aa r X i v : . [ m a t h . QA ] M a r ODD KHOVANOV HOMOLOGY FOR TANGLES
GR´EGOIRE NAISSE AND KRZYSZTOF PUTYRA
Abstract.
We extend the covering of even and odd Khovanov link homology to tangles,using arc algebras. For this, we develop the theory of quasi-associative algebras andbimodules graded over a category with a 3-cocycle. Furthermore, we show that a coveringversion of a level 2 cyclotomic half 2-Kac–Moody algebra acts on the bicategory of quasi-associative bimodules over the covering arc algebras, relating our work to a constructionof Vaz. Introduction
In the seminal paper [17], Khovanov constructed a homology theory for links that cat-egorifies the Jones polynomial, in the sense that the graded Euler characteristic of theformer coincides with the later. The construction relies on a 2d-TQFT obtained fromthe Frobenius algebra Z [ x ] / ( x ). Khovanov extended his construction to tangles with aneven number of endpoints in [18]. The invariant then takes the form, for a tangle T ,of the homotopy type of a complex Kh( T ) of graded bimodules over arc algebras H n .It respects a gluing property, meaning that given a pair of tangles T ′ and T , we haveKh( T ′ ) ⊗ H n Kh( T ) ∼ = Kh( T ′ T ) . Generalizations of arc algebras were introduced by Strop-pel [24] and independently by Chen–Khovanov [7], allowing to extend the construction toany tangle. These algebras are given an in-depth study by Brundan and Stroppel in theseries of paper [4, 3, 5, 6].Ozsv´ath, Rasmussen and Szab´o constructed in [21] a distinct version of Khovanov linkhomology. It is based on a projective TQFT, defined only up to sign. They called it oddKhovanov homology since it makes use of anticommutative variables instead of commu-tative ones. Their construction yields a distinct invariant of links from the usual ‘ even ’Khovanov homology. Odd Khovanov homology also categorifies the Jones polynomial,and both construction agree modulo 2. Furthermore, even Khovanov homology can bethought as obtained from the geometry of the complex projective line P ( C ) [25], whileodd Khovanov homology is obtained from the real one P ( R ) [9].The second author of this article built a framework [22] that allows a cobordism-type in-terpretation of Ozsv´ath–Rasmussen–Szab´o projective TQFT. The idea is to enrich thenotion of cobordisms with a chronology , i.e. a framed Morse function that separatescritical points [13]. These chronological cobordisms form a 2-category ChCob wherethe 2-morphisms are diffeotopies of the chronology maps. Then, the projective TQFTof [21] becomes a genuine 2-functor. In [22] is also introduced the notion of coveringKhovanov homology for links, coming from a complex of modules over the ground ring R := Z [ X, Y, Z ± ] / ( X = Y = 0). Specializing X = Y = Z = 1 recovers even Khovanovhomology, and X = Z = 1 and Y = − odd arc algebras are not associative, preventingthe construction of an odd invariant for tangles by simply mimicking [18]. Indeed, whileit is not hard to associate a chain complex of graded spaces to a tangle by considering allthe ways one can close it, and do the usual procedure of [17, 18, 21], the non-associativityprevents to have a gluing property. . In particular, it is not clear how to define the tensorproduct of modules over a non-associative algebra. Following an unpublished idea of thesecond author and Shumakovitch, it is explained in [12] how the odd arc algebras can bemade quasi-associative by grading them over some groupoid with an associator (i.e. a3-cocycle). This notion extends the definition of quasi-associative algebras of Albuquerqueand Majid [1]. Unfortunately, this quasi-associative structure is not enough to constructthe desired bimodules that are needed for the construction of a tangle invariant as in [18].Another issue is that the maps used to form the complex Kh( T ) do not preserve thegrading, thus requiring a notion of grading shift functor for quasi-associative bimodules.The present paper gives a solution to these problems, and we construct a covering versionof Khovanov homology for tangles. Moreover, we also connect our construction to an tangleodd invariant introduced by Vaz in [26]. Main results and structure of the paper.
We first recall in Section 2 the notion ofchronological cobordisms. We use them in Section 3 to give a covering version of the arcalgebras over R .In Section 4, we extend the definition of quasi-associative algebras and bimodules tostructures graded over a category with an associator. We also introduce the notion of a shifting system , which allows to construct grading shift functors for these objects. Thesecan be interpreted as a generalization of parity shift functors in the theory of supergradedstructures. We also explain how to construct ‘graded’ categories of bimodules, wherethe maps posses non-trivial degree, by enriching to notion of shifting-system to a shifting2-system . This can be seen as a generalization of the notion of supercategories of super-bimodules as in [2]. Finally, we develop basic notions of homological algebra for thesestructures.In Section 5, we construct a grading category G where the morphisms are given by apair of a flat tangle and an element in Z × Z . We show that the covering arc algebras arequasi-associative over G . Note that our grading category is different from the one in [12].We also construct a shifting-2-system over G , where the shifts are given by a pair of achronological cobordism together with a shift in the Z × Z -grading. All this allows us inSection 6 to mimic the construction of [18] in the quasi-associative context. We show ityields a tangle version of covering Khovanov homology, that agrees over a link with the onefrom [22], and respects the gluing property. Furthermore, by specializing the parameters,we get a tangle version of odd Khovanov homology from [21].In the last section, we relate our construction to Vaz tangle invariant [26]. This invariantis built using an odd version of a level 2 cyclotomic half 2-Kac–Moody algebra associated DD KHOVANOV HOMOLOGY FOR TANGLES 3 to gl n , in the same spirit as Khovanov–Lauda [19] and Rouquier [23], and odd categorifi-cation [10, 11, 14, 15]. We extend the 2-Kac–Moody of [26] to a convering version over R .We show it acts in a graded sense on the bicategory of quasi-associative bimodules over thecovering arcs algebras, giving an odd version of the 2-action constructed in [5] for the evencase. Furthermore, after applying this 2-functor, the complex associated to a tangle in [26]coincides with ours. We conjecture that this 2-functor is fully faithful, and therefore, thatVaz invariant is equivalent to our version of odd Khovanov homology. Acknowledgments.
The authors would like to thank Pedro Vaz for interesting discus-sions. The authors would also like to thank the Erwin Schr¨odinger International Institutefor Mathematics and Physics for their hospitality. G.N. is grateful to the Max Planck Insti-tute for Mathematics in Bonn for its hospitality and financial support. G.N. was a ResearchFellow of the Fonds de la Recherche Scientifique - FNRS, under Grant no. 1.A310.16 whilestarting working on this project.2.
Chronological cobordims
We recall the notion of chronological cobordisms and the 2-categories they form, follow-ing [22].2.1.
Chronological cobordims.
Let Σ and Σ be two compact oriented 1-manifoldsembedded in R . An (embedded) cobordism from Σ to Σ is a compact surface W ⊂ R × I such that ∂W ∼ = Σ ⊔ − Σ , where − Σ means we take Σ with reverse orientation.Given a compact manifold W , a generic function f : W → I has only Morse criticalpoints. A generic homotopy f t of Morse functions can have degenerate singularities of thesimplest form, modeled by a cubic polynomial: q ( x , . . . , x n ) = q (0) − x − · · · − x k + x k +1 + · · · + x n − + x n . Such a point p is called an A -singularity . It can be characterized by requiring Hess p ( f )to have a 1-dimensional kernel, on wich d f p does not vanish. A function that has onlyMorse and A -singularities is called an Igusa function [13].Choose a Riemannian metric on W , so that we can consider the Hessian of f as a linearfunction Hess p ( f ) : T p W → T p W . Because it is symmetric, the tangent space T p W ata critical point p decomposes into T p W = E + p ⊕ E − p ⊕ N p , where E ± p is the positive ornegative eigenspace, and N p is the nullspace of Hess p ( f ). A framing on f : W → I is achoice of an orthonormal bases of the negative eigenspace E − p , extended by a vector v ∈ N p for which d f p ( v, v, v ) >
0, if p is degenerate. Definition 2.1 (Cf. [22]) . An (embedded) chronological cobordism ( W, τ ) is a surface W ⊂ R × I with a framing τ on the restriction π | S of the canonical projection π : R × I → I ,such that • W is transverse to R × ∂I ; • ∂W ⊂ R × ∂I (it is a cobordism from ∂W ∩ R × { } to ∂W ∩ R × { } ); • π | W is a Morse function with at most one critical point on each level. GR´EGOIRE NAISSE AND KRZYSZTOF PUTYRA A change of chronology is a diffeotopy H t of R × I , such that each π H t ( W ) is a generichomotopy of Morse functions, together with a smooth family of framings on π | H t ( W ) .We say that two embedded chronological cobordisms are equivalent if they can be relatedby a diffeotopy H t for which π | H t ( W ) is separative Morse at every moment t ∈ I . This is forexample the case when π ◦ H t = π for all t ∈ I , that is when we deform the cobordism onlyalong the horizontal plane. We refer to such diffeotopies as horizontal . Another exampleis given by vertical diffeotopies, that is when H t ( p, z ) = ( p, h t ( z )) for some diffeotopy h t on I . This means we deform uniformly the cobordism along the vertical axis.Two changes of chronology between equivalent cobordisms are equivalent if, after com-posing with the equivalences of cobordisms, they are homotopic in the space of orientedIgusa functions.2.1.1. Locally vertical changes of chronology.
A important family of changes of chronologyis given by the locally vertical changes of chronology. This means they are vertical insidesome cylinders fixed in R × I . Definition 2.2 (Cf. [22]) . Let W be a chronological cobordism. Choose a family ofdisjoint disks D , . . . , D R in R , such that W intersects each ∂C i in vertical lines, where C i := D i × I . We say that a change of chronology H t on W is locally vertical with respectto the cylinders C i if it is vertical inside each C i , but fixes all points outside them, exceptsmall annular neighborhoods of ∂C i , in which we interpolate the two behaviors.The particularity of locally vertical change of chronologies is that they are unique up tohomotopy. Proposition 2.3 ([22, Proposition 4.4]) . Let H t and H ′ t be two locally vertical changes ofchronology with respect to the same cylinders. If H = H ′ , then they are homotopic in thespace of framed diffeotopies. The 2-category ChCob.
From now on, except if specified otherwise, all cobordismsare embedded chronological cobordisms.Given a cobordism ( W ′ , τ ′ ) from B to C and a cobordism ( W, τ ) from A to B , one cancompose them by gluing the cobordisms together along B (adding a collar if necessary,see [22][ § H ′ on ( W ′ , τ ′ ) and achange of chronology H on ( W, τ ), we can compose them horizontally , giving a change ofchronology H ′ ◦ H on ( W ′ ◦ W, τ ′ ◦ τ ).Let ( W , τ ) and ( W , τ ) be cobordisms with a change of chronology H : ( W , τ ) ⇒ ( W , τ ) and let and ( W ′ , τ ′ ) be a cobordism equivalent to H ( W , τ ). Let also H ′ :( W ′ , τ ′ ) ⇒ ( W , τ ) be a change of chronology on ( W ′ , τ ′ ). Then, we can vertically compose H ′ with H , giving a change of chronology H ′ ⋆ H : ( W , τ ) ⇒ ( W , τ ) as the composition( W , τ ) H −→ ( W , τ ) ≃ −→ ( W , τ ′ ) H ′ −→ ( W , τ ).We usually draw cobordisms as diagrams read from bottom to top, and we draw thechosen framing as an arrow written next to the critical point. Composition on the left istranslated by gluing from above. DD KHOVANOV HOMOLOGY FOR TANGLES 5
Definition 2.4.
Let
ChCob be the (strict) 2-category consisting of • objects are finite disjoint collections of circles in R ; • ChCob ( A, B ) are classes of embedded chronological cobordismsfrom A to B , up to equivalence; • composition of 1-morphism is given by gluing cobordisms; • • vertical and horizontal compositions of 2-morphisms are given by ⋆ and ◦ definedabove, respectively.The 2-category ChCob becomes Gray monoidal (that is a monoidal 2-category, see [22,Appendix B.2]) when equipped with the horizontal monoidal product of cobordims givenby right-then-left juxtaposition of cobordism, pushing all critical points of the cobordismon the left to the top: . . .. . . W ′ ⊗ . . .. . . W := . . .. . . W ′ . . .. . . W The unit is given by the empty cobordism. Given two changes of chronology H : W ⇒ W and H ′ : W ′ ⇒ W ′ , we obtain a change of chronology H ′ ⊗ H : W ′ ⊗ W ⇒ W ′ ⊗ W bysetting ( H ′ ⊗ H ) t ( x ) := H ′ t ( x ) , if x ∈ W ,H t ( x ) , if x ∈ W ′ ,x, otherwise , where the otherwise can occur because of the identity corbodisms we added below W ′ andabove W .The 1-morphisms in ChCob (as Gray monoidal category) are generated by the fiveelementary cobordisms:merge split birth positive death negative deathwith a twist acting as a strict symmetry (see [22, Definition B.9]).
Definition 2.5.
The Z × Z -degree of a cobordism W isdeg( W ) := | W | := ( − , − . Linearized R ChCob.
Let R := Z [ X, Y, Z ± ] / ( X = Y = 1). To each elementarychange of chronology, we assign a monomial in R as given by Fig. 1. Any change ofchronology can be decomposed as a composition of such elementary changes. Let ı be themap that associate to a change of chronology the resulting monomial after decomposing GR´EGOIRE NAISSE AND KRZYSZTOF PUTYRA
X YZ XY Figure 1.
Diagrams for elementary changes of chronologies, grouped byvalues of the function ı . Thin lines are the input circles and thick arrowsvisualize saddle points. Orientations of the arrows are omitted if ı does notdepend on them. For the Z it means that we go from doing the merge thenthe split to the split then the merge. The other direction gives Z − .it and applying the above mentioned rule. From [22], we know it is well-defined andmultiplicative in the sense that ı ( H ′ ◦ H ) = ı ( H ′ ) ı ( H ) and ı ( H ′ ⋆ H ) = ı ( H ′ ) ı ( H ). Definition 2.6.
The linearized category of cobordims R ChCob is the R -linear monoidalcategory consisting of • objects are the same as in ChCob ; • morphisms are R -linear combinations of morphisms in ChCob modulo the relation W ′ = ı ( H ) W for each change of chronology H : W → W ′ ; • the monoidal product is given by right-then-left juxtaposition of cobordisms.Let λ R : ( Z × Z ) → R be the bilinear map given by(1) λ R (cid:0) ( a ′ , b ′ ) , ( a, b ) (cid:1) := X a ′ a Y b ′ b Z a ′ b − b ′ a , where bilinear means λ R ( x, y + z ) = λ R ( x, y ) λ R ( z ) and λ R ( x + y, z ) = λ R ( x, z ) λ R ( y, z ). DD KHOVANOV HOMOLOGY FOR TANGLES 7
It appears that R ChCob admits a nice combinatorial description, with morphisms being R -linear combinations of cobordims modulo the following local relations:= X = Y = Y (2) = = =(3)and . . .. . . . . .. . . W ′ W = λ R ( | W | , | W ′ | ) . . .. . . . . .. . . W ′ W (4) . . .. . . W ′ . . .. . . W = λ R ( | W | , | W ′ | ) . . .. . . W ′ . . .. . . W (5)for all cobordisms W ′ and W .Note that in particular, two diffeomorphic cobordisms (with different chronology) areequal in Hom R ChCob up to a scalar in R . However, this scalar is in general not uniquesince, for example: = XY Cobordisms with corners.
We define a chronological cobordism with corners in thesame fashion as a chronological cobordism except that: • W is contained in I × R × I ; • ∂W ⊂ I × R × ∂I ∪ ∂I × R × I ; • ∂W ∩ ∂I × R × I = ⊔ i { p i × I } for a finite collection of disjoint p i ∈ ∂I × R × { } ; • W is transverse to I × R × ∂I and to ∂I × R × I ;Similarly, we ask for a change of chronology H t to restrict on ∂I ∩ R ∩ I as H t ( x, y, z ) =( x, h t ( y ) , h ′ t ( z )) for some diffeotopies h t , h ′ t on I . All this means the cobordism can haveborders, but these are vertical, and a change of chronology preserves that.We say that two cobordisms with corners W ′ and W are horizontally composable if { p i ∈ { } × I × { } ∩ W ′ } = { p i ∈ { } × I × { } ∩ W } . GR´EGOIRE NAISSE AND KRZYSZTOF PUTYRA
This means that W ′ has the same number of boundaries on the right than W on the left.Thus, in that situation we can horizontally compose W ′ with W by doing a right-then-leftcomposition W ′ • W , adding small curtains to smooth the composition if necessary, andrescaling. For example: • = Remark 2.7.
Note that the Z × Z -grading of cobordisms does not extend to cobordismswith corners. 3. Arc algebras and bimodules
We construct the non-associative covering arc algebras H n over R , as well as non-associative bimodules associated to flat tangles, using the procedure from [18] and thechronological TQFT from [22].3.1. Flat tangles.
Recall that a flat tangle is a tangle diagram with no crossing. Let B mn be the set of classes of flat tangles from 2 n fixed points on the horizontal line R ×{ } ⊂ R to2 m fixed points on R × { } ⊂ R , taken up to ambient isotopy that fixes the endpoints. Let¯ : B nm → B mn be the map that takes the mirror image along the horizontal line R × { / } .When m = 0, we simply write B n , and we refer to elements in B n as crossingless matchings .We put B • := ⊔ n ≥ B n , and we write | a | = n whenever a ∈ B n . There is a compositionmap B pn × B nm → B pm , ( t ′ , t ) t ′ t, where t ′ t is given by gluing t ′ on top of t , and rescaling. Let 1 n ∈ B nn be the identity tangleon 2 n strands: 1 n := . . . n It is a neutral element for the composition map.3.2.
Surgery.
For a ∈ B n , b ∈ B m and t ∈ B mn , we have that ¯ bta is a closed 1-manifold.Let 1 ba ( t ) : ¯ bta → ¯ bta be the identity cobordism given by ¯ bta × [0 , t for theidentity cobordism with corners t × [0 , t ′ ∈ B | c || b | and t ∈ B | b || a | ,there is a unique (up to homeomorphism) minimal cobordism ¯ ct ′ b ¯ bta → ¯ ct ′ ta given bysurgery over b , that is by contracting the symmetric arcs of ¯ bb using saddles. For eachsuch cobordism, we choose the chronology given by adding the saddle points from right toleft and orienting everything upward. We write W cba ( t ′ , t ) : ¯ ct ′ b ¯ bta → ¯ ct ′ ta the resultingchronological cobordism. Note that W cba ( t ′ , t ) has Euler characteristic −| b | . DD KHOVANOV HOMOLOGY FOR TANGLES 9
Example 3.1.
Take a = c = b =in B . Then, W cba (1 , ) is given by the movie:3.3. Chronological TQFT.
Let Mod R be the category of Z × Z -graded R -modules withmaps that preserve the degree. Recall the bilinear map λ R : ( Z × Z ) → R given by λ R (cid:0) ( a ′ , b ′ ) , ( a, b ) (cid:1) := X a ′ a Y b ′ b Z a ′ b − b ′ a , and note that λ − R ( x, y ) = λ R ( y, x ). The category Mod R is equipped with the usual notionof tensor product, but comes with a symmetry τ M,N : M ⊗ N → N ⊗ M given by τ M,N ( m ⊗ n ) := λ R (deg m, deg n ) n ⊗ m. We picture this formula as: m n = mn = λ R ( | m | , | n | ) mn Let A := Rv + ⊕ Rv − be the free R -module generated by two elements v + and v − withdegrees deg R ( v + ) := (1 , , deg R ( v − ) := (0 , − . Then, we consider the chronological TQFT F : R ChCob → Mod R from [22] mapping themonoidal product of R ChCob to the tensor product of Mod R , the twist cobordism to thesymmetry τ , a single circle to A , and F : A ⊗ A → A := ( v + ⊗ v + v + , v + ⊗ v − v − ,v − ⊗ v − , v − ⊗ v + XZv − , F : A → A ⊗ A := ( v + v − ⊗ v + + Y Zv + ⊗ v − ,v − v − ⊗ v − , F (cid:18) (cid:19) : R → A := n v + , F ! : A → R := ( v + ,v − . Dotted cobordisms.
There is also a notion of dotted cobordisms (see [22, § − , −
1) and move freely on the surface at the costof multiplying by scalars in R whenever we exchange dots with critical points, respect-ing Eq. (4) and Eq. (5). Let R ChCob • be the 2-category defined similarly as R ChCob but where cobordisms are dotted, modulo the following dot relations := 0 , = 1 , = + , = 0 , (6)and extending Eq. (4) and Eq. (5) to dots.The functor F extends to a functor F • : R ChCob • → Mod R with F • : A → A := ( v + v − ,v − . Arc spaces and arc algebras.
Consider a flat tangle t ∈ B mn . Define the space F ( t ) := M a ∈ B n ,b ∈ B m b F ( t ) a , b F ( t ) a := F (¯ bta ) . Given another flat tangle t ′ ∈ B km , define the composition map µ [ t ′ , t ] as d F ( t ′ ) c ⊗ R b F ( t ) a µ [ t ′ ,t ] −−−→ , whenever c = b , and as making the following diagram commutes otherwise: c F ( t ′ ) b ⊗ R b F ( t ) a c F ( t ′ t ) a F (¯ ct ′ b ) ⊗ R F (¯ bta ) F (¯ ct ′ b ¯ bta ) F (¯ ct ′ ta ) µ cba [ t ′ ,t ] ≃ ≃ F ( W cba ( t ′ ,t )) ≃ for c ∈ B k , b ∈ B m and a ∈ B n . Remark 3.2.
Note that the composition map µ [ t ′ , t ] does not preserve the Z × Z -grading.This will be fixed in Section 6 using the tools from Section 4. Definition 3.3.
The arc algebra H n is given by H n := M a,b ∈ B n b F (1 n ) a , with multiplication µ [1 n , n ]. Example 3.4.
As a simple example, we can take n = 1. Then, H ∼ = A with µ ( v + , v + ) = v + , µ ( v + , v − ) = v − , µ ( v − , v + ) = XZv − and µ ( v − , v − ) = 0. DD KHOVANOV HOMOLOGY FOR TANGLES 11
Remark 3.5.
Note that H n is not associative in the proper sense (see [12] for an examplewith n = 2) and also not unital. One of the main goal of Section 4 will be to constructa framework such that it becomes associative and unital as an algebra object in the rightmonoidal category. 4. C -graded structures Let C be a small category and k a unital commutative ring. We write C [ n ] := { X f ←− X f ←− · · · f n ←− X n } for the set of paths of length n in C . Definition 4.1. A grading category is pair ( C , α ) where α : C [3] → k × is a 3-cocycle,meaning it respects dα ( g, h, k, l ) := α ( h, k, l ) α ( gh, k, l ) − α ( g, hk, l ) α ( g, h, kl ) − α ( g, h, k ) = 1 , for all g ◦ h ◦ k ◦ l ∈ C [4] . We call α the associator .We assume for the rest of the section that we have fixed a grading category ( C , α ), anda unital commutative ring k . Definition 4.2. A C -graded k -module is a k -module M with a decomposition M ∼ = L g ∈C M g . Given x ∈ M g , we write | x | := g . A graded map f : M → N between two C -graded modules M, N is a k -linear map such that f ( M g ) ⊂ N g for all g ∈ C .We define the category Mod C of C -graded k -modules with objects being C -graded k -modules and morphisms being graded maps. It is a monoidal category with M ′ ⊗ M defined as M ′ ⊗ M := M g ∈C ( M ′ ⊗ M ) g , ( M ′ ⊗ M ) g := M g = g ◦ g M ′ g ⊗ k M g . The coherence isomorphism α : ( M ′′ ⊗ M ′ ) ⊗ M ≃ −→ M ′′ ⊗ ( M ′ ⊗ M ) is given by( z ⊗ y ) ⊗ x α ( | z | , | y | , | x | ) z ⊗ ( y ⊗ x ) , for homogeneous elements x ∈ M ′′| x | , y ∈ M ′| y | , z ∈ M | z | . The unit object is I C := M X ∈ Obj ( C ) ( k ) Id X , with left unitor λ : I C ⊗ M ∼ = M given by( k ⊗ m ) λ ( | k | , | m | ) km, λ ( | k | , | m | ) := α (Id Y , Id Y , | m | ) − , for m ∈ M g , k ∈ ( k ) Id Y and g : X → Y . The right unitor ρ is constructed similarly with( m ⊗ k ′ ) ρ ( | m | , | k ′ | ) mk ′ , ρ ( | m | , | k ′ | ) := α ( | m | , Id X , Id X ) , for k ′ ∈ ( k ) Id X , so that(7) λ (Id Y , x ) = ρ ( z, Id Y ) α ( z, Id Y , x ) . Remark 4.3.
Note that the definition of a grading category can be made more general byallowing arbitrary unitors respecting Eq. (7), and everything written below generalizes. C -graded algebras. Having a monoidal category at thand, constructing classicalalgebraic structures becomes straightforward.
Definition 4.4. A C -graded algebra is a monoid object in Mod C .Explicitly, it is C -graded k -module A ∼ = L g ∈C A g with a k -linear multiplication map µ A : A ⊗ A → A, usually written as yx := µ A ( y, x ) for x, y ∈ A , and a unit element 1 X ∈ A Id X for each X ∈ Obj ( C ), respecting: • µ A is graded µ A ( A g ′ , A g ) ⊂ A g ′ ◦ g ; • µ A is associative ( zy ) x = α ( | z | , | y | , | x | ) z ( yx ); • Y x = λ (Id Y , | x | ) x and x X = ρ ( | x | , Id X ) x for all x ∈ A | x | : X → Y .In particular, I C is a C -graded algebra with multiplication induced by k and xy = 0whenever | x | 6 = | y | . Definition 4.5.
Let A , A be two C -graded algebras. An A - A -bimodule is a bimoduleobject over the pair ( A , A ) in Mod C .Explicitly, it is a C -graded k -module M ∼ = L g ∈C M g with graded k -linear left and rightaction maps ρ L : A ⊗ M → M, ρ R : M ⊗ A → M, written as y · m := ρ L ( y, m ) and m · x := ρ R ( m, x ) for x ∈ A , y ∈ A and m ∈ M ,respecting: • ( y ′ y ) · m = α ( | y ′ | , | y | , | m | ) y ′ · ( y · m ); • ( m · x ′ ) · x = α ( | m | , | x ′ | , | x | ) m · ( x ′ x ); • ( y · m ) · x = α ( | y | , | m | , | x | ) y · ( m · x ); • Y · m = λ (Id Y , | m | ) m and m · X = ρ ( | m | , Id X ) m for all m ∈ M | m | : X → Y ,for all y ′ , y ∈ A , x ′ , x ∈ A and m ∈ M . We obtain the notion of a left (resp. right) A -module as a A - I C -bimodule (resp. I C - A -bimodule). A graded map of A - A -bimodules f : M → M ′ is a graded k -linear map preserving the actions f ( y · m ) = y · f ( m ) and f ( m · x ) = f ( m ) · x for all x ∈ A , m ∈ M and y ∈ A . We write Bimod C ( A , A ) for thecategory of A - A -bimodules with graded maps. For the particular case of A - I C -modules(i.e. left A -modules) we write it as Mod C ( A ).Take M ′ ∈ Bimod C ( A , A ) and M ∈ Bimod C ( A , A ). We first observe that M ′ ⊗ M has the structure of a ( A , A )-bimodule with left and right actions defined by making thefollowing diagrams commute: A ⊗ ( M ′ ⊗ M ) M ′ ⊗ M ( A ⊗ M ′ ) ⊗ M α − ρ L ⊗ ( M ′ ⊗ M ) ⊗ A M ′ ⊗ MM ′ ⊗ ( M ⊗ A ) α ⊗ ρ R DD KHOVANOV HOMOLOGY FOR TANGLES 13 that is y · ( m ′ ⊗ m ) := α ( | y | , | m ′ | , | m | ) − ( y · m ′ ) ⊗ m, ( m ′ ⊗ m ) · x := α ( | m ′ | , | m | , | y | ) m ′ ⊗ ( m · x ) , for all homogeneous x ∈ A , y ∈ A , m ∈ M and m ′ ∈ M ′ . Definition 4.6.
The tensor product of M ′ with M over A is the coequalizer( M ′ ⊗ A ) ⊗ M M ′ ⊗ M M ′ ⊗ A M,M ′ ⊗ ( A ⊗ M ) ≃ α ρ R ⊗ ⊗ ρ L in Mod C .Explicitly, we have M ′ ⊗ A M := M ′ ⊗ M/ (cid:0) ( m ′ · x ) ⊗ m − α ( | m ′ | , | x | , | m | ) m ′ ⊗ ( x · m ) (cid:1) . It is an A - A -bimodule with left and right actions induced by the ones on M ′ ⊗ M .Given a map of A - A -bimodules f : M → N and another one of A - A -bimodules f ′ : M ′ → N ′ we obtain an induced map ( f ′ ⊗ f ) : ( M ′ ⊗ A M ) → ( N ′ ⊗ A N ) given by( f ′ ⊗ f )( m ′ ⊗ m ) := f ′ ( m ′ ) ⊗ f ( m ).For M ∈ Bimod C ( A , A ) we construct a functor M ⊗ A − : Mod C ( A ) → Mod C ( A ) , N M ⊗ A N. Putting all these together, we obtain a bicategory B imod C with objects being C -gradedalgebras and 1-hom spaces Hom( A , A ) := Bimod C ( A , A ). The horizontal compositionHom( A , A ) × Hom( A , A ) is given by tensor product over A . Example 4.7.
As an easy example we can consider the category Z with one object ⋆ andHom Z ( ⋆, ⋆ ) := Z , with α = 1. Then, a Z -graded object is the same thing as a Z -gradedone.4.2. C -grading shift. A C -grading shift ϕ is a collection of maps ϕ = { ϕ Y X : D Y X ⊂ Hom C ( X, Y ) → Hom C ( X, Y ) } X,Y ∈ Obj ( C ) . We write ϕ ( g ) := ϕ Y X ( g ) whenever g ∈ D Y X . Definition 4.8.
Let Σ( C ) be the wide subcategory of C given byHom Σ( C ) ( X, Y ) := ( ∅ , whenever X = Y , Z (cid:0) End C ( X ) (cid:1) , otherwise,where Z denotes the center. Definition 4.9. A C -shifting system S = ( I, Φ) is the datum of • a monoid I with composition • : I × I → I and neutral element e ∈ I ; • a collection of C -grading shifts Φ = { ϕ i } i ∈ I ,respecting: • Φ contains the neutral shift ϕ e with D Y X e := Hom Σ( C ) ( X, Y ) and ϕ Y X e := Id D Y X e ; • for each pair ϕ ZYj and ϕ Y Xi we have D ZYj ◦ D Y Xi ⊂ D ZXj • i and D ZYj × D Y Xi D ZXj • i Hom C ( Y, Z ) × Hom C ( X, Y ) Hom C ( X, Z ) −◦− ( ϕ ZYj ,ϕ Y Xi ) ϕ ZXj • i −◦− commutes; • there is a subset I Id ⊂ I such that for each pair X, Y ∈ Obj ( C ) there is a partitionof Hom C ( X, Y ) = ⊔ i ∈ I Id D XYi , and ϕ i = Id D Y Xi for all i ∈ I Id ; • if I contains an absorbing element 0 ∈ I , then ϕ is a null shift with D XY = ∅ . Remark 4.10.
Note that the neutral shift ϕ e preserves only Σ( C ). In general, we cannothope to ask for it to preserve the whole hom-spaces. Indeed, suppose we have a non trivialshift ϕ i : Id X x where x = Id X ∈ End( X ), and consider an element y ∈ End( X )such that x ◦ y = y ◦ x . Then, we would have ϕ e • i ( y ◦ Id X ) = ϕ e ( y ) ◦ ϕ i (Id X ) = y ◦ x and ϕ i • e (Id X ◦ y ) = ϕ i (Id X ) ◦ ϕ e ( y ) = x ◦ y , but we also want ϕ e • i ( y ◦ Id X ) = ϕ i ( y ) = ϕ e • i (Id X ◦ y ),which is a contradiction. Since y / ∈ Z (End( X )), we actually have ϕ e ( y ) = 0.Let β = { β ZY Xj,i } be a collection of maps β ZY Xj,i : D ZYj × D Y Xi → k × , for all X, Y, Z ∈ Obj ( C ) and i, j ∈ I , such that β ZY X e , e = 1. Definition 4.11.
We say that a C -shifting system S is compatible with α through β if α ( g ′′ , g ′ , g ) β ZXWk • j,i ( g ′′ g ′ , g ) β ZY Xk,j ( g ′′ , g ′ )= β ZY Wk,j • i ( g ′′ , g ′ g ) β Y XWj,i ( g ′ , g ) α ( ϕ ZYk ( g ′′ ) , ϕ Y Xj ( g ′ ) , ϕ XWi ( g )) , (8)for all Z g ′′ ←− Y g ′ ←− X g ←− W ∈ D ZYk ◦ D Y Xj ◦ D XWi and i, j, k ∈ I . In that situation,we refer to β ZY Xj,i as compatibility maps . Also, we write β j,i ( g ′ , g ) := β XY Zj,i ( g ′ , g ) for all Z g ′ ←− Y g ←− X ∈ D ZYj ◦ D Y Xi .Let S = ( I, { ϕ i } i ∈ I ) be a C -shifting system compatible with α . For each i ∈ I we definethe grading shift functor ϕ i : Mod C → Mod C as the identity on morphisms, and for each C -graded k -module M we put ϕ i ( M ) := L g ∈ D i ϕ i ( M ) ϕ i ( g ) where ϕ i ( M ) ϕ i ( g ) := M g . Inother words, the grading shift functor sends elements in degree g ∈ D i to elements indegree ϕ i ( g ), and elements in degree not in D i to zero. The identity shift functor is givenby ϕ Id := L i ∈ I Id ϕ i . Note that ϕ Id ( M ) ∼ = M . Remark 4.12.
As already pointed out in Remark 4.10, in general we have ϕ Id = ϕ e . DD KHOVANOV HOMOLOGY FOR TANGLES 15
For
M, M ′ ∈ Mod C there is a canonical isomorphism β j,i : ϕ j ( M ′ ) ⊗ ϕ i ( M ) ≃ −→ ϕ j • i ( M ′ ⊗ M ) , m ′ ⊗ m β j,i ( | m ′ | , | m | ) m ′ ⊗ m, for all homogeneous m ∈ M, m ′ ∈ M ′ . These isomorphisms are compatible with α in thesense that the diagram(9) ( ϕ k M ′′ ⊗ ϕ j M ′ ) ⊗ ϕ i ( M ) ϕ k ( M ′′ ) ⊗ ( ϕ j M ′ ⊗ ϕ i M ) ϕ k • j ( M ′′ ⊗ M ′ ) ⊗ ϕ i ( M ) ϕ k ( M ′′ ) ⊗ ϕ j • i ( M ′ ⊗ M ) ϕ k • j • i (( M ′′ ⊗ M ′ ) ⊗ M ) ϕ k • j • i ( M ′′ ⊗ ( M ′ ⊗ M )) , αβ k,j ⊗ ⊗ β j,i β k • j,i β k,j • i ϕ k • j • i α commutes for all i, j, k ∈ I and M, M ′ , M ′′ ∈ Mod C , thanks to Eq. (8). Remark 4.13.
For the sake of simplicity, we will often use e I := I ⊔ { Id } . Then, when weuse a compatibility map β Id ,i , we mean β Id ,i ( g ′ , g ) := β j,i ( g ′ , g ) , for g ′ ∈ D j and j ∈ I Id . In other words, β Id ,i = P j ∈ I Id β j,i where we interpret β j,i ( g ′ , g ) = 0whenever g ′ / ∈ D j . We will also write ϕ Id • i := L j ∈ I Id ϕ j • i . The same applies for β j, Id . Wealso put β Id , Id := 1. Proposition 4.14.
The compatibility map β j,i defines a natural isomorphism β j,i : ϕ j ( − ) ⊗ ϕ i ( − ) ≃ −→ ϕ j • i ( − ⊗ − ) of bifunctors for all i, j ∈ e I .Proof. We need to check that ϕ j ( M ′ ) ⊗ ϕ i ( M ) ϕ j • i ( M ′ ⊗ M ) ϕ j ( N ′ ) ⊗ ϕ i ( N ) ϕ j • i ( N ′ ⊗ N ) β j,i f ′ ⊗ f f ′ ⊗ fβ j,i commutes for all f : M → N and f ′ : M ′ → N ′ . This is immediate since the grading shiftfunctors do not modify the morphisms, and β j,i depends only on the degree, and f and f ′ preserve the degrees. (cid:3) Shifting bimodules.
Consider a C -shifting system ( S = ( I, { ϕ i } i ∈ I ) , β , ) compatiblewith α . We assume that all C -graded algebras A are supported on Σ( C ) in the sense that A g = 0 whenever g / ∈ Hom Σ( C ) . Thus, we have ϕ e ( A ) ∼ = A . Definition 4.15.
For M ∈ Bimod C ( A , A ) we define the shifted bimodule ϕ i ( M ) as the C -graded k -module M shifted by ϕ i with left and right actions given by making the diagrams A ⊗ ϕ i ( M ) ϕ i ( M ) ϕ e ( A ) ⊗ ϕ i ( M ) ϕ e • i ( A ⊗ M ) ϕ e • i ( M ) ≃ ϕ i ρ L β e ,i ρ L ϕ i ( M ) ⊗ A ϕ i ( M ) ϕ i ( M ) ⊗ ϕ e ( A ) ϕ i • e ( M ⊗ A ) ϕ i • e ( M ) ≃ ϕ i ρ R β i, e ρ R commute.Explicitly, it gives us y · ϕ i ( m ) := β e ,i ( | y | , | m | ) ϕ i ( y · m ) ,ϕ i ( m ) · x := β i, e ( | m | , | x | ) ϕ i ( m · x ) , for all x ∈ A , y ∈ A and m ∈ M . Remark 4.16.
Note that we need to have A i supported in Σ( C )-degree, otherwise wecannot identify A i with ϕ e ( A i ). Proposition 4.17.
The shifted bimodule ϕ i ( M ) is an A - A -bimodule.Proof. It follows immediately from Eq. (8) together with the fact that A and A aresupported in Σ( C )-degree, and β e , e = 1. (cid:3) Therefore, we obtain a shifting endofunctor ϕ i : Bimod C ( A , A ) → Bimod C ( A , A ) , for each i ∈ I induced by ϕ i : Mod C → Mod C . Proposition 4.18.
Let M ′ ∈ Bimod C ( A , A ) and M ∈ Bimod C ( A , A ) . Then, there isan isomorphism of A - A -bimodules β j,i : ϕ j ( M ′ ) ⊗ A ϕ i ( M ) ≃ −→ ϕ j • i ( M ′ ⊗ A M ) , induced by the canonical isomorphism β j,i : ϕ j ( M ′ ) ⊗ ϕ i ( M ) ≃ −→ ϕ j • i ( M ′ ⊗ M ) . DD KHOVANOV HOMOLOGY FOR TANGLES 17
Proof.
We first show that β j,i induces an isomorphism as k -modules. Consider the followingdiagram: (cid:0) ϕ j ( M ′ ) ⊗ A (cid:1) ⊗ ϕ i ( M ) ϕ j ( M ′ ) ⊗ ϕ i ( M ) ϕ j ( M ′ ) ⊗ A ϕ i ( M ) ϕ j ( M ′ ) ⊗ (cid:0) A ⊗ ϕ i ( M ) (cid:1) ϕ j • e • i (cid:0) ( M ′ ⊗ A ) ⊗ M (cid:1) ϕ j • i ( M ′ ⊗ M ) ϕ j • i ( M ′ ⊗ A M ) ϕ j • e • i (cid:0) M ′ ⊗ ( A ⊗ M ) (cid:1) ϕ j ρ R ⊗ β j • e ,i ◦ ( β j, e ⊗ α ≃ β j,i ≃ ⊗ ϕ i ρ L β j, e • i ◦ (1 ⊗ β e ,i ) ρ R ⊗ α ⊗ ρ L The left square is equivalent to Eq. (9) and thus commutes. The other two squares commuteby definition of ϕ i ρ L and ϕ j ρ R . Since β j,i is an isomorphism, we obtain by universalproperty of the coequalizer an induced isomorphism.We show that the induced map β j,i preserves the left action, the proof for the right actionbeing similar. Consider the following diagram: (cid:0) A ⊗ ϕ j ( M ′ ) (cid:1) ⊗ ϕ i ( M ) ϕ e • j ( A ⊗ M ) ⊗ ϕ i ( M ) A ⊗ (cid:0) ϕ j ( M ′ ) ⊗ ϕ i ( M ) (cid:1) ϕ j ( M ′ ) ⊗ ϕ i ( M ) A ⊗ ϕ j • i ( M ′ ⊗ M ) ϕ j • i ( M ′ ⊗ M ) ϕ e • j • i (cid:0) A ⊗ ( M ′ ⊗ M ) (cid:1) ϕ e • j • i (cid:0) ( A ⊗ M ′ ) ⊗ M (cid:1) β e ,j ρ L ⊗ α − ⊗ β j,i ρ L β j,i ρ L β e ,j • i α − ρ L ⊗ where the upper and lower parts commute by definition of the left action on a tensor productof bimodules. Furthermore, the outer part of the diagram commutes thanks to Eq. (9).Thus, we conclude the inner part commutes, and β j,i preserves the left action. (cid:3) Example 4.19.
As in Example 4.7, we can consider Z . In this case, we can take I := Z (for the additive structure), ϕ n : Hom Z ( ⋆, ⋆ ) → Hom Z ( ⋆, ⋆ ) , m m + n for all n ∈ Z , and β n,m := 1 for all m, n ∈ Z . It matches trivially with the usual notion of Z -grading shift. Example 4.20.
A more interesting example is given by looking at the category Z with asingle object ⋆ and Hom Z ( ⋆, ⋆ ) := Z / Z , and α = 1. Then, we take I := { e , π } := Z / Z ,with ϕ e := Id and ϕ π ( m ) := m + 1 mod 2. We also put β e ,π ( x, y ) := ( − x , β π, e ( x, y ) := 1,and β π,π ( x, y ) := ( − x . Then, it corresponds with the classical notion of super-gradedstructures, with ϕ π being the parity shift. The sign appearing in the compatibility map β e ,π is the same as in the usual isomorphism M ′ ⊗ Π M ≃ −→ Π (cid:0) M ′ ⊗ M ′ ) , where Π is the parity shift functor, see for example [2]. Example 4.21.
Another similar example is given by the category Z with one object ⋆ and Hom Z ( ⋆, ⋆ ) := Z × Z , with α = 1. We take I := Z × Z , with ϕ ( n ,n ) ( x , x ) :=( x + n , x + n ), and β ( n ,n ) , ( m ,m ) (( y , y ) , ( x , x )) := λ R (( y , y ) , ( m , m )), where λ R is the bilinear map defined in Eq. (1). This constructs shifting functors on Mod R . Example 4.22.
Both Example 4.20 and Example 4.21 can be generalized as follow. Con-sider a monoid G and put C G to be the category with a single object ⋆ and Hom C G ( ⋆, ⋆ ) := G , and put α = 1. Consider a map λ G : G × G → k such that λ G ( x, a ) λ G ( xy, b ) = λ G ( y, b ) λ G ( x, ab ) for all x, y, a, b ∈ G (this happens whenever λ G is bilinear ; if we alsoask that λ G (1 ,
1) = 1, then the two conditions are equivalent). Then, put I := Z ( G ) (thecenter of G ), ϕ a ( x ) := ax , and β b,a ( y, x ) := λ G ( y, a ). Remark 4.23.
In Section 5, we will construct a more sophisticated grading category, witha non-trivial associator, and where I is very different from the hom-spaces in the gradingcategory.4.4. C -graded dg-modules. There is a natural notion of C -graded dg-modules (differen-tially graded modules) that is almost identitical to the usual notion (see for example [16]).Most results about dg-modules holds for C -graded dg-modules. Definition 4.24. A C -graded dg-bimodule ( M, d M ) over a pair of C -graded k -algebras A and A is a Z × C -graded A - A -bimodule M = L n ∈ Z ,g ∈C M ng , where we call the Z -grading homological , together with a differential d M such that: • d M ( M ng ) ⊂ M n +1 g ; • d M ( y · m ) = y · d M ( m ); • d M ( m · x ) = d M ( m ) · x ; • d M ◦ d M = 0,for all y ∈ A , m ∈ M and x ∈ A .A map of C -graded dg-bimodules preserves both the C -grading and the homological grading,and commutes with the differentials. DD KHOVANOV HOMOLOGY FOR TANGLES 19
We write Bimod C dg ( A , A ) for the abelian category of C -graded A - A -dg-bimodules, andMod C dg ( A ) for the abelian category of C -graded A - I C -dg-bimodules. We also write | m | C :=deg C ( m ) := g and | m | h := deg h ( m ) := n for respectively the C -degree and homologicaldegree of m ∈ M ng .The tensor product over A of a dg-bimodule ( M ′ , d M ′ ) ∈ Bimod C dg ( A , A ) with anotherone ( M, d M ) ∈ Bimod C dg ( A , A ) is defined as( M ′ , d M ′ ) ⊗ A ( M, d M ) := ( M ′ ⊗ A M, d M ′ ⊗ M ) , where deg h ( m ′ ⊗ m ) := deg h ( m ′ ) + deg h ( m ), and d M ′ ⊗ M ( m ′ ⊗ m ) := d M ′ ( m ′ ) ⊗ m + ( − deg h ( m ′ ) m ′ ⊗ d M ( m ) . Given an A - A -dg-bimodule ( M, d M ), its homology is H ( M, d M ) := ker( d M ) / img( d M ),which is a Z × C -graded bimodule. As usual, a map between two C -graded dg-bimodules f : ( M, d M ) → ( M ′ , d M ′ ) induces a map on homology f ∗ : H ( M, d M ) → H ( M ′ , d M ′ ),and we say that f is a quasi-isomorphism whenever f ∗ is an isomorphism. The derivedcategory D C ( A ) of a C -graded algebra A is given by localizing the category Mod C dg ( A )along quasi-isomorphisms. We also define similarly the derived category of dg-bimodules D C ( A , A ).The homological shift functor is the functor[1] : Bimod C dg ( A , A ) → Bimod C dg ( A , A ) , given by sending the dg-module ( M, d M ) to ( M [1] , d M [1] ) where • the homological degree is shifted up by one, ( M [1]) hg := M h − g ; • d M [1] := − d M ; • M [1] inherits the left and right actions of M , that is y · ( m )[1] := ( y · m )[1] and( m )[1] · x := ( m · x )[1],and the identity on morphisms. Remark 4.25.
There is a more general notion of C -graded dg-module over a C -graded dg-algebra, where the algebra itself carries a Z -grading and a differential. In that situation,the homological shifting functor would need to twist the left action as usual.Let f : ( M, d M ) → ( M ′ , d M ′ ) be a map of C -graded dg-bimodules. The mapping cone of f is Cone( f ) := ( M [ − ⊕ M ′ , d C ) , d C := (cid:18) − d M f d M ′ (cid:19) It is a C -graded dg-bimodule that fits in a short exact sequence0 → ( M ′ , d M ′ ) ı M ′ −−→ Cone( f ) π M −−→ ( M, d M )[ − → . The derived category D C ( A ) is a triangulated category with translation functor given bythe homological shift [ − the following form:( M, d M ) f −→ ( M ′ , d M ′ ) ı m ′ −−→ Cone( f ) π M −−→ ( M, d M )[ − . Definition 4.26.
We say that a C -graded dg-module over A is relatively projective if it isa direct summand of a direct sum of shifted copies (both in homological and in C -degree)of the free dg-module ( A, C -graded dg-module is cofibrant if it is a directsummand in Mod C dg ( A ) of the inverse limit of a filtration0 = F ⊂ F ⊂ F ⊂ · · · ⊂ F r ⊂ F r +1 ⊂ · · · , where each F r +1 /F r is isomorphic to a relatively projective dg-module.A nice property of cofibrant dg-modules is that taking a tensor product with such adg-module preserves quasi-isomorphisms: given a quasi-isomorphism f : M ∼ −→ M ′ and acofibrant dg-module P , then f ⊗ M ⊗ A P ∼ −→ M ′ ⊗ A P is a quasi-isomorphism. Bystandard arguments in homological algebra (see [16]), we obtain the following: Proposition 4.27.
For any C -graded dg-module M , there exists a cofibrant dg-module p ( M ) with a surjective quasi-isomorphism p ( M ) M. ∼ We call p ( M ) the bar resolution of M , and the assignment M → p ( M ) is functorial. This allows us to define for any M ∈ Bimod C dg ( A , A ) the derived tensor product functor M ⊗ L A − : D C ( A ) → D C ( A ) , M ⊗ L A X := M ⊗ A p ( X ) . Whenever M is cofibrant as right A -dg-module, then we have M ⊗ A − ∼ = M ⊗ L A − .4.5. Homogeneous maps.
We use the same hypothesis as in Section 4.3.
Definition 4.28.
Let M and N be C -graded A - A -bimodules. We say that a (non-graded)map f : M → N is purely homogeneous of degree i ∈ e I if for all m ∈ M we have • f ( m ) = 0 whenever | m | / ∈ D i ; • | f ( m ) | = ϕ i ( | m | ); • y · f ( m ) = β e ,i ( | y | , | m | ) f ( y · m ) for all y ∈ A ; • f ( m ) · x = β i, e ( | m | , | x | ) f ( m · x ) for all x ∈ A .A map f : M → N is homogeneous if it is a finite sum f = P j ∈ J ⊂ e I f j of purely homoge-neous maps f j : M → N of degree j .Note that if f : M → N is purely homogeneous of degree i , then the induced map¯ f : ϕ i ( M ) → N , where ¯ f ( ϕ i x ) := f ( x ), is graded. Let g : M ′ → N ′ be a homogeneousmap of A - A -bimodules and f : M → N be a homogeneous map of A - A -bimodules. Wedefine the tensor map g ⊗ f := P j ∈ I ( f ⊗ g ) j where( g ⊗ f ) j ( m ′ ⊗ m ) := X i ′ • i )= j β | g | , | f | ( | m ′ | , | m | ) − g i ′ ( m ′ ) ⊗ f i ( m ) , DD KHOVANOV HOMOLOGY FOR TANGLES 21 for all homogeneous elements m ′ ∈ M ′ , m ∈ M .We would like to define a (non-abelian) category BIMOD C ( A , A ) of C -graded bimoduleswith homogeneous maps, hence having hom-spacesHom BIMOD C ( A ,A ) ( M, N ) := M i ∈ e I Hom
Bimod C ( A ,A ) ( ϕ i ( M ) , N ) . However, the composition of homogeneous maps is in general not homogeneous, so thatBIMOD C ( A , A ) is in general not a category.4.6. C -shifting 2-system. In order to have that the composition of two purely homoge-neous maps is purely homogeneous, we need to define a way to vertically compose gradingshifts.
Definition 4.29. A C -shifting 2-system S = {I , Φ } is a C -shifting system { ( I , • , e ) , Φ } such that I is equipped with an associative vertical composition map − ◦ − : I × I → I , respecting • e ◦ e = e ; • D j ◦ i = D i ∩ ϕ i − ( D j ); • ϕ j | ϕ i ( D i ) ∩ D j ◦ ϕ i | D j ◦ i = ϕ j ◦ i ; • ϕ ( j ′ ◦ i ′ ) • ( j ◦ i ) = ϕ ( j ′ • j ) ◦ ( i ′ • i ) and they have the same domain,for all j ′ , j, i ′ , i ∈ I .As before, we need some compatibility maps. Thus, consider a collection of maps γ Y Xj,i : D Y Xi → k × , for all j, i ∈ I and Y, X ∈ C , and where γ Y X e , e = 1. We also require that γ Y Xj,i = 1 whenever i or j ∈ I Id . Consider as well a collection of invertible scalarsΞ ZY Xj ′ ,ji ′ ,i ∈ k × , respecting Ξ Y Xj ′ ,ji ′ ,i = 1 whenever ( j ′ ◦ i ′ ) • ( j ◦ i ) = ( j ′ • j ) ◦ ( i ′ • i ) (for example, whenever j ′ = i ′ = e ). Furthermore, if j ′ , i ′ , j or i is in I Id , then we can exchange it with anyother element of I Id and Ξ Y Xj ′ ,ji ′ ,i stays the same. We will write Ξ j ′ ,ji ′ ,i ( g ) := Ξ Y Xj ′ ,ji ′ ,i whenever | g | C ∈ Hom C ( X, Y ). Definition 4.30.
We say that a C -shifting 2-system S is compatible with α through ( β, γ, Ξ)if the underlying C -shifting system is compatible with α through β and(10) β ZY Xj ′ ◦ i ′ ,j ◦ i ( g ′ , g ) γ Y Xj ′ ,i ′ ( g ′ ) γ ZXj,i ( g )Ξ ZY Xj ′ ,ji ′ ,i = γ ZXj ′ • j,i ′ • i ( g ′ g ) β ZY Xi ′ ,i ( g ′ , g ) β ZY Xj ′ ,j ′ ( ϕ i ′ ( g ′ ) , ϕ i ( g )) , for all g ′ ∈ D ZYi , g ∈ D Y Xi , and(11) γ Y Xk ◦ j,i ( g ) γ Y Xk,j ( ϕ i ( g )) = γ Y Xk,j ◦ i ( g ) γ Y Xj,i ( g ) , for all g ∈ D Y Xi .This allows us to construct natural isomorphisms ϕ j ◦ ϕ i ≃ −→ ϕ j ◦ i ,ϕ ( j ′ ◦ i ′ ) • ( j ◦ i ) ≃ −→ ϕ ( j ′ • j ) ◦ ( i ′ • i ) , in Mod C given by ϕ j ◦ ϕ i ( M ) → ϕ j ◦ i ( M ) , m γ j,i ( | m | ) m,ϕ ( j ′ ◦ i ′ ) • ( j ◦ i ) ( M ) → ϕ ( j ′ • j ) ◦ ( i ′ • i ) ( M ) , m Ξ j ′ ,ji ′ ,i ( | m | ) m, for all homogenous element m ∈ M . Then, Eq. (10) means the following diagram com-mutes:(12) ϕ j ′ ◦ ϕ i ′ ( M ′ ) ⊗ ϕ j ◦ ϕ i ( M ) ϕ j ′ • j (cid:0) ϕ i ′ ( M ′ ) ⊗ ϕ i ( M ) (cid:1) ϕ j ′ • j ◦ ϕ i ′ • i ( M ′ ⊗ M ) ϕ j ′ ◦ i ′ ( M ′ ) ⊗ ϕ j ◦ i ( M ) ϕ ( j ′ ◦ i ′ ) • ( j ◦ i ) ( M ′ ⊗ M ) ϕ ( j ′ • j ) ◦ ( i ′ • i ) ( M ′ ⊗ M ) β j ′ ,j γ j ′ ,i ′ ⊗ γ j,i β i ′ ,i γ j ′• j,i ′• i β j ′◦ i ′ ,j ◦ i Ξ j ′ ,ji ′ ,i for all M ′ , M ∈ Mod C , and Eq. (11) the following one:(13) ϕ k ◦ ϕ j ◦ ϕ i ( M ) ϕ k ◦ ϕ j ◦ i ( M ) ϕ k ◦ j ◦ ϕ i ( M ) ϕ k ◦ j ◦ i ( M ) γ j,i γ k,j γ k,j ◦ i γ k ◦ j,i for all M ∈ Mod C .As before, we extend it to ˜ I := I ⊔ { Id } where putting Id in Ξ Y X − , −− , − means we can replaceit by any element j ∈ I Id . Since ϕ Id ◦ ϕ i ∼ = ϕ i ∼ = ϕ i ◦ ϕ Id by identity maps, we putId ◦ i = i = i ◦ Id. We also extend ϕ ( j ′ • j ) ◦ ( i ′ • i ) for j ′ , j, i ′ , i ∈ ˜ I by replacing any occurrenceof Id with a direct sum over elements in I Id . Proposition 4.31.
The natural isomorphisms ϕ j ◦ ϕ i ≃ −→ ϕ j ◦ i and ϕ ( j ′ ◦ i ′ ) • ( j ◦ i ) ≃ −→ ϕ ( j ′ • j ) ◦ ( i ′ • i ) both induce natural isomorphisms between endofunctors of Bimod C ( A , A ) .Proof. We only need to show the natural isomorphisms respect the bimodule structure. Itimmediately follows from Eq. (12), and using the fact that Ξ e ,j e ,i = Ξ j ′ , e i ′ , e = 1. (cid:3) Furthermore, given such a system and two purely homogeneous maps g : M ′ → M ′′ and f : M → M ′ in BIMOD C ( A , A ) of degree j and i respectively, we define their C -gradedcomposition as ( g ◦ C f )( m ) := γ j,i ( | m | ) − ( g ◦ f )( m ) . DD KHOVANOV HOMOLOGY FOR TANGLES 23
Proposition 4.32.
Equipped with the C -graded composition, BIMOD C ( A , A ) is an addi-tive category.Proof. First, we obtain that if g has degree j and f has degree i , then g ◦ G f has degree j ◦ i in BIMOD C ( A , A ). This follows from Eq. (10) using the fact that Ξ e ,j e ,i = Ξ j ′ , e i ′ , e = 1.Then, we need to show that the C -graded composition is associative, which follows fromEq. (11). (cid:3) Proposition 4.33.
Consider two pairs of purely homogeneous maps f : M → M , f : M → M ∈ BIMOD C ( A , A ) and g : N → N , g : N → N ∈ BIMOD C ( A , A ) . Wehave (cid:0) ( g ◦ C g ) ⊗ ( f ◦ C f ) (cid:1) ( m ⊗ A n ) = Ξ | g | , | f || g | , | f | ( | m | • | n | ) (cid:0) ( g ⊗ f ) ◦ C ( g ⊗ f ) (cid:1) ( m ⊗ A n ) , for all m ∈ M , n ∈ N .Proof. It follows from Eq. (12). (cid:3)
Example 4.34.
Consider again superstructures as in Example 4.20. We put j ◦ i = j + i mod 2, and we have γ ⋆⋆j,i := 1 and Ξ ⋆⋆j ′ ,ji ′ ,i := ( − i ′ j . Then, it coincides with the usual notionof composition of tensor product of supermaps. Example 4.35.
Recall Example 4.21. We put γ ⋆⋆ ( n ,n ) , ( m ,m ) := 1 and Ξ ⋆⋆ ( n ′ ,n ′ ) , ( n ,n )( m ′ ,m ′ ) , ( m ,m ) := λ R (( m ′ , m ′ ) , ( n , n )).4.7. Graded commutativity.
Suppose ( C , α ) is equipped with a compatible C -shifting2-system {I , Φ } . Definition 4.36. A commutativity system on {I , Φ } is a collection T := { (( j ′ , i ′ ) , ( j, i )) ∈I × I } , such that • if (( j ′ , i ′ ) , ( j, i )) ∈ T then ϕ j ′ ◦ i ′ = ϕ j ◦ i and (( j, i ) , ( j ′ , i ′ )) ∈ T; • if (( j ′ , i ′ ) , ( j , i )) ∈ T , (( j ′ , i ′ ) , ( j , i )) ∈ T then (( j ′ • j ′ , i ′ • i ′ ) , ( j • j , i • i )) ∈ T.Consider a collection of scalars τ Y Xj ′ ,ji ′ ,i ∈ k × , for all Y, X ∈ Obj ( C ) and (( j ′ , i ′ ) , ( j, i )) ∈ T, such that τ Y Xj ′ ,ji ′ ,i = 1 whenever j ′ ◦ i ′ = j ◦ i ,and ( τ Y Xj ′ ,ji ′ ,i ) − = τ Y Xj,j ′ i,i ′ for all (( j ′ , i ′ ) , ( j, i )) ∈ T. If (( j ′ , i ′ ) , ( j, i )) / ∈ T, then we put τ j ′ ,ji ′ ,i := 0.We write τ j ′ ,ji ′ ,i ( g ) := τ Y Xj ′ ,ji ′ ,i whenever g ∈ Hom C ( X, Y ). Definition 4.37.
We say that the commutativity system T is compatible with { α, β, Ξ } through τ if it respects the following compatibility condition:(14) β j ◦ i ,j ◦ i ( g ′ , g ) τ j ′ ,j i ′ ,i ( g ′ ) τ j ′ ,j i ′ ,i ( g )Ξ j ,j i ,i ( g ′ • g ) = β j ′ ◦ i ′ ,j ′ ◦ i ′ ( g ′ , g ) τ j ′ • j ′ ,j • j i ′ • i ′ ,i • i ( g ′ • g )Ξ j ′ ,j ′ i ′ ,i ′ ( g ′ • g ) , for all (( j ′ , i ′ ) , ( j , i )) , (( j ′ , i ′ ) , ( j , i )) ∈ T and g ′ ∈ D j ◦ i , g ∈ D j ◦ i . We also requirethat(15) τ Y X Id • i,j • Id j • Id , Id • i = (Ξ Y X Id ,ij, Id ) − Ξ Y Xj,
IdId ,i . For each (( j ′ , i ′ ) , ( j, i )) ∈ T we obtain a natural isomorphism of functors ϕ j ′ ◦ i ′ ≃ −→ ϕ j ◦ i , in Mod C given by ϕ j ′ ◦ i ′ ( M ) → ϕ j ◦ i ( M ) , m τ j ′ ,ji ′ ,i ( | m | ) m, for all homogenous element m ∈ M . Then, the compatibility condition ensures the follow-ing diagram commutes(16) ϕ j ′ ◦ i ′ ( M ) ⊗ ϕ j ′ ◦ i ′ ( M ) ϕ ( j ′ ◦ i ′ ) • ( j ′ ◦ i ′ ) ( M ⊗ M ) ϕ ( j ′ • j ′ ) ◦ ( i ′ • i ′ ) ( M ⊗ M ) ϕ j ◦ i ( M ) ⊗ ϕ j ◦ i ( M ) ϕ ( j ◦ i ) • ( j ◦ i ) ( M ⊗ M ) ϕ ( j • j ) ◦ ( i ◦ i ) ( M ⊗ M ) β j ′ ◦ i ′ ,j ′ ◦ i ′ τj ′ ,j i ′ ,i ⊗ τj ′ ,j i ′ ,i Ξ j ′ ,j ′ i ′ ,i ′ τj ′ • j ′ ,j • j i ′ • i ′ ,i • i β j ◦ i ,j ◦ i Ξ j ,j i ,i as well as the following one: ϕ (Id • i ) ◦ ( j • Id) ( M ) ϕ (Id ◦ j ) • ( i ◦ Id) ( M ) ϕ j • i ( M ) ϕ ( j • Id) ◦ (Id • i ) ( M ) ϕ ( j ◦ Id) • (Id ◦ i ) ( M ) τ Id • i,j • Id j • Id , Id • i ΞId ,ij, Id − Ξ j, IdId ,i Definition 4.38.
Given a C -shifting 2-system with a compatible commutativity systemT, we say that a diagram of purely homogeneous maps M M M M f ∗ f ∗ f ∗ τf ∗ DD KHOVANOV HOMOLOGY FOR TANGLES 25 is C -graded commutative if (( | f ∗ | , | f ∗ | ) , ( | f ∗ | , | f ∗ | )) ∈ T and τ | f ∗ | , | f ∗ || f ∗ | , | f ∗ | . ( f ∗ ◦ f ∗ ) = ( f ∗ ◦ f ∗ ) . We extend the definition linearly to homogeneous maps.
Proposition 4.39.
Given two C -graded commutative diagrams N N N N g ∗ g ∗ g ∗ g ∗ and M M M M f ∗ f ∗ f ∗ f ∗ then the diagram N ⊗ A M N ⊗ A M N ⊗ A M N ⊗ A M g ∗ ⊗ f ∗ g ∗ ⊗ f ∗ g ∗ ⊗ f ∗ g ∗ ⊗ f ∗ is C -graded commutative.Proof. It is a consequence of Proposition 4.33 and Eq. (14). (cid:3)
Proposition 4.40.
The natural isomorphism τ j ′ ,ji ′ ,i : ϕ j ′ ◦ i ′ ≃ −→ ϕ j ◦ i induces a natural iso-morphism as endofunctors over Bimod C ( A , A ) .Proof. It follows from the commutativity of Eq. (16) and the fact that τ e , ee , e = 1, andΞ e ,j ′ e ,i ′ = Ξ e ,j e ,i = 1 and Ξ j ′ , e i ′ , e = Ξ j, e i, e = 1. (cid:3) We also define the grading shift functor ϕ i : BIMOD C ( A , A ) → BIMOD C ( A , A ) , as the usual grading shift by i on objects, and acting on maps as ϕ i ( f )( m ) := τ i, | f || f | ,i ( | m | ) f ( m ) , for f : M → N purely homogeneous and m ∈ M . Note that ϕ i ( f ) = 0 whenever(( i, | f | ) , ( | f | , i )) / ∈ T. Example 4.41.
Still for superstructures as in Example 4.20, we put T := { (( j ′ , i ′ ) , ( j, i )) | j ′ + i ′ ≡ j + i mod 2 } with τ ⋆,⋆π,ππ,π := − τ ⋆,⋆j ′ ,ji ′ ,i := 1 otherwise. Example 4.42.
Recall Example 4.21. We take T := { (cid:0) (( n , n ) , ( m , m )) , (( m , m ) , ( n , n )) (cid:1) ).We put τ ⋆,⋆ ( n ,n ) , ( m ,m )( m ,m ) , ( n ,n ) := λ R (( n , n ) , ( m , m )). Dg- C -graded modules. Suppose ( C , α ) is equipped with a compatible C -shifting2-system {I , Φ } with a commutativity system T.In Section 4.4, we have introduced the notion of C -graded dg-(bi)module, which comeswith a differential that preserves the C -grading. We now introduce a notion of dg-(bi)modulewith a differential that is C -homogeneous. Definition 4.43. A dg- C -graded bimodule ( M, d M ) over a pair of C -graded k -algebras A and A is a Z × C -graded A - A -bimodule M = L n ∈ Z ,g ∈C M ng , with a differential d M = P j ∈ J ⊂ I d M,j , where J is finite, such that: • d M,j ( M ng ) ⊂ M n +1 ϕ j ( g ) if g ∈ D j and d M,j ( M ng ) = 0 otherwise; • d M,j ( y · m ) = β e ,j ( | y | C , | m | C ) − y · d M,j ( m ); • d M,j ( m · x ) = β j, e ( | m | C , | x | C ) − d M,j ( m ) · x ; • d M ◦ d M = 0,for all y ∈ A , m ∈ M and x ∈ A .A map of dg- C -graded bimodules is an homogeneous map of C -graded bimodules thatpreserves the homological grading and C -graded commutes with the differentials.Let BIMOD C dg ( A , A ) be the additive category of dg- C -graded A - A -bimodule. Thehomology is defined as before, but it is not C -graded anymore, and therefore not a bimodule.However, it is still a Z -graded space, and a map of dg- C -graded bimodules induces a Z -graded map in homology. The homological shift functor is also constructed as before. Let f : M → M ′ be a map of dg- C -graded bimodules. The mapping cone of f isCone( f ) := ( M [ − ⊕ M ′ , d C ) , d C := − τ | d M ′ | , | f || f | , | d M | d M f d M ′ ! Note that d C is an homogeneous map, and it yields a complex since f C -graded com-mutes with d M and d M ′ . The tensor product of M ′ ∈ BIMOD C dg ( A , A ) with M ∈ BIMOD C dg ( A , A ) is defined as before, except that d M ′ ⊗ M,j ( m ′ ⊗ m ) :=( d M ′ ,j ⊗ m ′ ⊗ m ) + ( − τ Id • i,i ′ • Id i ′ • Id , Id • i ( | m ′ | • | m | )) deg h ( m ′ ) (1 ⊗ d M,j )( m ′ ⊗ m ) . Note that since τ Id • i,i ′ • Id i ′ • Id , Id • i = (Ξ Id ,ij, Id ) − Ξ j, IdId ,i the definition is independent from a choice ofcommutative system. Also, it gives a chain complex thanks to Proposition 4.33. Remark 4.44.
Note that BIMOD C dg ( A , A ) is not a monoidal category in the usual sense.Indeed, the tensor product is not a bifunctor because of Proposition 4.33. We can think ofit as a C -graded bifunctor.We say that two maps f, g : M → M ′ are homotopic f ∼ g if there is a map h : M [ − → N in BIMOD C Z ( A , A ) (thus preserving the homological grading, homogeneous w.r.t. the DD KHOVANOV HOMOLOGY FOR TANGLES 27 C -grading and not necessarily commuting with the differentials) such that f − g = d M ′ ◦ h + τ | d M ′ | , | h || h | , | d M | h ◦ d M . Note that obviously homotopic maps induce equivalent maps in homology. Also recallthat a homotopy equivalence is pair of maps f : M → M ′ and g : M ′ → M such that g ◦ f ∼ Id M and f ◦ g ∼ Id M ′ , thus inducing an isomorphism in homology. Definition 4.45.
Let K OM C ( A , A ) be the additive category with the same objects asBIMOD C dg ( A , A ) and hom-spaces given byHom K OM C ( A ,A ) ( M, N ) := Hom
BIMOD C dg ( A ,A ) ( M, N ) / ∼ . We refer to it as the C -graded homotopy category of A - A -bimodules.5. The grading category G Consider the subset b B mn ⊂ B mn of reduced flat tangles (i.e. without free loop). Let b : B mn → b B mn be the map that remove free loops. Define also the map s cba ( − , − ) : B | c || b | × B | b || a | → Z for a, b, c ∈ B • as given for t ′ ∈ B | c || b | and t ∈ B | b || a | by s cba ( t ′ , t ) := deg( W cba ( t ′ , t )) , where W cba ( t ′ , t ) is the canonical cobordism defined in Section 3.2. Lemma 5.1.
For any triple of flat tangles t ′′ ∈ B | d || c | , t ′ ∈ B | c || b | and t ∈ B | b || a | , we have s dca ( t ′′ , t ′ t ) + s cba ( t ′ , t ) = s dcb ( t ′′ , t ′ ) + s dba ( t ′′ t ′ , t ) .Proof. It follows by minimality condition on the Euler characteristic of the canonical cobor-disms W cba ( t ′ , t ). (cid:3) Definition 5.2.
We define the category G as: • objects are given by elements in B • ; • hom-spaces areHom G ( a, b ) := b B mn × Z , Id a = (1 n , ( n, , for a ∈ B n and b ∈ B m ; • composition is ( t ′ , p ′ ) ◦ ( t, p ) := ( b t ′ t, p + p ′ + s cba ( t ′ , t )) , for ( t, p ) ∈ Hom G ( a, b ) and ( t ′ , p ′ ) ∈ Hom G ( b, c ).Note that the composition in G is associative thanks to Lemma 5.1.We turn G into a grading category by equipping it with the associator α : G [3] → R givenfor d ( t ′′ ,p ′′ ) ←−−−− c ( t ′ ,p ′ ) ←−−− b ( t,p ) ←−− a ∈ G [3] , by α (( t ′′ , p ′′ ) , ( t ′ , p ′ ) , ( t, p )) := α α , where(17) ¯ dt ′′ t ′ b ⊗ ¯ bta ¯ dt ′′ c ⊗ ¯ ct ′ b ⊗ ¯ bta ¯ dt ′′ t ′ ta ¯ dt ′′ c ⊗ ¯ ct ′ ta W dba ( t ′′ t ′ ,t ) W dcb ( t ′′ ,t ′ ) ⊗ ba ( t )1 dc ( t ′′ ) ⊗ W cba ( t ′ ,t ) ⇒ α W dca ( t ′′ ,t ′ t ) in the sense that there exists a unique (up to homotopy) locally vertical change of chronol-ogy H : W dca ( t ′′ , t ′ t ) ◦ (cid:0) dc ( t ′′ ) ⊗ W cba ( t ′ , t ) (cid:1) ⇒ W dba ( t ′′ t ′ , t ) ◦ (cid:0) W dcb ( t ′′ , t ′ ) ⊗ ba ( t ) (cid:1) , and we take α := ı ( H ), and where α := λ R ( s cba ( t ′ , t ) , p ′′ ) , with λ R being the bilinear map defined in Eq. (1).In terms of pictures we can explain α as coming from( t ′′ , p ′′ )( t ′ , p ′ ) ( t, p ) = α ( t ′′ , p ′′ )( t ′ , p ′ ) ( t, p ) = α α ( t ′′ , p ′′ )( t ′ , p ′ ) ( t, p )where the trivalent vertices represent the canonical cobordisms used to form the composi-tion maps µ . Remark 5.3.
Note that the definition of α also make sense for non-reduced flat tangles.However, taking the reduced version of the tangle would give the same value α ( t ′′ , t ′ , t ) = α ( b t ′′ , b t ′ , b t ) since W cba ( t ′ , t ) is the identity on the extra loops in t ′ and t . Proposition 5.4.
The map α : G [3] → R defined above is a 3-cocycle. DD KHOVANOV HOMOLOGY FOR TANGLES 29
Proof.
By definition, dα ( g ′′′ , g ′′ , g ′ , g ) computes the difference between taking the two pos-sible path in the following diagram:(18) g ′′′ g ′′ g ′ g g ′′′ g ′′ g ′ gg ′′′ g ′′ g ′ g g ′′′ g ′′ g ′ g g ′′′ g ′′ g ′ g α ( g ′′′ ,g ′′ g ′ ,g ) α ( g ′′ ,g ′ ,g ) α ( g ′′′ ,g ′′ ,g ′ ) α ( g ′′′ g ′′ ,g ′ ,g ) α ( g ′′′ ,g ′′ ,g ′ g ) ⇒ dα Since we are considering two locally vertical changes of chronology with same source andtarget, the following diagram of cobordisms commutes by Proposition 2.3: t ′′′ t ′′ t ′ t t ′′′ t ′′ t ′ tt ′′′ t ′′ t ′ t t ′′′ t ′′ t ′ tt ′′′ t ′′ t ′ t t ′′′ t ′′ t ′ t α ( t ′′′ , d t ′′ t ′ ,t ) α ( t ′′ ,t ′ ,t ) α ( t ′′′ ,t ′′ ,t ′ ) α ( d t ′′′ t ′′ ,t ′ ,t ) λ R ( s cba ( t ′ ,t ) ,s edc ( t ′′′ ,t ′′ )) α ( t ′′′ ,t ′′ , c t ′ t ) Therefore, we have that the contribution of α in Eq. (18) is(top) = λ R (cid:0) s cba ( t ′ , t ) , s edc ( t ′′′ , t ′′ ) (cid:1) (bottom) . For the α contribution, we have for the top that λ R (cid:0) s dcb ( t ′′ , t ′ ) , p ′′′ (cid:1) λ R (cid:0) s dba ( t ′′ t ′ , t ) , p ′′′ (cid:1) λ R (cid:0) s cba ( t ′ , t ) , p ′′ (cid:1) = λ R (cid:0) s dcb ( t ′′ , t ′ ) + s dba ( c t ′′ t ′ , t ) , p ′′′ (cid:1) λ R (cid:0) s cba ( t ′ , t ) , p ′′ (cid:1) , and for the bottom that λ R (cid:0) s cba ( t ′ , t ) , p ′′′ + p ′′ + s edc ( t ′′′ , t ′′ ) (cid:1) λ R (cid:0) s dca ( t ′′ , b t ′ t ) , p ′′′ (cid:1) = λ R (cid:0) s cba ( t ′ , t ) , s edc ( t ′′′ , t ′′ ) (cid:1) λ R (cid:0) s cba ( t ′ , t ) , p ′′′ (cid:1) λ R (cid:0) s cba ( t ′ , t ) , p ′′ (cid:1) λ R (cid:0) s dca ( t ′′ , b t ′ t ) , p ′′′ (cid:1) = λ R (cid:0) s cba ( t ′ , t ) , s edc ( t ′′′ , t ′′ ) (cid:1) λ R (cid:0) s dca ( t ′′ , b t ′ t ) + s cba ( t ′ , t ) , p ′′′ (cid:1) λ R (cid:0) s cba ( t ′ , t ) , p ′′ (cid:1) , using the bilinearity of λ R . By Lemma 5.1, we conclude that the α contribution in (18) is λ R (cid:0) s cba ( t ′ , t ) , s edc ( t ′′′ , t ′′ ) (cid:1) (top) = (bottom) . Putting all these together, we conclude that dα = 0. (cid:3) G -grading shift. Our goal is to construct a G -shifting system compatible with α .First, we note that Σ( G ) is the subcategory generated by identity tangles. Explicitly, for a ∈ B n and b ∈ B m , we haveHom Σ( G ) ( a, b ) = ( { (Id n , p ) | p ∈ Z } , if n = m , ∅ , otherwise.For a chronological cobordism with corners W : t → t ′ , with t, t ′ ∈ B mn , and for v ∈ Z ,we write W v for the pair ( W, v ). Then, we define the shifting map ϕ W v with domain D baW v := { ( b t, p ) ∈ Hom G ( a, b ) | p ∈ Z } and ϕ W v ( b t, p ) := (cid:0)b t ′ , p + v + deg(1 ¯ b W a ) (cid:1) , for ( b t, p ) ∈ Hom G ( a, b ).Given W : t → t ′ and W : t → t ′ with t , t ′ ∈ B mn and t , t ′ ∈ B n ′ m we obtain acobordism W • W : t t → t ′ t ′ by right-then-left horizontal composition. In this case weput W v • W v := ( W • W ) v + v . Otherwise, we put W v • W v := 0. Then, we define themonoid I := { W v } W,v ⊔ { e , } with composition • defined above, neutral element e andan absorbing element 0. Finally, we take I Id := { ( t ) } t ∈ b B •• ⊂ I , where we recall t is theidentity cobordism on t . Proposition 5.5.
The datum of S := ( I, { ϕ W v } W,v ⊔ { ϕ e , ϕ } ) forms a G -shifting system.Proof. Straightforward. (cid:3)
We construct compatibility maps for S by putting β cbaW v ,W v (( b t , p ) , ( b t , p )) := β β β ′ β ′′ , DD KHOVANOV HOMOLOGY FOR TANGLES 31 with β ∈ R given by making the following diagram commute(19) ¯ ct ′ b ¯ bt ′ a ¯ ct b ¯ bt a ¯ ct ′ t ′ a ¯ ct ′ ta W cba ( t ′ ,t ′ ) W cba ( t ,t )1 ¯ c W b ¯ b W a ¯ c ( W • W )1 a ⇒ β in the same sense as in Eq. (17), and β := λ R ( s cba ( t ′ , t ′ ) , v + v ) ,β ′ := λ R (deg(1 ¯ c W b ) , v ) ,β ′′ := λ R ( p , deg(1 ¯ b W a ) + v ) . In terms of pictures, we think of it as( t , p ) W v ( t , p ) W v = β ′′ ( t , p ) W v ( t , p ) W v = β ′ β ′′ ( t , p ) W v ( t , p ) W v = β β ′ β ′′ ( t , p ) W v + v ( t , p ) W = β β β ′ β ′′ ( t , p ) W • W v + v ( t , p )(20)The compatibility maps with the neutral shift ϕ e are computed in a similar fashion, bytreating it as the identity cobordism on the identity tangle, and with v = 0. Remark 5.6.
We computed β , by thinking of a degree shift as shifting first by the chrono-logical cobordism W , then shifting the Z × Z -degree by v . We could also have done it in the reverse order, modifying only β := λ R ( s cba ( t , t ) , v + v ) and β ′ := ( v , W ). It wouldgive a different set of compatibility maps. Lemma 5.7.
We have deg(1 ¯ c ( W • W )1 a ) + s cba ( t , t ) = deg(1 ¯ c W b ) + deg(1 ¯ b W a ) + s cba ( t ′ , t ′ ) . Proof.
There is a diffeomorphism t W t W ∼ = t t W • W between the cobordisms. Thus, they have the same degree. (cid:3) Proposition 5.8.
The construction above forms a G -shifting system compatible with α .Proof. We fix d ( b t ,p ) ←−−− c ( b t ,p ) ←−−− b ( b t ,p ) ←−−− a ∈ G [3] and W i : t i → t ′ i and v i ∈ Z for i ∈ { , , } . Since it is clear from the context, we willwrite below | W | instead of deg(1 ¯ b W a ) and so on.We first look at the contribution of α and β in the left and right side of Eq. (8). ByProposition 2.3, the following diagram of cobordisms commutes: • • • •
13 2 1 3 2 1 3 2 • • • β α λ R ( s dcb ( t ,t ) , | χ | ) β α λ R ( s cba ( t ′ ,t ′ ) , | W | ) β β where a box with label i is W i . Therefore, we have(21) λ R ( s dcb ( t ,t ) , | W | ) × (left) = λ R ( s cba ( t ′ , t ′ ) , | W | ) × (right) . We now compute the contribution of α , β , β ′ and β ′′ . The left side of Eq. (8) yields λ R ( s cba ( t , t ) , p ) × λ R ( s dba ( t ′ t ′ , t ′ ) , v + v + v ) λ R ( | W • W | , v ) λ R ( p + p + s dcb ( t , t ) , | W | + v ) × λ R ( s dcb ( t ′ , t ′ ) , v + v ) λ R ( | W | , v ) λ R ( p , | W | + v ) , (22) DD KHOVANOV HOMOLOGY FOR TANGLES 33 and the right side yields λ R ( s dca ( t ′ , t ′ t ′ ) , v + v + v ) λ R ( | W | , v + v ) λ R ( p , v + v + | W • W | ) × λ R ( s cba ( t ′ , t ′ ) , v + v ) λ R ( | W | , v ) λ R ( p , | W | + v ) × λ R ( s cba ( t ′ , t ′ ) , p + | W | + v ) . (23)We use Lemma 5.7 to get λ R ( | W • W | , v ) = λ R ( | W | + | W | , v ) λ R ( s dcb ( t ′ , t ′ ) , v ) λ − R ( s dcb ( t , t ) , v ) , in Eq. (22). We also decompose λ R ( | W | + v , p + p + s dcb ( t , t )) in Eq. (22), getting aterm λ R ( s dcb ( t , t ) , v ) that cancels with λ − R ( s dcb ( t , t ) , v ) above, so that remains λ R ( s cba ( t , t ) , p ) × λ R ( | W | + | W | + p + p , v ) × λ R ( s dba ( t ′ t ′ , t ′ ) , v + v + v ) λ R ( p + p + s dcb ( t , t ) , | W | ) × λ R ( s dcb ( t ′ , t ′ ) , v + v + v ) λ R ( | W | , v ) λ R ( p , | W | + v ) , (24)in Eq. (22). Playing a similar game on Eq. (23), we obtain λ R ( p , | W | + | W | ) λ R ( s cba ( t , t ) , p ) × λ R ( s dca ( t ′ , t ′ t ′ ) , v + v + v ) λ R ( | W | , v + v ) λ R ( p , v + v ) × λ R ( s cba ( t ′ , t ′ ) , v + v + v ) λ R ( | W | , v ) λ R ( p , | W | + v ) × λ R ( s cba ( t ′ , t ′ ) , | W | ) , (25)where we also use the fact that λ R ( x, y ) = λ − R ( y, x ). By Lemma 5.1, we have λ R ( s dba ( t ′ t ′ , t ′ ) ,v + v + v ) λ R ( s dcb ( t ′ , t ′ ) , v + v + v = λ R ( s dca ( t ′ , t ′ t ′ ) , v + v + v ) λ R ( s cba ( t ′ , t ′ ) , v + v + v ) , so that a careful examination of the remaining terms yields λ R ( s cba ( t ′ , t ′ ) , | W | ) × Eq. (24) = λ R ( s dcb ( t , t ) , | W | ) × Eq. (25) . Together with Eq. (21), it concludes the proof. (cid:3)
Vertical composition.
We extend the G -shifting system to a G -shifting 2-system.For this, we define the vertical composion on I as follows: W v ◦ W v := ( ( W ◦ W ) v + v , if t ′ = t ,0 , otherwise,for W : t → t ′ and W : t → t ′ . Then, we construct compatibility maps by setting γ baW v ,W v (( t, p )) := λ R (deg(1 ¯ b W a ) , v ) , and Ξ cbaW ′ v ′ ,W ′ v ′ W v ,W v (( t, p )) := ı ( H ca ) λ R (cid:0) v , v ′ (cid:1) , where H : ( W ′ ◦ W ) • ( W ′ ◦ W ) ⇒ ( W ′ • W ′ ) ◦ ( W • W ) is a locally vertical change ofchronology. This is if the cobordism are vertically composable, otherwise the compatibilitymaps are zero. Proposition 5.9.
The G -shifting 2-system S defined above is compatible with α through ( β, γ, Ξ) also defined above.Proof. By Proposition 5.8 we already know the underlying shifting system is compatiblewith α through β . Thus, we only need to verify that Eq. (10) and Eq. (11) both hold. Thesecond one is straightforward using the linearity of λ R . The first one can be proven usingsimilar arguments as in Proposition 5.8. We leave the details to the reader. (cid:3) Changes of chronology.
Consider a change of chronology H : W ⇒ W ′ . We canextend it trivially to a change of chronology b H a : (1 ¯ b W a ) ⇒ (1 ¯ b W ′ a ). Therefore, weobtain a natural transformation ϕ H : ϕ W ⇒ ϕ W ′ as Mod G -functors, given by ϕ H ( M ) : ϕ W ( M ) → ϕ W ′ ( M ) , ϕ W ( m ) ϕ H ( | m | ) ϕ W ′ ( m ) , where ϕ H ( | m | ) := ı ( b H a ) − , for all M ∈ Mod G and m ∈ M g : a → b . Proposition 5.10.
For M ∈ Bimod G ( A , A ) , the map ϕ H ( M ) defined above is a map ofbimodules. In particular, ϕ H gives a natural transformation of Bimod G ( A , A ) -functors.Proof. We will show a slightly stronger statement, that is the following diagram commutes: ϕ W ( M ′ ) ⊗ ϕ W ( M ) ϕ W • W ( M ′ ⊗ M ) ϕ W ′ ( M ′ ) ⊗ ϕ W ′ ( M ) ϕ W ′ • W ′ ( M ′ ⊗ M ) β W W ϕ H ⊗ ϕ H ϕ H • H β W ′ ,W ′ for all changes of chronologies H i : W i → W ′ i . Take m ′ ∈ M ′ , m ∈ M with | m ′ | ∈ Hom G ( b, c ) and | m | ∈ Hom G ( a, b ). The only part of β W ′ ,W ′ ( | m ′ | , | m | ) that could differ from β W ,W ( | m ′ | , | m | ) is β . Thus, we have β W ′ ,W ′ ( | m ′ | , | m | ) ı (cid:0) c ( H ⊗ H ) a : 1 ¯ c ( W ⊗ W a ⇒ ¯ c ( W ′ ⊗ W ′ )1 a (cid:1) = β W ,W ( | m ′ | , | m | ) ı (cid:0) c ( H ) b ⊗ b ( H ) a : 1 ¯ c W b ¯ b W a ⇒ ¯ c W ′ b ¯ b W ′ a (cid:1) , by applying the change of chronologies on Eq. (19). But we also have ı (cid:0) c ( H ) b ⊗ b ( H ) a (cid:1) − = ı (cid:0) c ( H ) b (cid:1) − ı (cid:0) b ( H ) a (cid:1) − = ϕ H ⊗ ϕ H ( | m ′ | ⊗ | m | ) ,ı (cid:0) c ( H • H ) a (cid:1) − = ϕ H • H ( | m ′ || m | ) , concluding the proof. (cid:3) Proposition 5.11.
We have ϕ H ′ ◦ ϕ H ∼ = ϕ H ′ ⋆H , and ϕ H ′ ⊗ ϕ H ∼ = ϕ H ′ ◦ H , DD KHOVANOV HOMOLOGY FOR TANGLES 35 for any pair of changes of chronology H : W ⇒ W ′ and H ′ : W ′ ⇒ W ′′ .Proof. It follows immediately from ı ( b ( H ′ ) a ⋆ b H a ) = ı ( b ( H ′ ) a ) ı ( b H a ) = ı ( b ( H ′ ) a ◦ b H a ). (cid:3) Take a cobordism with corners W : t → t ′ . Consider a minimal cobordism c W : t → t ′ .We define b χ ( W ) := χ ( c W ) − χ ( W )2 , χ ( c W ) − χ ( W )2 ! ∈ Z × Z , where χ is the Euler characteristics. Proposition 5.12.
Let W : t → t ′ be a cobordism. Let c W : t → t ′ be a minimal corbordism.There is a natural isomorphism ϕ W v ∼ = ϕ c W v + b χ ( W ) , for all v ∈ Z .Proof. If W is not minimal, it means that there is a change of chronology H : W ⇒ W ′ where W ′ is obtained from c W by attaching tubes in the top region. These tubes locallylook like: orand we can always do a change of chronology so that we keep only the one on the rightside. Note that in this case, the first saddle point is a split that spawns an isolated circlecomponent, and the second one a merge. Therefore, the tube part behaves like a degreeshift by ( − , −
1) in Z × Z thanks to Eq. (4). Thus, we have a natural isomorphism givenby the identity ϕ W ′ v ∼ = ϕ c W v + b χ ( W ) . Composition with the natural isomorphism ϕ H gives thedesired isomorphism. (cid:3) Note that, in particular, for any pair ( W : t → t ′ , v ), it means ϕ W v admits locally a leftinverse (up to natural isomorphism) given by( ϕ W v ) − := ϕ W ( − v − b χ ( W ◦ W )) , where W is obtained from W by taking the mirror image with respect to the horizontalplane. Indeed, by Proposition 5.12, we have( ϕ W v − ) ◦ ϕ W v ∼ = ϕ t . For example we have (cid:0) ϕ ( v ,v (cid:1) − = ϕ (1 − v , − v . Proposition 5.13.
Let W : t → t ′ be a cobordism. Suppose t contains a free loop S . Thenthere is a natural isomorphism ϕ W ( v ,v ∼ = ϕ W ′ ( v − ,v , m λ R ((1 , , deg(1 ¯ b W a )) m, for | m | ∈ Hom G ( a, b ) , and where W ′ : t \ S → t ′ is given by gluing a birth under the freeloop in W . Similarly if t ′ contains a free loop we have ϕ W v ,v ∼ = ϕ W ′′ ( v ,v − where W ′′ isgiven by gluing a positive death, and the isomorphism is given by the identity map.Proof. Gluing a birth change the degree of the cobordism 1 ¯ b W a by (1 ,
0) for all a, b ∈ B • .Because we added this element at the bottom and the compatibility maps are computedwith the idea that the Z -grading shift v is at the top of the cobordism (see Eq. (20)), weneed to compensate by λ R ((1 , , deg(1 ¯ b W a )). (cid:3) Z × Z -grading shift. We define the Z × Z -grading shift functor:( − ) { v , v } : Mod G → Mod G , by setting M { v , v } := M t ∈ B •• ϕ ( v ,v t ( M ) . It takes an element of degree ( t, ( p , p )) to an element of degree ( t, ( p + v , p + v )).When we write ϕ W v ( M ) { v , v } , we will mean we first shift by W v then by ( v , v ),i.e. (cid:0) ϕ W v ( M ) (cid:1) { v , v } . The Z × Z -grading shift functors define a Z × Z -grading onBimod G ( A , A ).5.5. G -graded commutativity. We equip the G -grading shifting 2-system S with a com-mutativity system T given by all pairs { (( W ′ v ′ , W ′ v ′ ) , ( W v , W v )) } whenever there is alocally vertical change of chronology H : W ′ ◦ W ′ ⇒ W ◦ W and v ′ = v , v ′ = v . In thatcase, we put τ baW ′ v ′ ,W v W ′ v ′ ,W v := ı ( b H a ) − λ R ( v , v ) . Proposition 5.14.
The commutativity system T defined above is compatible with the G -grading shifting 2-system S , also defined above, through τ .Proof. For the sake of simplicity, we assume that v ′ = v ′ = v = v . They can be added inthe proof without much effort. Then, we have that Eq. (14) holds from the commutativity DD KHOVANOV HOMOLOGY FOR TANGLES 37 of the following diagram: m i ′ j ′ m i ′ j ′ m • i ′ • j ′ i ′ • j ′ • m m • i ′ i ′ • • j ′ j ′ • m m i j m i j m • i • j i • j • m m • i i • • j j • m β j ′ ◦ i ′ ,j ′ ◦ i ′ τj ′ ,j i ′ ,i .τj ′ ,j i ′ ,i Ξ j ′ ,j ′ i ′ ,i ′ τj ′ • j ′ ,j • j i ′ • i ′ ,i • i β j ◦ i ,j ◦ i Ξ j ,j i ,i where a box with label i represent the cobordism W i . The commutativity comes from thefact the two path are given by applying ı on locally vertical changes of chronology withsame outputs. Also, Eq. (15) holds thanks to Eq. (4) and Eq. (5). (cid:3) Symmetric structure.
The monoidal category Mod G admits a symmetry τ M,N : M ⊗ N → M ⊗ N, τ
M,N ( m ⊗ n ) := λ R ( | m | R , | n | R ) n ⊗ m, where | m | R = p ∈ Z × Z if | m | = ( t, p ) ∈ Hom G . Proposition 5.15.
The symmetry τ M,N turns
Mod G into a symmetric monoidal category.Proof. The proof use similar arguments as in Proposition 5.4, picturing the symmetry as m n τ M,N −−−−→ nm = n m = λ R ( | m | R , | n | R ) n m We leave the details to the reader. (cid:3)
Whenever we have a symmetric monoidal category ( M , ⊗ , τ ) , it makes sense to definethe notion of center of a monoid object A ∈ M (and it can be made more general, see [8]).Consider Z ( A ) the category of morphisms y : Y → A in M such that the diagram Y ⊗ A A ⊗ A AA ⊗ Y A ⊗ A y ⊗ τ Y,A µ ⊗ y µ commutes. The center of A is the terminal object Z ( A ) → A ∈ Z ( A ). Note that, byuniversal property of the terminal object in Z ( A ), Z ( A ) is a commutative monoid objectin M . Concretely, in the case of Mod C , it means the center of a G -graded algebra A isgiven by Z ( A ) := { z ∈ A | µ ( z ⊗ x ) = µ ( τ Z ( A ) ,A ( z ⊗ x )) , ∀ x ∈ A } , as usual. In the case of G , it translates to Z ( A ) := { z ∈ A | zx = λ R ( | z | R , | x | R ) xz, ∀ x ∈ A } . Tangle invariant
We first show that H n is a G -graded algebra. Then, we explain how to adapt theconstruction from [18] to our case.6.1. G -graded arc algebra and arc bimodules. Recall the space F ( t ) from Section 3.5.We can think of it as an object in Mod G wheredeg G ( m ) := (cid:0)b t, deg R ( m ) (cid:1) ∈ Hom G ( a, b ) , for m ∈ b F ( t ) a . Then, we have that the composition map µ [ t ′ , t ] preserves the G -grading. Lemma 6.1.
We have µ [ t ′′ t ′ , t ] (cid:0) µ [ t ′′ , t ′ ]( z, y ) , x (cid:1) = α ( | z | , | y | , | x | ) µ [ t ′′ , t ′ t ] (cid:0) z, µ [ t ′ , t ]( y, x ) (cid:1) , for all x ∈ F ( t ) , y ∈ F ( t ′ ) , z ∈ F ( t ′′ ) .Proof. This is immediate by construction of α in G and the fact that µ [ t ′ , t ] is constructedusing F ( W cba ( t ′ , t )). (cid:3) Proposition 6.2. H n is a unital, associative G -graded R -algebra with units given by a a .Proof. The associativity is givenb by Lemma 6.1.For the unitality, we compute for all m ∈ F ( t ) with | m | = ( t, p ) ∈ Hom G ( a, b ) that α (Id b , Id b , | c | ) = X | b | X | b | = 1 ,α ( | m | , Id a , Id a ) = X | a | λ R (( −| a | , , p ) , DD KHOVANOV HOMOLOGY FOR TANGLES 39 since the cobordisms involved consist only of merges, recalling that Id a = (1 | a | , ( | a | , µ [1 | b | , t ]( b b , m ) = m,µ [ t, | a | ] m, ( a a ) = λ R ( p, ( | a | , X | a | , thanks to Eq. (3). This is best explained by the following pictures:Id b ( t, p ) = ( t, p )and ( t, p ) Id a = λ R ( p, ( | a | , t, p ) Id a = X | a | λ R ( p, ( | a | , t, p )We conclude that H n is unital. (cid:3) Remark 6.3.
When specializing X = Y = Z = 1 or X = Z = 1 and Y = −
1, the algebra H n coincides with the usual Khovanov arc algebra [18] or the odd one [12]. Moreover,the center Z ( H n ) then coincides with usual notion of center or the notion of odd centerfrom [12].For t ∈ B mn , the composition maps µ [1 m , t ] and µ [ t, n ] turn F ( t ) into a H m - H n -bimodulein Mod G , by the same arguments as in the proof of Proposition 6.2. Moreover, any cobor-dism with corners W : t → t ′ induces a graded map ϕ W ( F ( t )) F ( W ) −−−→ F ( t ′ ) . Let ⊗ n denotes the ( G -graded) tensor product ⊗ H n . Proposition 6.4.
The H m - H n -bimodule F ( t ) is projective as left H m -module and as right H n -module.Proof. The proof is essentially the same as in [18, Proposition 3]. We leave the details tothe reader. (cid:3)
Proposition 6.5.
For t ′ ∈ B m ′ n and t ∈ B nm we have an isomorphism F ( t ′ ) ⊗ n F ( t ) ∼ = F ( t ′ t ) , induced by µ [ t ′ , t ] . Proof.
We first note that µ [ t ′ , t ] : F ( t ′ ) ⊗ R F ( t ) induces a map F ( t ′ ) ⊗ n F ( t ) → F ( t ′ t ) , by the universal property of the coequalizer, since by Lemma 6.1 we have µ [ t ′ , t ]( m ′ · x, m ) = α ( | m ′ | , | x | , | m | ) µ [ t ′ , t ]( m ′ , x · m ) . The remaining of the proof is exactly the same as in [18, Theorem 1]. (cid:3)
Lemma 6.6.
The following diagram (cid:0) F ( t ′′ ) ⊗ H F ( t ′ ) (cid:1) ⊗ H F ( t ) F ( t ′′ ) ⊗ H (cid:0) F ( t ′ ) ⊗ H F ( t ) (cid:1) F ( t ′′ t ′ ) ⊗ H F ( t ) F ( t ′′ ) ⊗ H F ( t ′ t ) F ( t ′′ t ′ t ) F ( t ′′ t ′ t ) αµ [ t ′′ ,t ′ ] ⊗ ⊗ µ [ t ′ ,t ] µ [ t ′′ t ′ ,t ] µ [ t ′′ ,t ′ t ] commutes for all t ′′ , t ′ , t ∈ B •• .Proof. Immediate by definition of α and µ [ t ′ , t ]. (cid:3) Tangle resolution.
An ( n, m )-tangle T is a tangle in R × I connecting 2 m pointson the bottom to 2 n -points on the top. A plane diagram of such a tangle is a genericprojection of the tangle on R × I , marking the order of superposition in the crossings.Given a crossing in a plane diagram of a tangle, one can resolve it in to possible ways:ith-crossing0-resolution 1-resolution ξ i =0 ξ i =1 A resolution of a plan diagram is given by resolving all its crossings. It yields a flat tanglein B nm . Suppose T has k crossings, that we order by reading T from bottom to top (wecan suppose there are no pair of crossings at the same height). For each ξ = ( ξ k , . . . , ξ ) ∈{ , } k , we write T ξ ∈ B nm for the resolution of T given by resolving the i -th crossing asgiven by ξ i . We also write 0 := (0 , . . . , ∈ { , } k and 1 := (1 , . . . , ∈ { , } k . DD KHOVANOV HOMOLOGY FOR TANGLES 41
We can suppose all crossings in T are pivoted to look like above (or mirror), and weassociate to them an arrow pointing upward or leftward:orFor each ξ i = 0 in ξ , we write ξ + i := ( ξ k , . . . , ξ i +1 , , ξ i − , . . . , ξ ). We construct achronological cobordism W ξ,i : T ξ → T ξ + i by putting a saddle above the 0-resolution of the i -th crossing, with orientation given by the arrow on the crossing. This defines a map: ϕ W ξ,i (cid:0) F ( T ξ ) (cid:1) F ( W ξ,i ) −−−−→ F ( T ξ + i ) . Note that we need to shift F ( T ξ ) in order to get a graded map of bimodules.Let | ξ | = P ki =0 ξ i be the weight of ξ . For each ξ , we define recursively a cobordism W ξ := W ξ + ℓ ◦ W ξ,ℓ , where ℓ is the lowest integer such that ξ ℓ = 0, and W is the identity. In other words, thereis a unique (non-chronological) cobordism from T ξ to T given changing all 0 to 1, and wegive it a chronology by stacking the saddles in the order given by reading the crossings in T from bottom to top.We write C ( T ) r := M | ξ | = r C ( T ) ξ [ r ] , C ( T ) ξ := ϕ W ξ (cid:0) F ( T ξ ) (cid:1) , where we recall [ r ] = [1] r is a shift up by r units in the homological degree. For each ξ j = 0,we consider the change of chronology H ξ,j : W ξ ⇒ W ξ + j ◦ W ξ,j , that consists in taking the saddle above the j -th crossing and pushing it to the bottom.This allows us to build a graded map of bimodules d ξ,j := F ( W ξ,j ) ◦ ϕ H ξ,j (cid:0) F ( T ξ ) (cid:1) : C ( T ) ξ → C ( T ) ξ + j . Lemma 6.7.
The diagram C ( T ) ξ + i C ( T ) ξ C ( T ) ξ + i + j C ( T ) ξ + j d ξ + i,j d ξ,i d ξ,j d ξ + j,i commutes for all ξ and i, j such that ξ i = ξ j = 0 . Proof.
We have d ξ + i,j ◦ d ξ,i = F ( W ξ + i,j ◦ W ξ,i ) ◦ ϕ H ( T ξ ) , where H : W ξ ⇒ W ξ + i + j ◦ W ξ + i,j ◦ W ξ,i is a locally vertical change of chronology. Moreover,we have F ( W ξ + i,j ◦ W ξ,i ) = F ( W ξ + j,i ◦ W ξ,j ) ◦ ϕ H ′ ( T ξ ) , where H ′ : W ξ + i + j ◦ W ξ + j,i ◦ W ξ,i ⇒ W ξ + i + j ◦ W ξ + j,i ◦ W ξ,j . Thus, d ξ + i,j ◦ d ξ,i = F ( W ξ + j,i ◦ W ξ,j ) ◦ ϕ H ′ ( T ξ ) ◦ ϕ H ( T ξ )= d ξ + j,i ◦ d ξ,j ◦ ϕ H ′′ ( T ξ ) ◦ ϕ H ′ ( T ξ ) ◦ ϕ H ( T ξ ) , for H ′′ : W ξ + i + j ◦ W ξ + i,j ◦ W ξ,i ⇒ W ξ . Since H ′′ ⋆ H ′ ⋆ H : W ξ ⇒ W ξ , by Proposition 2.3 itis homotopic to the identity change of chronology, and we obtain ϕ H ′′ ( T ξ ) ◦ ϕ H ′ ( T ξ ) ◦ ϕ H ( T ξ ) = ϕ H ′′ ⋆H ′ ⋆H ( T ξ ) = ϕ Id Wξ ( T ξ ) = Id , so that d ξ + i,j ◦ d ξ,i = d ξ + j,i ◦ d ξ,j . (cid:3) Remark 6.8.
This is the first difference that appears between the usual construction in[21, 22] and our framework of G -graded modules. Indeed, in the references, they obtainmaps that are commutative only up to some sign (or scalar) assignment, that is shownafterward to always exists. With our approach, the grading shift already assign the correctscalar to each map in order to have a commutative cube of resolutions.Let p ( ξ, j ) := { k ≥ ℓ > j | ξ ℓ = 1 } for all ξ and 1 ≤ j ≤ k . We construct a map d r : T r → T r +1 , given by d r | C ( T ) ξ : C ( T ) ξ → M { j | ξ j =0 } C ( T ) ξ + j ⊂ T r +1 , d r | C ( T ) ξ := X { j | ξ j =0 } ( − p ( ξ,j ) d ξ,j , for all | ξ | = r . Note that d r +1 ◦ d r = 0. We write F ( T ) := (cid:0) k M r =0 C ( T ) r , d = X r d r (cid:1) ∈ Bimod G dg ( H n , H m ) . Remark 6.9.
Note that if T is made of a single crossing, then F ( T ) ∼ = Cone (cid:16) ϕ W (cid:0) F ( T ) (cid:1) F ( W ) −−−→ F ( T ) (cid:17) [1] , where W is the saddle from the 0-resolution of the crossing to its 1-resolution. DD KHOVANOV HOMOLOGY FOR TANGLES 43
Composition of tangles.
The dg-bimodule F ( T ) is nicely behaved with respect tocomposition of tangles. Proposition 6.10.
Let T ′ be an ( m ′ , n ) -tangle and T an ( n, m ) -tangle. There is an iso-morphism F ( T ′ ) ⊗ n F ( T ) ≃ −→ F ( T ′ T ) , induced by the composition maps µ [ T ′ ξ ′ , T ξ ] .Proof. Let k and k ′ be the number of crossings in T and T ′ respectively. Take ξ ∈ { , } k and ξ ′ ∈ { , } k ′ . Consider the sequence of isomorphisms C ( T ′ ) ξ ′ ⊗ n C ( T ) ξ = ϕ W ξ ′ (cid:0) F ( T ′ ξ ′ ) (cid:1) ⊗ n ϕ W ξ (cid:0) F ( T ξ ) (cid:1) β Wξ ′ ,Wξ −−−−−→ ϕ W ξ ′ • W ξ (cid:0) F ( T ′ ξ ′ ) ⊗ n F ( T ξ ) (cid:1) µ [ T ′ ξ ′ ,T ξ ] −−−−−→ ϕ W ξ ′ • W ξ (cid:0) F ( T ′ ξ ′ T ξ ) (cid:1) ∼ = ϕ W ξ ′⊔ ξ (cid:0) F ( T ′ ξ ′ T ξ ) (cid:1) = C ( T ′ T ) ξ ′ ⊔ ξ , where the last isomorphism comes from the fact that W ξ ′ • W ξ ∼ = W ξ ′ ⊔ ξ . It is an isomorphismthanks to Proposition 6.5. Thus, we only need to show that both diagrams C ( T ′ ) ξ ′ ⊗ n C ( T ) ξ C ( T ′ T ) ξ ′ ⊔ ξ C ( T ′ ) ξ ′ + i ⊗ n C ( T ) ξ C ( T ′ T ) ξ ′ ⊔ ξ +( i + k ) µd ξ ′ ,i ⊗ d ξ ′⊔ ξ,i + k µ C ( T ′ ) ξ ′ ⊗ n C ( T ) ξ C ( T ′ T ) ξ ′ ⊔ ξ C ( T ′ ) ξ ′ ⊗ n C ( T ) ξ + j C ( T ′ T ) ξ ′ ⊔ ξ + jµ ⊗ d ξ,j d ξ ′⊔ ξ,j µ (26)commute.In this proof, we write: C := F ( T ξ ) , C ′ := F ( T ′ ξ ′ ) , C ′′ := F ( T ′ ξ ′ T ξ ) , C ′ := F ( T ξ ′ + i ) , C ′′ := F ( T ′ ξ ′ + i T ξ ) ,ϕ := ϕ W ξ , ϕ ′ := ϕ W ξ ′ , ϕ ′′ := ϕ W ξ ′⊔ ξ , ϕ ′ := ϕ W ξ ′ + i , ϕ ′′ := ϕ W ( ξ ′ + i ) ⊔ ξ , and also: ϕ i ′ := ϕ W ξ ′ ,i , F ′ i := F ( W ξ ′ ,i ) , F ′′ i + k := F ( W ξ ′ ⊔ ξ,i + k ) . We also write ϕ ′ • ϕ := ϕ W ξ ′ + i • W ξ etc. Note that ϕ ′′ = ϕ ′ • ϕ and ϕ ′′ = ϕ ′ • ϕ . Then, the diagram on the left of Eq. (26) factorizes as: ϕ ′ ( C ′ ) ⊗ n ϕ ( C ) ϕ ′ ◦ ϕ i ′ ( C ′ ) ⊗ n ϕ ( C ) ϕ ′ ( C ′ ) ⊗ n ϕ ( C ) ϕ ′ • ϕ (cid:0) ϕ i ′ ( C ′ ) ⊗ n C ) (cid:1) ϕ ′ • ϕ (cid:0) C ′ ⊗ n C (cid:1) ϕ ′ • ϕ ( C ′ ⊗ n C ) ( ϕ ′ • ϕ ) ◦ ( ϕ i ′ • ϕ Tξ ) (cid:0) C ′ ⊗ n C (cid:1) ϕ ′ • ϕ (cid:0) C ′ ⊗ n C (cid:1) ϕ ′′ ( C ′ ⊗ n C ) ϕ ′′ ◦ ( ϕ i ′ • ϕ Tξ ) (cid:0) C ′ ⊗ n C (cid:1) ϕ ′′ (cid:0) C ′ ⊗ n C (cid:1) ϕ ′′ ( C ′′ ) ϕ ′′ ◦ ( ϕ i ′ • ϕ Tξ )( C ′′ ) ϕ ′′ ( C ′′ ) ϕ Hξ ′ ,i ⊗ β Wξ ′ ,Wξ F ′ i ⊗ β Wξ ′ + i,Wξ f β Wξ ′ + i,Wξ f F ′ i ⊗ β Wξ ′ ,i, f ϕ H f F ′ i ⊗ f ϕ H µ f F ′ i ⊗ µ f µϕ H F ′′ i + k where H : W ξ ′ • W ξ ⇒ ( W ξ ′ + i • W ξ ) ◦ ( W ξ ′ ,i •
1) is a locally vertical change of chronology.We claim the exterior square commutes. We can suppose we are applying the maps onsome element x ′ ⊗ x with | x ′ | = ( T ′ ξ ′ , p ′ ) ∈ Hom G ( b, c ) and | x | = ( T ξ , p ) ∈ Hom G ( a, b ). Wecompute each face of the diagram: • For the face f , we first observe that the contribution from β in the definition of β W ξ ′ ,W ξ together with ϕ H computes ı (cid:0) W cba ( T ′ , T ) ◦ (1 ¯ c W ξ ′ b ⊗ ¯ b W ξ a ) ⇒ (1 ¯ c W ξ ′ + i • W ξ a ) ◦ (1 ¯ c W ξ ′ ,i • T ξ a ) ◦ W cba ( T ′ ξ ′ , T ξ ) (cid:1) ). Moreover, the contribution from β in β W ξ ′ + i ,W ξ and in β W ξ ′ ,i , together with ϕ H ξ ′ ,i compute the same change of chronology.Therefore only remains the contribution from β , β ′ and β ′′ . Both β and β ′ arezero in all cases, and β ′′ is zero for β W ξ ′ ,i , . Then, we observe that the contributionof β ′′ is the same in both β W ξ ′ + i ,W ξ and β W ξ ′ ,W ξ , being λ R ( p ′ , deg(1 ¯ b W ξ a ). Thus, f commutes. • The face f commutes by functoriality of the bifunctor β W ξ ′ + i ,W ξ . • The face f commutes up to the factor β W ξ ′ ,i , ( | x ′ | , | x | ) − = ı (cid:0) (1 ¯ c W ξ ′ ,i • T ξ a ) ◦ W cba ( T ′ ξ ′ , T ξ ) ⇒ W cba ( T ′ ξ ′ + i , T ξ ) ◦ (1 ¯ c W ξ ′ ,i b ⊗ ¯ b T ξ a ) (cid:1) − , in the sense that the top path equal the bottom path times this factor. DD KHOVANOV HOMOLOGY FOR TANGLES 45 • The faces f and f trivially commute. • The face f commutes by naturality of ϕ H . • The face f commutes up to the factor ı (cid:0) ¯ c W ξ ′ ⊔ ξ,i + k a ◦ W cba ( T ′ ξ ′ , T ξ ) ⇒ W cba ( T ′ ξ ′ + i , T ξ ) ◦ (1 ¯ c W ξ ′ ,i b ⊗ ¯ b T ξ a ) (cid:1) , since both path are related by this locally vertical change of chronology.Since W ξ ′ ⊔ ξ,i + k = W ξ ′ ,i • T ξ , we obtain that the contributiuon of f cancels with the oneof f . All other faces being commutative, we conclude the outer square commutes.The commutativity of the second square follows from a similar argument, with a few ad-ditional subtleties. First, the computation of f differs a little bit, but is still commutativethanks to the fact that: λ R ( p ′ , deg(1 ¯ b W ξ a )) = λ R ( p ′ , deg(1 ¯ b W ξ + j a ) + deg(1 ¯ b W ξ,j a ))= λ R ( p ′ , deg(1 ¯ b W ξ + j a )) λ R ( p ′ , deg(1 ¯ b W ξ,j a )) . Secondly, for f , we obtain β T ′ ξ ′ ,W ξ,j = ı ( H ′′ ) λ R ( p ′ , deg(1 ¯ b W ξ,j a )) , for some change of chronology H ′′ : (1 ¯ c T ′ ξ ′ • W ξ,j a ) ◦ W cba ( T ′ ξ ′ , T ξ ) ⇒ W cba ( T ′ ξ ′ , T ξ + j ) ◦ (1 ¯ c T ′ ξ ′ b ⊗ ¯ b W ξ,j a ). The face f is computed as x ′ xW ξ,j = λ R ( p ′ , deg(1 ¯ b W ξ,j a )) x ′ xW ξ,j = ı ( H ′′ ) λ R ( p ′ , deg(1 ¯ b W ξ,j a )) ′ x ′ T ξ • W ξ,j x thus giving a factor cancelling with the one of f . This ends the proof. (cid:3) Reidemeister moves.
Write ⊗ H := ⊗ L n ≥ H n . Let T be an oriented ( n, m )-tangle.We can decomposes it as a composition of elementary tangles T = T k · · · T . If T i is acup or cap elementary tangle, then we put Kh( T i ) := F ( T i ). If T i is a crossing, then weassociate to it the dg-bimodule given by the following rule:Kh ! := Cone ( ϕ ) F ! F (cid:0) (cid:1) −−−−−−−−→ F ! {− , } , Kh ! := Cone F ! F (cid:0) (cid:1) ◦ ϕ H −−−−−−−−−−→ (cid:0) ϕ (cid:1) − F ! [1] { , } , where ϕ H : Id ⇒ (cid:0) ϕ (cid:1) − ◦ ϕ . Then, we defineKh( T ) := Kh( T k ) ⊗ H · · · ⊗ H Kh( T ) . Proposition 6.11.
For each T , there exists a shifting functor ϕ W v and an ℓ ∈ Z such that ϕ W v (Kh( T ))[ ℓ ] ∼ = F ( T ) .Proof. If T is an elementary tangle then we haveKh ! { , } ∼ = F ! , (cid:0) ϕ (cid:1) Kh ! [ − { , − } ∼ = F ! . Note that we obtain the second isomorphism thanks to the fact we twist the operator F (cid:0) (cid:1) by ϕ H , so that the following diagram commutes: (cid:0) ϕ (cid:1) F ! (cid:0) ϕ (cid:1) ◦ (cid:0) ϕ (cid:1) − F !(cid:0) ϕ (cid:1) F ! F ! F (cid:0) (cid:1) ◦ ϕ H F (cid:0) (cid:1) ϕ H Then, the general result follows from Proposition 6.10. (cid:3)
Lemma 6.12.
Consider two (plane diagrams of ) oriented ( n, m ) -tangles T and T ′ suchthat T ′ is obtained from T by a planar isotopy. Then Kh( T ′ ) ∼ = Kh( T ) in D G ( H n , H m ) .Proof. Suppose that we exchange the i -th crossing with the ( i + 1)-th one in T to obtain T ′ . Take ξ ′ := σ i ξ where σ i ∈ S k is the simple transposition that acts by exchanging ξ i with ξ i +1 in ξ . Then, T ′ ξ ′ is equivalent to T ξ . Thus, we have ϕ H : C ( T ) ξ = ϕ W ξ (cid:0) F ( T ξ ) (cid:1) ≃ −→ ϕ W ′ ξ ′ (cid:0) F ( T ′ ξ ′ ) (cid:1) = C ( T ′ ) ξ ′ , DD KHOVANOV HOMOLOGY FOR TANGLES 47 where H : W ξ ⇒ W ′ ξ ′ is a locally vertical change of chronology. For ξ j = 0 and j ′ := σ i ( j ),consider the following diagram: C ( T ) ξ C ( T ) ξ + j ϕ W ξ + j ◦ ϕ W ξ,j ( F ( T ξ )) ϕ W ′ ξ ′ + j ′ ◦ ϕ W ′ ξ ′ ,j ′ ( F ( T ′ ξ ′ )) C ( T ′ ) ξ ′ C ( T ′ ) ξ ′ + j ′ , d ξ,j ϕ Hξ,j ϕ H ϕ H ′ F ( W ξ,j ) ϕ H ′ F ( W ′ ξ ′ ,j ′ ) ϕ H ′ ξ ′ ,j ′ d ′ ξ ′ ,j ′ where H ′ : W ξ + j ⇒ W ′ ξ ′ + j ′ is a locally vertical change of chronology. The left part com-mutes thanks to Proposition 2.3. The upper and lower part commute by definition of thedifferential. The right part commutes since W ξ,j is equivalent to W ′ ξ ′ ,j ′ . Thus, the wholediagram commutes, and F ( T ′ ) ∼ = F ( T ). By consequence, Kh( T ′ ) ∼ = Kh( T ).Suppose that T ′ is obtained from T by the following local planar isotopy: → Suppose the crossing in the pictured disk is the i -th one. Then we have T ξ ∼ = T ′ ξ for all ξ . Moreover, for ξ i = 0, there is a change of chronology H ξ : W ξ ⇒ W ′ ξ for all ξ , givenby reversing the orientation of the saddle above the i -th crossing. For ξ i = 1 we have anequivalence W ξ ∼ = W ′ ξ ′ . Similarly, there is H W ξ,j : W ξ,j ⇒ W ′ ξ,j , which is trivial whenever j = i . Then, consider the following diagram: C ( T ξ ) C ( T ξ + j ) ϕ W ξ + j ◦ ϕ W ξ,j ( F ( T ξ )) ϕ W ′ ξ + j ◦ ϕ W ′ ξ,j ( F ( T ′ ξ )) C ( T ′ ξ ) C ( T ′ ξ + j ) , d ξ,j ϕ Hξ,j ϕ Hξ ϕ Hξ + j F ( W ξ,j ) ϕ Hξ + j ◦ HWξ,j F ( W ′ ξ,j ) ϕ H ′ ξ,j d ′ ξ,j It commutes for j = i for the same reasons as above. For j = i , it also commutes since F ( W ′ ξ + j ) ◦ ϕ H Wξ,j = F ( W ξ,j ). The other cases are similar, finishing the proof. (cid:3) Lemma 6.13.
Consider two oriented ( n, m ) -tangles T and T ′ . Suppose T differs from T ′ only in a small region where we have T ⊃ and T ′ ⊃ Then, there is an isomorphism
Kh( T ) ∼ = Kh( T ′ ) in D G ( H n , H m ) .Proof. We can decompose T such that T = T T C T and T ′ = T T I T where T C =and T I = . We will show that T C ∼ = T I and the result will from fact that Kh( T i ) iscofibrant both as left and as right module, thanks to Proposition 6.4.No matter the orientation on T , we haveKh ∼ = Cone ( ϕ ) F F (cid:0) (cid:1) −−−−−−−−−→ F {− , } . Then, as in [22, Lemma 7.3], we have a quasi-isomorphism Kh( T C ) ≃ −→ Kh( T I ) given by thevertical map in the following diagram:( ϕ ) F {− , } F {− , } F F (cid:0) (cid:1) F (cid:0) (cid:1) Note that the vertical map is graded. (cid:3)
Lemma 6.14.
Consider two oriented ( n, m ) -tangles T and T ′ . Suppose T differs from T ′ only in a small region where we have T ⊃ and T ′ ⊃ Then, there is an isomorphism
Kh( T ) ∼ = Kh( T ′ ) in D G ( H n , H m ) .Proof. The proof is similar to the one of Lemma 6.13 except thatKh ∼ = Cone F F (cid:0) (cid:1) ◦ ϕ H −−−−−−−−−−−→ (cid:0) ϕ (cid:1) − F [1] { , } , DD KHOVANOV HOMOLOGY FOR TANGLES 49 and the quasi-isomorphism looks like F F { , } (cid:0) ϕ (cid:1) − F { , } F (cid:0) (cid:1) ◦ ϕ H F (cid:0) (cid:1) which is again graded. (cid:3) Lemma 6.15.
Let T and T ′ be two oriented ( n, m ) -tangles such that T ′ differs from T only in a small region where T ⊃ and T ′ ⊃ Then, there is an isomorphism T ∼ = T ′ {− , } in D G ( H n , H m ) .Proof. Using Proposition 5.13, we obtain that Kh( ) looks like the first row in thediagram below, and we construct a graded map Kh( T ′ ) → Kh( T ) by the vertical arrows: F {− , } (cid:0) ϕ ( − , (cid:1) F (cid:0) ϕ (0 , (cid:1) F (cid:0) ϕ ( − , (cid:1) F F {− , } −F where the middle part is in homological degree zero. Up to composing the vertical mapswith a change of chronology, we obtain a commutative diagram. By [22, Lemma 7.5], weknow the resulting vertical maps yield a quasi-isomorphism. (cid:3) Lemma 6.16.
Let T and T ′ be two oriented ( n, m ) -tangles such that T ′ differs from T only in a small region where T ⊃ and T ′ ⊃ Then, there is an isomorphism T ∼ = (cid:0) ϕ (0 , (cid:1) T ′ in D G ( H n , H m ) .Proof. We obtain that Kh( ) looks like (cid:0) ϕ (0 , (cid:1) FF {− , } F { , }F {− , } where the middle part is in homological degree zero. Then, we verify we can use a similarunderlying quasi-isomorphism as in Lemma 6.15, and it is graded. (cid:3) Lemma 6.17.
Let T and T ′ be two oriented ( n, m ) -tangles such that T ′ differs from T only in a small region where T ⊃ and T ′ ⊃ Then, there is an isomorphism T ∼ = T ′ in D G ( H n , H m ) .Proof. We will suppose all strands are oriented upward, the other cases being similar.Then, we haveKh ∼ = Cone F F (cid:0) (cid:1) ◦ ϕ H −−−−−−−−−−−−→ ( ϕ ) − F [1] { , } , Kh ∼ = Cone F F (cid:0) (cid:1) ◦ ϕ H ′ −−−−−−−−−−−−→ ( ϕ ) − F [1] { , } . By Lemma 6.15 and Lemma 6.16, we obtainKh ∼ = Cone F {− , } F L −→ ( ϕ ) − F [1] { , } , Kh ∼ = Cone F {− , } F R −→ ( ϕ ) − F [1] { , } , for some maps F L and F R obtained by composition. DD KHOVANOV HOMOLOGY FOR TANGLES 51
Consider the following diagram ( ϕ ) − FF {− , } ( ϕ ) − F fF L F R where f is the obvious graded isomorphism obtained after decomposing ( ϕ ) − F and ( ϕ ) − F , and applying the corresponding changes of chronology. We ob-serve that the diagram is commutative, since the underlying maps are given by the samecobordisms, which can be thought as all being normalized through changes of chronology.This implies that Kh( T ) ∼ = Kh( T ′ ). (cid:3) Theorem 6.18. If T and T ′ are isotopic ( n, m ) -tangles, then there exists a shifting functor ϕ W v such that there is an isomorphism ϕ W v (cid:0) Kh( T ) (cid:1) ≃ −→ Kh( T ′ ) , in D G ( H n , H m ) .Proof. This follows immediatly from Lemma 6.13, Lemma 6.14, Lemma 6.15, Lemma 6.16,and Lemma 6.17. (cid:3)
Tangle invariant.
Consider the degree collapsing map κ : Hom G → Z given by κ ( t, ( p , p )) = p + p for ( t, ( p , p )) ∈ Hom G ( a, b ).For t ∈ B mn , let F q ( t ) be the Z × G -graded space given by the same elements as F ( t ) butwhere m ∈ F q ( t ) has degreedeg Z ×G ( m ) := ( κ (deg G ( m )) + n, deg G ( m )) . We denote deg q ( m ) for the Z -degree of m , and we refer to it as quantum degree .Note that for a cobordism W : t → t ′ , then F ( W ) : F q ( t ) → F q ( t ′ ) is homogenenouswith respect to the quantum degree, and is of degreedeg q ( F ( W )) = { − } . Moreover, the composition map µ [ t, t ′ ] preserves the quantum degree for all t, t ′ ∈ B •• . We extend the shifting functor ϕ W ( v ,v to a shifting functor for the Z × G -grading bysetting deg q (cid:0) ϕ W ( v ,v ( m )) := deg q ( m ) + v + v + deg q ( F ( W )) , for all m ∈ M ∈ Mod Z ×G . All the results in Section 5 extend directly to the Z × G -case,using the same compatibility maps, in the sense that all isomorphisms involved are gradedwith respect to the quantum degree.All this means H n is Z -graded, and F q ( t ) is a Z -graded H m - H n -bimodule (in the G -graded sense). Furthermore, the differential d ξ,j of Section 6.2 also preserves the quantumgrading. Thus, we can define a F q ( T ) in a similar fashion, using F q ( t ) instead of F ( t ), andit is a Z -graded dg-bimodule. We define similarly Kh q ( T ).Consider the additive subcategory BIMOD q ( H m , H n ) ⊂ BIMOD G ( H m , H n ) with ob-jects being Z × G -bimodules and maps are homogeneous with respect to the G -gradingand preserve the quantum grading. Let K OM q ( H m , H n ) be the corresponding homotopycategory. We can think of Kh q ( T ) as an object of K OM q ( H m , H n ). Remark 6.19.
Note that since we consider maps that do not preserve the G -grading, itmeans the homology of an object in K OM q ( H m , H n ) is a Z × Z -graded (for the homologicaland the quantum grading) R -module. Theorem 6.20 ( Gluing property ) . Let T ′ be an oriented ( m ′ , n ) -tangle and T an oriented ( n, m ) -tangle with compatible orientations. There is an isomorphism Kh q ( T ′ ) ⊗ n Kh q ( T ) ∼ = Kh q ( T ′ T ) , in K OM q ( H m , H n ) .Proof. This follows from Proposition 6.10 together with the fact that the composition maps µ [ t ′ , t ] preserve the quantum degree. (cid:3) Theorem 6.21 ( Invariant property ) . If T and T ′ are isotopic ( n, m ) -tangles, then thereis an isomorphism Kh q ( T ) ≃ −→ Kh q ( T ′ ) , in K OM q ( H m , H n ) .Proof. By [22, § {− , } isinvariant for the quantum grading, and so is (cid:0) ϕ (0 , (cid:1) . (cid:3) Proposition 6.22. If T is a link, then H (Kh q ( T )) is isomorphic to the covering Khovanovhomology H cov ( T ) of T , as constructed in [22] .Proof. The cube of resolution we obtain is the same as the one constructed in [22, § DD KHOVANOV HOMOLOGY FOR TANGLES 53 the definition of d ξ,i , we obtain the same cube). Since the cube we have is anticommutive,it means the scalars we get from the changes of chronology yield a sign assignment (in thesense of [22, § § H (Kh q ( T )) ∼ = H cov ( T ). (cid:3) In particular, it means that if we take X = Y = Z = 1 (before computing homology),we recover the usual Khovanov homology of the tangle [18], and if we take X = Z = 1and Y = −
1, we obtain a tangle version of odd Khovanov homology [21], thanks to [22,Proposition 10.8]. 7.
A chronological half 2-Kac–Moody
We generalize the level 2 cyclotomic odd half 2-Kac–Moody from [26] to one that corre-sponds with the covering TQFT of [22]. Then, we construct a graded 2-action from it onBIMOD q ( H • , H • ) similar the one constructed by Brundan–Stroppel [5] in the even case.7.1. Chronological cyclotomic half 2-Kac–Moody.
Let k be a unital, commutativering. Fix some natural integer n >
0. Fix a set of type A n +1 simple roots Π := { α , . . . , α n } .We consider gl n +1 -type weights of the form w = ( w , . . . , w n ) where w i ∈ { , , } . For1 ≤ i ≤ n , we say there is an arrow F i : w → w ′ , whenever w ′ i − = w i − w ′ i = w i + 1 and w ′ j = w i for j = i − , i . Also, there is an arrowF (2) i : w → w ′′ , whenever there are arrows F i : w → w ′ and F i : w ′ → w ′′ . These arrows can be composedin the obvious way (but F i F i = F (2) i ). DefineCSeq := (cid:8) i ( ε r ) r · · · i ( ε )1 | ε ℓ ∈ { , } , i ℓ ∈ { , . . . , n } (cid:9) . To i = i ( ε r ) r · · · i ( ε )1 ∈ CSeq we assign a composition of arrows F i := F ( ε r ) i r · · · F ( ε ) i , whereF (1) i := F i .We also define a map p : Π × Π → Z × Z , p ( α i , α j ) := , if | i − j | > , ( − , , if j = i + 1 , (0 , − , if j = i − , (1 , , if j = i. In order to shorten notations, we will simply write p ij := p ( α i , α j ). Definition 7.1.
Let R be the R -algebra generated by string diagrams where there canbe simple and double strands, and they both carry a label in Π. Simple strands can alsocarry decorations in the form of dots and a double strand can split in two strands andvice versa (preserving the labels). These diagrams are equipped with a height functionsuch that there can not be two critical points (i.e. crossing, dot or splitter) at the same height. Composition of a diagram D ′ with a diagram D is given by putting D ′ on top of D , if the labels and the strands corresponds (i.e. we cannot connect a double strand witha single strand) and zero otherwise. Furthermore, these diagrams are Z × Z -graded wherethe generators are:deg R α i = ( − , − , deg R α i = (1 , , deg R α i α i = (0 , , deg R α i α j = p ij , deg R α i α j = 2 p ij , deg R α i α j = 2 p ij , deg R α i α j = 4 p ij . We consider these diagrams modulo graded planar isotopy , meaning that we can applyambient isotopies on the diagram at the cost of multiplying by a scalar whenever weexchange the height of critical points: . . .. . . W ′ . . . . . .. . . W = λ R ( | W | , | W ′ | ) . . .. . . W . . . . . .. . . W ′ Finally, let R be the quotient of R by the following local relations: α i = 0 , α i α j = i = j , − XY Z α i α i +1 + XY Z α i α i +1 if j = i + 1, Y Z α i α i − − Y Z α i α i − if j = i − α i α j otherwise,(27) α i α i − XY α i α i = α i α i = α i α i − XY α i α i (28) DD KHOVANOV HOMOLOGY FOR TANGLES 55 α i α j = λ R (( − , − , p ij ) α i α j α i α j = λ R ( p ij , ( − , − α i α j (29) α i α i ± + XY α i α i ± = 0 , (30) α i α i α i = α i α i α i (31) α i α j α k = λ R ( p jk , p ik ) λ R ( p jk + p ik , p ij ) α i α j α k for i = k = j = i or | i − k | > α i α j α k = Xλ R ( p jk , p ik ) λ R ( p jk + p ik , p ij ) α i α j α k for i = j and k = ± i or j = k and i = k ± − Y Z − α i α i +1 α i + Z − α i α i +1 α i = α i α i +1 α i XY Z − α i α i − α i − XZ − α i α i − α i = α i α i − α i (34) α i α i = α i α i α i = 0 , (35) α i = α i = − XY α i α i = 0 = α i α i (36) α i α j = λ R (cid:0) (1 , , p ij (cid:1) α i α j α i α j = λ R (cid:0) (1 , , p ij (cid:1) α i α j (37) α i α j = λ R (cid:0) (0 , , p ij (cid:1) α i α j α i α j = λ R (cid:0) (0 , , p ij (cid:1) α i α j (38)and the mirror along the vertical axis of Eq. (37) and Eq. (38). Remark 7.2.
Since whenever there are two dots on the same strand it is zero, we alsohave that α i α i α i = 0 , for all α i ∈ Π, and similarly if we have two double strands, or a double strand next to asingle strand.The algebra R also possesses a quantum grading given by collapsing the R -grading:deg q ( x ) = p + p whenever deg R ( x ) = ( p , p ). For i ∈ CSeq, we define the idempotent1 i ∈ R given by only vertical strands, with label determined by i ℓ and it is a simple strandwhenever ε ℓ = 1 or double strand when ε ℓ = 2. For example with i = i (1) j (1) i (2) k (1) we DD KHOVANOV HOMOLOGY FOR TANGLES 57 have: 1 i = α i α j α i α k Consider a weight w and i , j ∈ CSeq such that w ′ F j w and w ′ F i w for the sameweight w ′ . Any diagram D in 1 j R i can be decomposed as a composition of diagrams i r D r i r − D r − i r − · · · i D i with i = i and i r = j , and where there is exactly one generator(i.e. dot, splitter or crossing) in each D ℓ . We say that D is an illegal diagram for w ′ , w whenever there are no arrow w ′ F i ℓ w for some ℓ . For example with n = 2, the followingdiagram is illegal α α α for w = (2 , ,
0) and w ′ = (0 , , w ′ F F F w (it factorizesthrough a weight (3 , , i , F j ) := 1 j R i /I, I := (cid:0) { D ∈ j R i | D is illegal for w , w ′ } (cid:1) , where F i , F j : w → w ′ .We define the category Hom( w , w ′ ) where objects are given by formal direct sums offormal Z × Z -shifts of the compositions of the arrows F i , F (2) i and Id w going from w to w ′ .The morphisms are matrices of elements in HOM(F i , F j ) of quantum degree zero, with thecomposition given by multiplication in R . Definition 7.3.
Let U −R be the 2-category with: • objects are ( n + 1)-uples w = ( w , . . . , w n ) where w i ∈ { , , } ; • Hom U −R ( w , w ′ ) is given by Hom( w , w ′ ) as above; • horizontal composition of 1-morphism is concatenation of arrows; • horizontal composition of 2-morphisms is given by right-then-left juxtaposition ofdiagrams: . . .. . . W ′ ◦ . . .. . . W := . . .. . . W ′ . . .. . . W ∈ Hom(F i ′ F i , F j ′ F j ) , for W ′ ∈ Hom(F i ′ , F j ′ ) , W ∈ Hom(F i , F j ).As usual, specializing X = Y = Z = 1 recovers a level 2 cyclotomic half 2-Kac–Moodyalgebra (as in [20, 5]), and taking Y = − Ladder diagrams.
Given a weight w = ( w , . . . , w n ), we associate to it a stringdiagram with n + 1 vertical strands that are labeled by the elements w i . We picture theselabels by writing a strand with label 0 as dashed, label 1 as solid, and label 2 as doubledashed. We refer to the dashed and double dashed strands as invisible , and to the solidstrands as visible . For example, if we take w = (0 , , , ,
0) we obtain:
Then, to the arrow F i : w → w ′ we associate the following ladder diagram: w w . . . w i − w i − w i w i − w i +1 w i +1 +1 w i +2 w i +2 . . . w n w n where we consider the rung as a solid line, and thus as visible. To the arrow F (2) i : w → w ′ we associate w w . . . w i − w i − w i w i − w i +1 w i +1 +2 w i +2 w i +2 . . . w n w n where the rung is double dashed, and thus invisible. The composition of arrows translatesto stacking ladder diagram on top of each other, from bottom to top. Let L (F i ) for i ∈ CSeqbe the ladder diagram associated to F i .Because w i ∈ { , , } and w ′ i ∈ { , , } , there are only 4 possibilities for what theladder diagram of F i : w → w ′ looks like on the i -th and ( i + 1)-th strands:(39)For F (2) i : w → w ′ , there is only one possibility: Example 7.4.
Here is an example of a ladder diagramassociated to w ′ F F F (2)2 F w where w = (2 , , ,
1) and w ′ = (0 , , , DD KHOVANOV HOMOLOGY FOR TANGLES 59
The 2-functor.
Given a weight w we write s ( w ) := { w i ∈ w | w i = 1 } . Weconsider the full sub-2-category U −R of U −R with objects w being such that s ( w ) ≡ q where objects are H n for all n ≥
2- BIMOD q ( H n , H m ) := BIMOD q ( H n , H m ). The horizontal composition is given bytensor product ⊗ H . We will construct a 2-action U −R →
2- BIMOD q , similar to the onethat can be found in [5] (when X = Y = Z = 1 our 2-action coincides with the one in thereference). Remark 7.5.
A construction of a generalized covering arc algebra (in the sense of [7, 24])is possible, allowing to act with the whole U −R instead of U −R . For the sake of simplicity,we do not consider this case in this paper.Because of Eq. (39), we know that there is at most 2 visible lines meeting at each vertexof L (F i ) for all i ∈ CSeq. In particular, it means we can turn it into a flat tangle byremoving all the invisible lines and smoothing if necessary, so that we turn (39) into:We write L (F i ) for the corresponding flat tangle. Example 7.6.
For example, we have L −−→ This allows us to define a Z × Z -graded additive 1-map F L : U −R →
2- BIMOD q , with w H s ( w ) / , and F i : w → w ′ is sent to F ( L (F i )). Remark 7.7.
We use the term to emphasize the fact it is not a functor (the objectsand 1-morphisms in 2- BIMOD q do not form a category since 2- BIMOD q is a bicategory).The idea is to look at F (cid:0) L (F i ) (cid:1) , and to assign an homogeneous map of bimodules (inthe form of a cobordism) to each generating element of R , depending on the weight w .First, let us define Γ w ( i ) ∈ R asΓ w ( i ) := ( − XY ) { w j ∈ w | w j =1 for j ≤ i } , where the exponent is the number of visible lines on the left or equal to the i th verticalline in the ladder diagram associated to w . Then, we define F L as follow, supposing alldiagrams involved to be not illegal: • To a dot on a strand labeled α i , we associate Γ w ( i ) times the cobordism given bythe identity on L (F i ) with a dot above the rung. For example: F L α i : {− , − } → := Γ w ( i ) F : F ! → F ! • To the splitters we associate
F L α i : → := F ( ∅ ) F ! −−−−−−→ F ! and F L α i α i : → := F ! ( − XY )Γ w ( i ) F −−−−−−−−−−−−−−→ F ( ∅ ) • The crossings with at least a double strand are all sent to the identity cobordism. • To a crossing between two (single) strands of the same color, we associate:
F L α i α i : → := ( − XY )Γ w ( i ) F : F ! → F ! • To a crossing between two strands with | i − j | >
1, we associate the identitycobordism, since at the level of the tangles we are exchanging distant rungs, andthus the source and target are equivalent tangles.
DD KHOVANOV HOMOLOGY FOR TANGLES 61 • To a crossing between two strands with j = i − F L α i α i − : → F (cid:16) (cid:17) : F → F where the lines we drawn as dotted could be either dashed, solid or double dashed(depending on the weight w ). The arrow gives the orientation of the saddle point.Similarly, for j = i + 1 we put: F L α i α i +1 : → ( − XY )Γ w ( i ) F (cid:16) (cid:17) : F → F However, there is an issue here if we want
F L to be a map of R -algebra. Indeed, if weisotopy two distant crossings with labels j = i ±
1, it will produce a global scalar given bythe graded planar isotopy. On the other side, changing the chronology at the level of thecobordisms will be different, and it will depend on the closure of the tangle (the saddlepoint could be either a split or a merge). Hopefully, by introducing some grading shift on F ( L (F i )), we can fix this. The idea is to observe that if we shift the target F ! F (cid:0) (cid:1) −−−−−−−−→ ( ϕ (1 , ) F ! or the source (cid:0) ϕ (1 , (cid:1) F ! F (cid:0) (cid:1) −−−−−−−−→ F ! then the first map is of degree (0 , − j = i −
1, and the second map isof degree ( − , j = i + 1. Also, given a pair of maps g : M ′ → N ′ , f : M → N that are purely homogeneous and with degree that does not change the underlying tangle, we have that the following 2-diagram(40) N ′ ⊗ H MM ′ ⊗ H M N ′ ⊗ H NM ′ ⊗ H N ⊗ fg ⊗ ⊗ f g ⊗ ⇒ λ R ( | f | , | g | )commutes. Moreover, the degree shift introduced preserve the quantum grading, so thatthe spaces are isomorphic to their shifted version in BIMOD q ( H • , H • ).7.3.1. Normalized tangle.
We give an algorithm to construct the shifted version of
F L (F i ).Fix a ladder diagram L (F i : w → w ′ ). Let T := L (F i ) and ϕ := Id. Let k := 0 be acounting variable (i.e. a variable that will increase at each step of the algorithm). We read L (F i ) from bottom to top, decomposing it as L (F i ) = w r F i r w r − · · · w F i w , where w ′ = w r and w = w . Whenever we encounter the following situation:(41)i.e. w ℓ F i ℓ w ℓ − with ( w ℓ − ) i ℓ = 1, we follow the dashed strand until we possibly encounterone of the two following situations:(42)In the first case, we consider the chronological cobordism W k +1 : T k → T k +1 given byputting a saddle realizing a surgery over the arcs of the tangle diagrams corresponding to(41) and (42), with orientation to the top left, and we put ϕ k +1 := ϕ W (1 , k +1 ◦ ϕ k , k := k + 1 . Then, we continue reading L (F i ) from just above (41), that is starting at F i ℓ +1 . In thesecond case, we jump one case to the left and continue to follow the dashed strand, untilwe either attain the end of the ladder diagram or Eq. (42), repeating. When we reach theend of L (F i ), we put ϕ F i := ϕ k . Finally, we define F L (F i ) := ϕ F i (cid:0) F L (F i ) (cid:1) . Example 7.8.
For example, we have FL (cid:0) ϕ (1 , (cid:1) F DD KHOVANOV HOMOLOGY FOR TANGLES 63
We define
F L on objects as
F L . For the 2-morphisms, we need to throw in someextra changes of chronology in the recipe. First, we need to modify the 2-morphism for acrossing τ ij with | i − j | >
1. Suppose F j is obtained from F i by exchanging F i ℓ with F i ℓ +1 and | i ℓ − i ℓ +1 | >
1. In this situation, we put(43)
F L α i ℓ +1 α i ℓ := ϕ H τij ( F L (F i )) F L α i ℓ +1 α i ℓ , where ϕ H τij : ϕ F i ⇒ ϕ F j is the natural transformation obtained from Proposition 5.10.Furthermore, we need to add some change of chronology for a crossing τ ij with j = i +1. Consider the natural isomorphism ϕ H τij : ϕ F i ⇒ ϕ F j ◦ ϕ (1 , given by change ofchronology, and where the saddle corresponds with the saddle above the rungs associatedto the strands involved in the crossing. Then, we define F L ( τ ij ) by making the followingdiagram commutes: ϕ F i (cid:0) F L (F i ) (cid:1) ϕ F j (cid:0) F L (F j ) (cid:1) ϕ F j ◦ ϕ (1 , (cid:0) F L (F i ) (cid:1) FL ( τ ij ) ∼ ϕ Hτij ϕ F j ( FL ( τ ij )) where we recall that the grading shift functor twists the morphisms since they carry anon-trivial G -degree (see Section 4.7 and Section 5.5). Similarly, for j = i −
1, we put(44) ϕ F i (cid:0) F L (F i ) (cid:1) ϕ F j (cid:0) F L (F j ) (cid:1) ϕ F i ◦ ϕ (1 , ◦ ϕ (0 , (cid:0) F L (F i ) (cid:1) ϕ F i ◦ ϕ (1 , (cid:0) F L (F j ) (cid:1) FL ( τ ij ) ∼ ϕ H ϕ F i ◦ ϕ (1 , ( FL ( τ ij )) ∼ ϕ Hτij where ϕ H τji : ϕ F i ◦ ϕ (1 , ⇒ ϕ F j , and ϕ H : Id ⇒ ϕ (1 , ◦ ϕ (0 , . If we considera splitter γ : F i → F j , then there exists a natural isomorphism ϕ H γ : ϕ F i ⇒ ϕ F j (becauseat least one of the potential saddles above the spawned rungs is a merge and annihilatedwith the degree shift (1 , ϕ F i (cid:0) F L (F i ) (cid:1) ϕ F j (cid:0) F L (F j ) (cid:1) ϕ F i (cid:0) F L (F j ) (cid:1) FL ( γ ) ϕ F i ( FL ( γ )) ∼ ϕ Hγ Finally, for the other cases where f : F i → F i , we define F L ( f ) := ϕ F i (cid:0) F L ( f ) (cid:1) , where again the grading shift functor twists the morphism. Example 7.9.
Let us do some detailed example of such a computation. Let n = 5, w = (2 , , , , ,
0) and i = 5412. We have L (F i ) = and L (F i ) ∼ =We want to apply F L ( τ ) and F L ( τ ) in the two possible order, and see what happens.In particular, we want to compare it to the equality τ τ = Zτ τ obtained from gradedplanar isotopy. We have F (cid:16) (cid:17) ϕ (1 , F (cid:16) (cid:17) ϕ (1 , F (cid:16) (cid:17) ϕ (1 , ◦ ϕ (1 , F (cid:16) (cid:17) FL ( τ ) FL ( τ ) FL ( τ ) FL ( τ ) where the thick line represents a saddle cobordism as in Fig. 1, and they are all oriented tothe top or to the left. For the rest of the example, we suppose we have chosen an arbitraryclosure of the tangles, so that we can make actual computations. We start by computingthe maps on the top path. For τ , we have ϕ H τ = 1 and thus F L ( τ ) : ϕ (1 , F (cid:16) (cid:17) ( − XY ) F (cid:18) (cid:19) −−−−−−−−−−→ F (cid:16) (cid:17) . For τ , we have again ϕ H τ = Id, and thus we obtain F L ( τ ) : F (cid:16) (cid:17) ϕ H −−→ ϕ (1 , ◦ ϕ (0 , F (cid:16) (cid:17) ϕ (1 , F (cid:18) (cid:19) −−−−−−−−−−−−→ ϕ (1 , F (cid:16) (cid:17) , DD KHOVANOV HOMOLOGY FOR TANGLES 65 where ϕ H is computed as the product of the scalars obtained by the sequence of changesof chronology: −→ (1 , λ −→ (1 , λ R ((0 , ,C ) −−−−−−→ (1 , ◦ (0 , where the saddle is above the pair of arcs at the right, and where λ = 1 and C = ( − ,
0) ifthe saddle is a merge, and λ = Z and C = (0 , −
1) if it is a split. These are computedusing Fig. 1. Then, note that (1 , has degree ((1 ,
0) + C ), thus ϕ (1 , F (cid:16) (cid:17) = λ R (cid:0) (0 , − , (1 ,
0) + C ) (cid:1) F (cid:16) (cid:17) . Putting all this together, we obtain
F L ( τ τ ) = λ ( − XY Z ) F (cid:16) ◦ (cid:17) . Now, we do the bottom path. We start by τ giving F L ( τ ) : ϕ (1 , F (cid:16) (cid:17) ϕ H ′ −−→ ϕ (1 , ◦ ϕ (1 , ◦ ϕ (0 , F (cid:16) (cid:17) ϕ (1 , ◦ ϕ (1 , F (cid:18) (cid:19) −−−−−−−−−−−−−−−−−−−→ ϕ (1 , ◦ ϕ (1 , F (cid:16) (cid:17) ϕ Hτ −−−→ ϕ (1 , ◦ ϕ (1 , F (cid:16) (cid:17) We compute ϕ H ′ as we did for ϕ H yielding ϕ H ′ = λ ′ λ R ((0 , , C ′ ) , where λ ′ = 1 , C ′ = ( − ,
0) if the saddle is a merge, and λ ′ = Z, C ′ = (0 , −
1) otherwise.Then, we have ϕ (1 , ◦ ϕ (1 , F (cid:16) (cid:17) = λ R (cid:0) (0 , − , C ′′ + C ′ + (2 , (cid:1) F (cid:16) (cid:17) , where C ′′ = deg (cid:16) (cid:17) . Then, we compute ϕ τ as ϕ τ = λ R (cid:0) (1 , , C ) ◦ ϕ H ′′ ◦ λ R (cid:0) C ′′ + (1 , , (1 , (cid:1) , where ϕ H ′′ is the change of chronology exchanging the two saddles in . For τ we have F L ( τ ) : ϕ (1 , ◦ ϕ (1 , F (cid:16) (cid:17) ( − XY ) ϕ (1 , F (cid:18) (cid:19) ) −−−−−−−−−−−−−−−−→ ϕ (1 , F (cid:16) (cid:17) where ϕ (1 , F (cid:16) (cid:17) = λ R (cid:0) ( − , , C + (1 , (cid:1) F (cid:16) (cid:17) . Putting all these together, we obtain
F L ( τ τ ) = λ ′ ϕ H ′′ λ R ( C ′′ , (1 , − XY Z ) F (cid:16) ◦ (cid:17) . In order to be able to compare the two results, we will now assume the tangle is closed byputting two arcs at the bottom and at the top. Then, we have F (cid:16) ◦ (cid:17) = X F (cid:16) ◦ (cid:17) , and λ = Z , λ ′ = Z , C ′′ = ( − ,
0) and ϕ H ′′ = Z . Thus, we obtain F L ( τ τ ) = Z F L ( τ τ ), which agree as expected with τ τ = Zτ τ . We leave as an exerciseto the reader the computation of other cases.7.3.2. R -relations. Fix two weights w and w ′ with s ( w ) ≡ s ( w ′ ) ≡ Lemma 7.10.
We have that
F L : Hom U −R ( w , w ′ ) → BIMOD q ( H s ( w ) / , H s ( w ′ ) / ) is a well-defined 1-functor.Proof. We need to show that the image of
F L respects the relations Eq. (27) to Eq. (38),and the graded planar isotopies. Thanks to Lemma 7.11 below and Proposition 4.39, wecan work locally since for f : F i → F j the following diagram G -graded commutes: F L (F i F i F i ) F L (F i ) ⊗ H F L (F i F i ) F L (F i ) ⊗ (cid:0) F L (F i ) ⊗ F L (F i ) (cid:1) F L (F i F j F i ) F L (F i ) ⊗ H F L (F j F i ) F L (F i ) ⊗ (cid:0) F L (F i ) ⊗ F L (F i ) (cid:1) FL ( i , ii ) − FL (1 ◦ f ◦
1) 1 ⊗FL ( i , i ) − FL (1) ⊗FL ( f ◦ FL (1) ⊗FL ( f ) ⊗FL (1) ⇐ ⇐ FL ( i , ji ) 1 ⊗FL ( j , i ) Therefore, if we can show that
F L ( f ) = F L ( g ) for some g and |F L ( f ) | G = |F L ( g ) | G , thenwe have F L (1 ◦ f ◦
1) =
F L (1 ◦ g ◦ F L is stableunder graded planar isotopy, thanks to Eq. (40). Also, for similar reasons as in [26, Proofof Proposition 2.11], it is enough to show that
F L is stables under the relations Eq. (27) -Eq. (35), and the remaining ones will follow. It appears that we can already verify (mostof) the relations at the level of R ChCob • , so that we can do most of our computationsusing dotted cobordisms.We have that F L α i = 0 , F L α i α i = 0 , immediately from (6). DD KHOVANOV HOMOLOGY FOR TANGLES 67
We obtain
F L α i α i = F L α i α i for | i − j | > ϕ τ ji ◦ ϕ τ ij = 1.In order to compute the case j = i −
1, we first suppose the diagrams involved are notillegal. Then, we compute using Eq. (6) that in R ChCob • : λ = = XZ + Y Z where λ = Z if the first saddle in the left-most term is a merge, and λ = 1 if it is a split,and all saddles are oriented with arrows going either left or forward. Moreover, it is nothard to see by drawing the underlying ladder diagrams that we always haveΓ w ( j ) = ( − XY )Γ w ( i ) . Therefore, we conclude that
F L ( τ ji τ ij ) = λ − (cid:0) ( − XY ) XZ.
F L ( x j ) + Y Z.
F L ( x i ) (cid:1) , where we write τ ji and τ ij for the two crossings and x i , x j for the dots on the strands labeled α i and α j respectively. Looking at Eq. (44), we first note that we can compute H as thecomposition of changes of chronology −→ (1 , λ −→ (1 , λ R ((0 , ,C ) −−−−−−→ (1 , ◦ (0 , where C = (0 , −
1) if λ = Z , and C = ( − ,
0) if λ = 1. Recall that by definition of ϕ H ,we multiply by the coefficients written on the arrows. Moreover, the change of chronologyinvolved in the defintion of ϕ F i ◦ ϕ (1 , ( F L ( τ ij )) gives λ R (cid:0) deg( F L ( τ ij )) , (1 ,
0) + C ) (cid:1) = λ R (cid:0) (0 , − , (1 ,
0) + C ) (cid:1) = ( Y Z if C = (0,-1) , . Finally, ϕ τ ij = ϕ τ ji = 1, so that by putting all this together we obtain F L ( τ ji τ ij ) = − Y Z . F L ( x j ) + Y Z . F L ( x i ) . We now suppose that the diagram involving two crossings in Eq. (27) is illegal, but theones on the right part are not. We must show that the right part acts as zero. In this case,the ladder diagram must looks like one of the followings:In either ways, we have Γ w ( i ) = Γ w ( j ) and F L ( x i ) = F L ( x j ) because the two rungs areconnected by a visible strand. Thus F L ( x i ) = F L ( x j ) and the right part of Eq. (27) iszero.The case j = i + 1 is similar, but still carries enough differences to be worth explaininga bit. First, we have F L ( x i ) = ϕ (1 , (cid:0) F L ( x i ) (cid:1) , giving a scalar 1 when the first saddle is a merge, and XY Z when it is a split. The sameapplies for x j . Moreover, keeping the same conventions as before, the saddles carry now adifferent orientation, so that λ = Z when the first saddle is a merge, and λ = XY when itis a split. The factors Γ w ( i ) are also different so that now F L ( τ ji τ ij ) = λ − (cid:0) XY Z.
F L ( x j ) + ( − XY ) Z. F L ( x i ) (cid:1) . Also, ϕ H multiplies by λ ′ λ R ( C ′ , (0 , − , where λ ′ = 1 and C ′ = ( − ,
0) if the first saddleis a merge, and λ ′ = Z and C ′ = (0 , −
1) in the other case. Putting all this together, weobtain
F L ( τ ji τ ij ) = XY Z.
F L ( x j ) − XY Z.
F L ( x i ) . The proof for the illegal diagrams is essentially the same, and we leave the details to thereader.We note that Eq. (28) immediately follows from Eq. (6) and the definition of Γ w ( i ).For Eq. (29), if | i − j | >
1, then it is immediate. If j = i + 1 and considering the firstcase, then we first observe that Γ w ( j ) = Γ w ′ ( j ) where w ′ F j w . Each side of Eq. (29) issent by F L to one of the two path in the following diagram: ϕ (1 , F (cid:18) (cid:19) F (cid:18) (cid:19) ϕ (1 , F (cid:18) (cid:19) F (cid:18) (cid:19) F ! ϕ (1 , F F F ! DD KHOVANOV HOMOLOGY FOR TANGLES 69
By Eq. (4), we have that at the level of the underlying cobordims there is a difference bya factor of λ R ( C, ( − , − C depends on whether the saddle is a merge or a splitafter closing the tangle. Moreover, the grading shift functor on the left downard arrowmultiply by a factor of λ R (( − , − , (1 ,
0) + C ), so that in the end we have Eq. (29). Theother case and the cases j = i − Z × Z -degreesgiven by p ij because of the changes of chronology.Both Eq. (31) and Eq. (33) are trivially verified because they factorize through invalidweights.For Eq. (34) and j = i + 1, we first note that the ladder diagram must have one of thetwo following forms (otherwise all diagrams are illegal):or(45)In the first case, the right part of the left term in Eq. (34) acts by zero and the other onedecomposes as (cid:0) ϕ W (1 , (cid:1) F L Γ . FL ( τ ij ) −−−−−−→ F L ( − XY )Γ . FL ( τ ii ) −−−−−−−−−−→ F L ϕ W (1 , FL ( τ ji ) ◦ ϕ H −−−−−−−−−−−−→ (cid:0) ϕ W (1 , (cid:1) F L where Γ := Γ w ( i ) and W := . At the level of the underlying cobordism, we have byEq. (2) and Eq. (3) that = X where the death is negative as usual. Then, we compute using the same kind of argumentsas before that ϕ H multiplies by a factor Y Z and the grading shift functor ϕ W (1 , twists F L ( τ ji ) by λ R ((0 , − , (1 , − Y Z . In conclusion, we get
F L ( τ ji τ ii τ ij ) = − Y Z − w . In the second case of ladder diagram from Eq. (45), the left part of the left term in Eq. (34)gives zero. For the right part, we get
F L FL ( τ ij ) ◦ ϕ H −−−−−−−→ (cid:0) ϕ W (1 , (cid:1) F L Γ .ϕ W (1 , FL ( τ ii ) −−−−−−−−−−→ (cid:0) ϕ W (1 , (cid:1) F L Γ . FL ( τ ji ) −−−−−−→ F L For the underlying cobordism, we have =Moreover, ϕ H gives a factor Z and ϕ W (1 , basically does nothing since W is a merge. Inconclusion, we obtain F L ( τ ji τ ii τ ij ) = Z − w . The case j = i − F L . (cid:3) Horizontal composition.
Fix F j : w ′ → w ′′ and F i : w → w ′ . We write β F j , F i for thecompatibility map obtained from ϕ F j and ϕ F i .Consider the map F L ( j , i ) defined by making the following diagram commutes: F L (F j ) ⊗ H F L (F i ) F L (F j F i ) ϕ F j (cid:0) F (cid:0) L (F j ) (cid:1)(cid:1) ⊗ H ϕ F i (cid:0) F (cid:0) L (F i ) (cid:1)(cid:1) ϕ F ji (cid:0) F (cid:0) L (F j ) L (F j ) (cid:1)(cid:1) ϕ F j • ϕ F i (cid:0) F (cid:0) L (F j ) (cid:1) ⊗ H ( F (cid:0) L (F i ) (cid:1)(cid:1) ϕ F j • ϕ F i (cid:0) F (cid:0) L (F j ) L (F i ) (cid:1)(cid:1) FL ( j , i ) β F j , F i µ [ L (F j ) , L (F i )] h j , i where h j , i is given by is given by first adding a grading shift ϕ W v j , ij , i corresponding with thefact we potentially generated new pairs of (41)/(42) by gluing the two ladder diagramstogether, followed by a change of chronology to put it in normalized form. Note that h j , i is an isomorphism in BIMOD q ( H • , H • ). Therefore, since all the maps involved areisomorphism, we have that F L ( j , i ) is an isomorphism (potentially carrying a non-trivial G -degree though). DD KHOVANOV HOMOLOGY FOR TANGLES 71
Lemma 7.11.
The isomorphism
F L ( j , i ) : F L (F j ) ⊗ H F L (F i ) ≃ −→ F L (F j F i ) , is G -graded natural in F i and F j .Proof. Suppose we have g : F j → F j ′ and f : F i → F i ′ . We claim the following diagramcommutes:(46) ϕ F i ′ • F i (cid:0) F L (F i ′ ) ⊗ H F L (F i ) (cid:1) ϕ F i ′ • F i (cid:0) F L (F i ′ i ) (cid:1) ϕ F i ′ F L (F i ′ ) ⊗ H ϕ F i F L (F i ) ϕ F i ′ i (cid:0) F L (F i ′ i ) (cid:1) ϕ F j ′ F L (F j ′ ) ⊗ H ϕ F j F L (F j ) ϕ F j ′ j (cid:0) F L (F j ′ j ) (cid:1) ϕ F j ′ • F j (cid:0) F L (F j ′ ) ⊗ H F L (F j ) (cid:1) ϕ F j ′ • F j (cid:0) F L (F j ′ j ) (cid:1) µ h i ′ , i β F ′ i , F i FL ( g ) ⊗ H FL ( f ) FL ( g ◦ f ) β F ′ j , F j µ h j ′ , j up to ϕ H : | h j ′ , j | ◦ ( | g | • | f | ) ⇒ | g ◦ f | ◦ | h i ′ , i | . It is easier to restrict to the case where f or g is a generator (and the other one is the identity), and the general result will follow.The difference of contribution of µ and of the F L part of f and g in the upper andlower path of Eq. (46) is given by the following equation: yW g xW f = β W g ,W f ( | y | , | x | ) y xW g • W f where W f (resp. W g ) is the underlying cobordism of F L ( f ) (resp. F L ( g )). Moreover, by coherence of the compatibility maps, the following diagram commutes: yW i ′ xW i | g | • | f | W j ′ , j yW i ′ xW i W i ′ , i | g ◦ f | y xW i ′ i yW g W j ′ xW f W j W j ′ , j yW g xW f W j ′ j y xW g • W f W j ′ j ϕ H ω h i ′ , i ◦ β F ′ i , F i ω h j ′ , j ◦ β F j ′ , F j β Wg,Wf
Note that for a dot, splitter or crossing with j = ±
1, then ω coincides with the changeof chronology given by the definition of tensor product of maps in BIMOD G together withthe twist in the definition of the grading shift functors on ϕ F i ′ ( F L ( g )) and ϕ F i ( F L ( g )).Similarly, ω coincides with the twist in ϕ F i ′ i (( F L ( g ◦ f ))). Therefore, we conclude thatEq. (46) G -graded commutes in these cases. When f (or g ) is a crossing with j = i ± ω and ω also carry a coefficient coming from non-trivial changes of chronology thatcoincides with the ϕ H τji . (cid:3) Inspired by the definition of a weak 2-functor (also called pseudofunctor), we introducethe following:
Definition 7.12.
We say that a R -linear bicategory D is G -graded if any 2-morphismdecomposes as a finite sum of 2-morphisms carrying a degree W v ∈ I , and it is compatiblewith respect to the horizontal and vertical composition.A G -graded 2-functor P : C → D from a bicategory C to a G -graded bicategory D consistsof • for each object X ∈ C , and object P X ∈ D ; • for each hom-category Hom C ( Y, X ), a functor P Y,X : Hom C ( Y, X ) → Hom D ( P Y , P X ); • for each object X ∈ C , an invertible 2-morphism P Id X : Id P X → P X,X (Id X ); DD KHOVANOV HOMOLOGY FOR TANGLES 73 • for each each triple Z, Y, X ∈ C an isomorphism P Z,Y,X ( f, g ) : P Z,Y ( g ) ◦ P Y,X ( f ) ⇒ P Z,X ( g ◦ f ) , G -graded natural in g : Z → X and in f : Y → X , meaning that the followingdiagram P Z,Y ( g ) ◦ P Y,X ( f ) P Z,X ( g ◦ f ) P Z,Y ( g ′ ) ◦ P Y,X ( f ′ ) P Z,X ( g ′ ◦ f ′ ) P Z,Y,X ( f,g ) P Z,Y ( θ ) ◦ P Y,X ( η ) P Z,X ( θ ◦ η ) P Z,Y,X ( f ′ ,g ′ ) G -graded commutes for any pair of 2-morphisms θ : g → g ′ and η : f → f ′ .such that • the diagrams(47) Id P Y ◦ P Y,X ( f ) P Y,X ( f ) P Y,Y (Id Y ) ◦ P Y,X ( f ) P Y,X (Id Y ◦ f ) λ D P Id Y ◦ Id P Y,Y,X (Id Y ,f ) P Y,X ( λ C ) and(48) P Y,X ( f ) ◦ Id P X P Y,X ( f ) P Y,X ( f ) ◦ P X,X (Id X ) P Y,X ( f ◦ Id X ) ρ D P Y,X ( f ) ◦ P Id X P Y,X,X ( f, Id X ) P Y,X ( ρ C ) both G -graded commute; • the diagram(49) (cid:0) P Z,Y ( h ) ◦ P Y,X ( g ) (cid:1) ◦ P X,W ( f ) P Z,Y ( h ) ◦ (cid:0) P Y,X ( g ) ◦ P X,W ( f ) (cid:1) P Z,X ( h ◦ g ) ◦ P X,W ( f ) P Z,Y ( h ) ◦ P Y,W ( g ◦ f ) P Z,W (( h ◦ g ) ◦ f ) P Z,W ( h ◦ ( g ◦ h )) α D P Z,Y,X ( h,g ) ◦ Id Id ◦ P Y,X,W ( g,f ) P Z,X,W ( h ◦ g,f ) P Z,Y,W ( h,g ◦ f ) P Z,W ( α C ) G -graded commutes for each quadruple Z, Y, X, W ∈ C .with λ C , ρ C and α C are the left unitor, the right unitor and the associator in C respectively. Theorem 7.13.
The assignment
F L : U −R → - BIMOD q is a G -graded 2-functor. Proof.
If we denote
F L as P in the definition of a G -graded 2-functor, then we have • P w := H s ( w ) / , • P w ′ , w is the 1-functor from Lemma 7.10, • P Id w : H s ( w ) / → F L (1 w ) ∼ = H s ( w ) / is basically the identity, • P w ′′ , w ′ , w (F j , F i ) := F L ( j , i ).Commutativity of Eq. (47) and of Eq. (48) is immediate by definition of left and rightaction on a shifted bimodule (see Definition 4.15).By the coherence conditions of the compatibility maps, we obtain that the followingdiagram commutes: i ′′ i ′ i ( i ′′ , i ′ ) • i ′′ i ′ , i ) i ′′ i ′ i • ( i ′ , i )( i ′′ , i ′ i ) i ′′ i ′ i i ′′ i ′ i h i ′′ i ′ , i ◦ h i ′′ , i ′ ◦ β i ′′ i ′ , i ◦ β i ′′ , i ′ α G ◦ ϕ H h i ′′ , i ′ i ◦ h i ′ , i ◦ β i ′′ , i ′ i ◦ β i ′ , i α G where we denoted by i the grading shift ϕ F i and i ′ , i for W v i ′ , i i ′ , i etc, and where ϕ H : W v i ′′ i ′ , i i ′′ i ′ , i ◦ ( W v i ′ , i i ′ , i • ⇒ W v i ′′ , i ′ i i ′′ , i ′ i ◦ (1 • W v i ′ , i i ′ , i ) . Since these are all the coefficients appearing in the definition of
F L ( j , i ), we concludeEq. (49) G -graded commutes. (cid:3) Proposition 7.14.
The G -graded 2-functor F L is faithful (at the level of 2-hom spaces).Proof.
By the results in [5], we know that
F L is fully faithful after specializing X = Y = Z = 1. Thus, it is injective at the level of 2-morphisms before specialization. (cid:3) Odd tangle invariant.
In particular, by specializing X = Z = 1 and Y = −
1, weobtain a 2-action of the cyclotomic odd half 2-Kac–Moody from [26] on the bicategoryof quasi-associative bimodules over the odd arc algebra. Moreover, the complex of 1-morphisms constructed in the reference for an ( m, n )-tangle T coincides with Kh( T ) afterapplying the 2-functor F L . Thus, showing that
F L is full would be enough to prove
DD KHOVANOV HOMOLOGY FOR TANGLES 75 that the invariant constructed in [26] coincides with ours, and in particular with the oddKhovanov homology [21] for links thanks to Proposition 6.22.
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