK − , (A.1)∆( F ) = F ⊗ K − ⊗ F, ǫ ( F ) = 0 , S ( F ) = − KF, (A.2)∆( K ± ) = K ± ⊗ K ± , ǫ ( K ± ) = 1 , S ( K ± ) = K ∓ . (A.3)Using this Hopf algebra structure, we can regard the category of finite-dimensional rep-resentations of U q ( sl ) as a monoidal category with duals. The unit in this monoidalcategory is given by the trivial representation V = k , on which U q ( sl ) acts by multipli-cation by ǫ ( X ) for any X ∈ U q ( sl ).We write V := span k { v , v − } and V ∗ := span k { v ∗ , v ∗− } for the fundamental repre-sentation and its dual. We identify both with the rank two module V := span k { v + , v − } using the isomorphisms (cid:26) v v + v − v − and (cid:26) v ∗ v − v ∗− q − v + . (A.4)This equips V with two actions of U q ( sl ) that differ by signs:The action on V ∼ = V The action on V ∼ = V ∗ Ev + = 0 Ev − = v + Ev + = 0 Ev − = − v + F v + = v − F v − = 0 F v + = − v − F v − = 0 Kv + = qv + Kv − = q − v − Kv + = qv + Kv − = q − v − (A.5)The duality between V and V ∗ comes with the evaluation and coevaluation maps ev : V ⊗ V k coev : k V ⊗ Vv + ⊗ v + v + ⊗ v − q v + ⊗ v − + q − v − ⊗ v + v − ⊗ v − v − ⊗ v + U q ( sl ) if V ⊗ V is identified with either V ⊗ V ∗ or V ∗ ⊗ V .The full subcategory of U q ( sl ) generated by tensor powers of V admits a graphicalrepresentation. Let TL be the Temperly–Lieb category , the linear category with objectsfinite collections of points on a real line and morphisms generated by flat loopless tan-gles, i.e. collection of disjoint intervals in R × I with endpoints on the boundary lines.Composition is defined by stacking pictures one onto another and trading each closedcomponent for q + q − . For example, ◦ = ( q + q − ) (A.7)There is a functor F TL : TL R ep ( U q ( sl )) that assigns V ⊗ n to a collection of n points,whereas caps and cups are sent to the evaluation and evaluation homomorphisms. It isknown that F TL is faithful [Th99]. 77 .2 Knots and tangles Let M be an oriented smooth 3–manifold. A proper 1–submanifold T ⊂ M is calleda tangle . We call it a link if it has no boundary, and a knot if in addition it has onecomponent. All tangles and links in this paper are assumed to be oriented unless statedotherwise. An isotopy of tangles T and T ′ is a smooth map Φ : M × I M such thateach Φ t := Φ( − , t ) is a diffeomorphism fixed at the boundary, Φ = id, and Φ ( T ) = T ′ .If T and T ′ are oriented, then we require that the orientation is preserved by Φ .Denote by − T the tangle T with reversed orientation of all its components. A tanglecobordism from a tangle T to T is an oriented surface S ⊂ M × I with boundary ∂S = − T × { } ∪ T × { } ∪ ( − ∂T × I ). We shall consider cobordisms only up toan isotopy, in which case they form a category: composition is given by gluing cobordisms,and the identity morphism on a tangle T is represented by the cylinder T × I ⊂ M × I .When M = F × I is a thickened surface, then isotopy classes of oriented tangles in M form a category, with the product induced by stacking, M ∪ M ∼ = M , and tangles withtangle cobordisms form a 2–category. Notation.
We shall write
Links ( M ) for the set of isotopy classes of oriented links in M ,and L inks ( M ) for the category of oriented links in M and cobordisms between them.Isotopy classes of oriented tangles in a thickened surface F × I form a category T an ( F )with the composition induced by stacking, and similarly cobordisms between tangles in F × I form a 2–category Tan ( F ). We write simply T an and Tan when F = R × I .Assume M is a line bundle over a surface F and consider the projection M F onto the zero section. It maps a generic tangle T to an immersedcollection of intervals and circles ˜ T ⊂ F with only finitely many multiplepoints, each a transverse intersection of two arcs. A diagram of T isconstructed from ˜ T by breaking one of the arcs at each double pointas follows. When F is oriented, then fibers of M admit a canonicalorientation and we break the lower arc at each double point of ˜ T (see an example abovefor F = R × I ). In case F is nonorientable, choose a minimal collection of curves γ thatcuts F into an orientable surface. Then there is a normal field over F − γ and we canconstruct the diagram as before. Then Reidemeister moves and planar isotopies relatediagrams between isotopic tangles if a crossing is switched when moved through γ : γ = γ This follows, because the normal field is reversed at points of γ . A.3 Constructions on categories
Below we review definitions of certain constructions on categories that appear throughoutthe paper.
Additive closure
We say that a linear category is additive if it has finite direct sums. Each category C admits the additive closure C ⊕ , the smallest additive category containing C . It isconstructed by introducing formal direct sums:78 an object of C ⊕ is a finite sequence ( x , . . . , x r ) with x i ∈ Ob( C ), possibly empty, • a morphism from ( x , . . . , x r ) to ( y , . . . , y s ) is a matrix ( f ij ) of morphisms f ij ∈ C ( x i , y j ), and • composition is defined by the matrix multiplication rule.There is an inclusion C C ⊕ , which takes an object x to the 1-element sequence ( x ).It is an equivalence of categories if C is already additive. Idempotent completion
An endomorphism p ∈ C ( x, x ) satisfying p ◦ p = p is an idempotent . We say that p splits if it decomposes p = s ◦ r such that r ◦ s is an identity morphism. In such a case r isan epimorphism and its codomain is called the image of p .A category is idempotent complete if all its idempotents split. Each category C admitsits idempotent completion Kar( C ), also called also the Karoubi envelope of C , which isthe smallest idempotent complete category containing C . It is constructed by taking allidempotents of C as objects, and defining morphisms from e to e ′ as those morphisms f from C that e ′ ◦ f ◦ e is well-defined and equal to f . The identity morphism on e is givenby the idempotent itself. Formal complexes and homotopy category A formal complexes over a category C is a sequence of objects and morphisms from C ( C • , d ) = (cid:0) . . . C i d i C i +1 d i +1 C i +2 . . . (cid:1) (A.8)satisfying d i +1 d i = 0 at each place. The morphisms d i are called the differential . We saythat ( C • , d ) is bounded if C i = 0 except finitely many indices.A formal chain map from ( C • , d ) to ( D • , d ) is a collection of morphisms f i : C i D i fitting into a commuting ladder C i C i +1 C i +2 D i D i +1 D i +2 d df i f i +1 f i +2 d d . . . . . .. . . . . . (A.9)Finally, a formal chain homotopy from f • to g • , both chain maps from ( C • , d C ) to( D • , d D ), is a collection of morphisms h i : C i D i − satisfying d i − D ◦ h i + h i +1 ◦ d iC = g i − f i . In such case we say that f • and g • are homotopic , which we write f ∼ g .Formal complexes (reps. bounded formal complexes) and chain maps modulo chainhomotopies constitute the homotopy category of complexes Com /h ( C ) (resp. Com b/h ( C )).Isomorphism in these categories are called homotopy equivalences and we usually write C • ≃ D • for complexes that are homotopically equivalent.The categories Com /h ( C ) and Com b/h ( C ) are triangulated [GM, Wei95], which meansthat they come with a homological degree shift functor and a collection of distinguishtriangles satisfying certain axioms. The degree shift functor is usually denoted by [1] andit shifts a complex leftwards, negating the differential at the same time: C [1] i := C i +1 , d [1] i := − d i +1 . (A.10)Distinguished triangles are of the form C • f • D • in • cone • ( f ) pr • C [1] • , (A.11)79here cone • ( f ) stands for the mapping cone of f , the formal complexcone i ( f • ) := C i +1 ⊕ D i , d i = (cid:18) − d i +1 C f i d iD (cid:19) . (A.12)The morphisms in i : D i cone i ( f • ) and pr i : cone i ( f • ) C [1] i = C i +1 are the inclu-sion and projection respectively. A.4 Bicategories A bicategory C is a ‘higher level’ analogue of a category. It consists of • a class of objects Ob( C ), • a category C ( x, y ) for each pair of objects ( x, y ), whose objects and morphismsare called and respectively and represented by single and doublearrows, • a unit id x ∈ C ( x, x ) for each object x , • a functor ◦ : C ( y, z ) × C ( x, y ) C ( x, z ) for each triple of objects ( x, y, z ), and • natural isomorphisms a : f ◦ ( g ◦ h ) ∼ = ( f ◦ g ) ◦ h l : id y ◦ f ∼ = f r : f ◦ id x ∼ = f (A.13)called associators and unitors , which satisfy the pentagon and triangle axioms[Be67].A bicategory is called strict or a if the natural isomorphisms are identities.We call C a locally small bicategory when each category C ( x, y ) is small, and C is small when also Ob( C ) is a set.Associators and unitors are often omitted for clarity. According to the MacLane’sCoherence Theorem [ML98, Chapter VII.2] there is only one way how to insert theseisomorphisms back when necessary. Notation. x y fg α
In this paper we denote categories with calligraphic letters C , D , etc., whereasbold letters C , D , etc. are reserved for bicategories. Identity morphisms are written asid x , and identity 2–morphisms as f . If C is a bicategory, then the compo-sition in C ( x, y ) is denoted by ∗ and called vertical , whereas ◦ is the hor-izontal composition . These come from the common convention to draw1–morphisms horizontally and 2–morphisms vertically, see the diagram tothe right of a 2–morphism α : f g .Choose bicategories C and D . A bifunctor F : C D consists of a function ofobjects Ob( C ) Ob( D ) and a collection of functors C ( x, y ) D ( F x, F y ), togetherwith natural 1–morphisms m : F ( g ) ◦ F ( f ) F ( g ◦ f ) i : id F x F (id x ) (A.14)satisfying certain coherence axioms. They are called morphisms of bicategories in [Be67],whereas the word homomorphism is reserved for the case when m and i are invertible. Insuch case we say that F is a strong bifunctor .Choose bifunctors F , G : C D . A natural transformation η : F G is a col-lection of 1–morphisms η x : F ( x ) G ( x ), one per object x ∈ C , and 2–morphisms80 f : G ( f ) ◦ η x η y ◦ F ( f ), one per 1–morphisms f ∈ C ( x, y ), such that F ( x ) G ( x ) F ( y ) G ( y ) η x G ( f ) η y F ( f ) F ( g ) η f F ( α ) = F ( x ) G ( x ) F ( y ) G ( y ) η x F ( g ) η y G ( f ) G ( g ) η g G ( α ) (A.15)for every 2–morphism α : f g . Moreover, η f must be coherent with all the othercanonical 2–isomorphisms (associators, unitors, the structure 2–isomorphisms of F and G ), see [Lei98]. We say that η is strong if each η f is invertible.Finally, let η, ν : F G be two natural transformations. A modification
Γ : η ν is a collection of 1–morphisms Γ x : η x ν x , such that ν f ∗ ( G f ◦ Γ x ) = (Γ y ◦ F f ) ∗ η f for every 1–morphism f : x y . Duals
A bicategory C has left duals if each f ∈ C ( x, y ) admits ∗ f ∈ C ( y, x ) together withcoevaluation and evaluation 2–morphismsid y coev ∗ f ◦ f f ◦ ∗ f ev id x (A.16)fitting into commuting triangles f f ◦ ∗ f ◦ ff f ◦ coev f ev ◦ f ∗ f ∗ f ◦ f ◦ ∗ f ∗ f coev ◦ ∗ f ∗ f ∗ f ◦ ev (A.17)where for clarity associators and unitors are omitted. The morphism ∗ f is called the leftdual to f . We define the right dual f ∗ of f by reversing the order of the horizontalcomposition in (A.16) and (A.17). If a dual 1–morphism exists, then it is unique up toan isomorphism. In particular, dual pairs are preserved by strong bifunctors. Examples
Small categories, functors, and natural transformations form a strong bicategory
Cat .A left (resp. right) dual to a functor F is its left (resp. right) adjoint. Hence, not all1–morphisms in Cat are dualizable.Tangles in a thickened surface F × I constitute a bicategory Tan ( F ), which objects arefinite collections of points in F , 1–morphisms are tangles with endpoints on F × ∂I , and 2–morphisms are tangle cobordisms. This bicategory has both left and right duals. Indeed,the mirror image T ! of a tangle T , obtained by flipping F × I , is both the left and rightdual of T . The evaluation 2-morphism is obtained by revolving T in four dimensions alongthe input surface F ×{ } , i.e. it is the image of the map ( p, t, s ) ( p, t cos( sπ ) , t sin( sπ ))with ( p, t ) ∈ T and s ∈ I , suitably normalized (see Fig. 12). The coevaluation is defineddually by a rotation along the output surface F × { } .Rings, bimodules, and bimodule maps form a bicategory Bimod , with horizontalcomposition given by the tensor product: N ◦ M := M ⊗ B N for an ( A, B )–bimodule M and a ( B, C )–bimodule N . This formula comes from interpreting an ( A, B )–bimodule M T T Figure 12: The evaluation cobordism
T T ! for a tangle T ∈ Tan (1 , − ) ⊗ A M : M od ( A ) M od ( B ) between the categories of right modules.This bicategory does not have duals. Indeed, an ( A, B )–bimodule M has a left (resp.right) dual if and only if it is finitely generated and projective as a right B –module (resp.left A –module). If so, the left and right dual modules are given as the modules of right B –linear and left A –linear morphisms respectively: ∗ M := Hom B ( M, B ) M ∗ := Hom A ( M, A )For this reason we usually restrict either to
Rep (bimodules with left duals) or
Birep (bimodules with both left and right duals).
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