EEQUIVARIANT ANNULAR KHOVANOV HOMOLOGY
ROSTISLAV AKHMECHET
Abstract.
We construct an equivariant version of annular Khovanov homology via theFrobenius algebra associated with U (1) × U (1)-equivariant cohomology of CP . Motivatedby the relationship between the Temperley-Lieb algebra and annular Khovanov homology,we also introduce an equivariant analogue of the Temperley-Lieb algebra. Contents
1. Introduction 12. Some link homology theories 32.1. Khovanov homology 32.2. U (2)-equivariant Khovanov homology 52.3. U (1) × U (1)-equivariant Khovanov homology 62.4. Inverting the discriminant and Lee homology 73. Annular Khovanov homology 84. Equivariant annular Khovanov homology 114.1. A preliminary observation 114.2. The equivariant annular TQFT G α D in equivariant annular homology 165. Dotted Temperley-Lieb category 19References 241. Introduction
In [Kh1] Khovanov introduced a categorification of the Jones polynomial by assigning achain complex
CKh ( D ) of graded abelian groups to a diagram D of an oriented link L ⊂ R .Reidemeister moves between link diagrams induce chain homotopy equivalences between thechain complexes, and the graded Euler characteristic of CKh ( D ) is the Jones polynomial of L . The chain complex is built by forming the so-called cube of resolutions and applying thetwo-dimensional TQFT corresponding to the Frobenius algebra H ∗ ( S ; Z ) = Z [ X ] / ( X ) . A crossingless diagram D is assigned a chain complex supported in homological degree zero byapplying the TQFT directly to D . In particular, the empty link is assigned H ∗ ( {∗} ; Z ) = Z ,while the unknot is assigned Z [ X ] / ( X ).Varying the TQFT has been explored extensively, [BN2, Kh3, Le, KR], and has provento be fruitful for topological applications, [Ra]. Of particular interest is the equivariant or universal theory, built using the Frobenius algebra A = Z [ E , E , X ] / ( X − E X + E ) a r X i v : . [ m a t h . QA ] A ug R. AKHMECHET with ground ring R = Z [ E , E ]. This is the Frobenius algebra associated with U (2)-equivariant cohomology of CP [Kh3]. It specializes to the original theory by setting E = E = 0 and to the Lee deformation [Le] by setting E = 0 , E = −
1. An equivariant versionof sl -homology was constructed in [MV], and a generalization to sl n -homology can be foundin [Kr].In another direction, Asaeda-Przytycki-Sikora [APS] defined homology for links in I -bundles over surfaces. The present paper concerns links in the solid torus, identified with A × [0 ,
1] where A = S × [0 ,
1] is the annulus. The APS construction in this case is knownas annular Khovanov homology or annular APS homology . It is a triply graded theory; inaddition to homological and quantum gradings, there is a third grading arising from thepresence of non-contractible circles in A . The APS annular chain complex may be obtainedby applying to the cube of resolutions the annular TQFT G : BN ( A ) → Z − ggmod , where BN ( A ) is the Bar-Natan category of the annulus, and k − ggmod denotes the categoryof bigraded modules over a ring k .This paper extends annular Khovanov homology to the equivariant setting. We work withthe Frobenius algebra A α = Z [ α , α , X ] / (( X − α )( X − α ))with ground ring R α = Z [ α , α ], which are the U (1) × U (1)-equivariant cohomology of CP and of a point, respectively [KR]. The Frobenius pair ( R α , A α ) is an extension of ( R, A ) byidentifying E , E with elementary symmetric polynomials in α , α , E (cid:55)→ α + α , E (cid:55)→ α α , so that the polynomial X − E X + E ∈ R [ X ] splits as ( X − α )( X − α ) in R α [ X ]. Weobserve in Section 4.1 that working over ( R, A ) cannot produce an equivariant version ofannular APS homology. There is a natural U (1) × U (1)-equivariant analogue BN α ( A ) of BN ( A ) where the local relations are dictated by the structure of A α .Our main construction is a TQFT G α , which, when applied to the cube of resolutions ofan annular link diagram, gives a U (1) × U (1)-equivariant version of annular homology. Theorem 1.1.
There exists a functor G α : BN α ( A ) → R α − ggmod such that the followingdiagram commutes BN α ( A ) R α − ggmod BN ( A ) Z − ggmod G α G where the vertical arrows are obtained by setting α = α = 0 . We define G α by choosing a suitable basis and using a filtration induced by an additionalannular grading, as in [Ro]. Given a collection of disjoint simple closed curves C ⊂ A ,each circle in C is assigned the module A α , with the module assigned to a trivial circleconcentrated in annular degree zero. The essential circles in C are naturally ordered. Foreach essential circle C in C we equip its module A α with a distinguished homogeneous basis,either { , X − α } or { , X − α } , depending in an alternating manner on the position of C .We show that maps assigned to cobordisms are non-decreasing with respect to the annulargrading. QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 3
A feature of the equivariant theory is that the dotted product cobordism on a non-contractible circle in A , , is not sent to the zero map. Algebraically, this says that multiplication by X on an essentialcircle is nonzero in the equivariant theory. On the other hand, this cobordism evaluates tozero in APS homology and also in the quantum annular homology [BPW].The paper is organized as follows. In Section 2.1 we review Khovanov homology using theframework of the Bar-Natan cobordism category [BN2]. The remaining parts of Section 2give an overview of Frobenius algebras A , A α , and A α D , following [KR]. Section 3 reviewsannular Khovanov homology. Our equivariant theory is defined in Section 4.2. In Section 4.3we study a further extension appearing in [KR], which is obtained by inverting an element D ∈ A α . We prove an analogue of [Le, Theorem 4.2], that for a k -component annular link, thehomology obtained by inverting D is free of rank 2 k . In Section 5 we recall the Temperley-Liebcategory and its relation to annular Khovanov homology, following observations in [BPW].This perspective leads to a natural equivariant analogue of the Temperley-Lieb category andalgebra, where strands may carry dots. Acknowledgements.
I would like to thank Mikhail Khovanov for suggesting this project,for many helpful discussions, and for comments on earlier versions of the paper. I would alsolike to thank my advisor Slava Krushkal. The author was supported by NSF RTG GrantDMS-1839968. 2.
Some link homology theories
We review Bar-Natan’s approach to Khovanov homology and describe four Frobeniusalgebras, all of which have appeared in the literature and yield homology for links in R .2.1. Khovanov homology.
We start with a brief overview of the Bar-Natan category BN and the construction of the chain complex [[ D ]] assigned to a link diagram D ; for a completetreatment see [BN2].First, recall the (dotted) Bar-Natan category BN . Let I := [0 ,
1] denote the unit interval.Objects of BN are formal direct sums of formally graded collections of simple closed curvesin the plane R . Morphisms are matrices whose entries are formal Z -linear combinations ofdotted cobordisms properly embedded in R × I , modulo isotopy relative to the boundary,and subject to the local relations shown in Figure 1. For the remainder of the paper, allcobordisms are assumed to possibly carry dots, unless specified otherwise.Let A = Z [ X ] / ( X ). The trace ε : A → Z , (cid:55)→ , X (cid:55)→ A a Frobenius algebra, with comultiplication A → A ⊗ A (cid:55)→ X ⊗ ⊗ XX (cid:55)→ X ⊗ X This is the Frobenius algebra underlying sl link homology [Kh1]. R. AKHMECHET (a)
Sphere (b)
Neck-cutting (c)
Dotted sphere (d)
Two dots
Figure 1.
Relations in BN The Bar-Natan relations (Figure 1) can be seen as arising from the structure of A in thefollowing way. Interpret the cup cobordism as the unit map η : Z → A , (cid:55)→ , the cap as the trace ε , and a dot as multiplication by X ∈ A . Then the sphere relationcorresponds to ε ( η (1)) = 0while the dotted sphere comes from ε ( X ) = 1. The two dots relation corresponds to therelation X = 0 in A . Neck-cutting is a topological incarnation of the algebraic relation y = Xε ( y ) + ε ( Xy ) , which holds for every y ∈ A .For a cobordism S ⊂ R × I , let d ( S ) denote the number of dots on S , and set the degreeof S to be(1) deg( S ) = − χ ( S ) + 2 d ( S ) . Note that the relations in Figure 1 are homogeneous. Define the quantum grading qdeg on A by setting(2) qdeg(1) = − X ) = 1 . Remark . The grading elsewhere in the literature [Kh1, BN1, BN2] is opposite that ofthe one appearing here. Moreover, viewing A as an algebra, it is more natural to set 1 and X in degrees 0 and 2, respectively, to make the multiplication grading-preserving. However,when viewing A as a module, degrees are balanced around 0 as above.For a ring k , let k − gmod denote the category of Z -graded k -modules and graded maps (ofany degree) between them. The Frobenius algebra A defines a (1 + 1)-dimensional TQFT,and it descends to a graded, additive functor(3) F : BN → Z − gmod . which is Z -linear on each morphism space. In fact, due to delooping [BN3], any such functoris determined by its value on the empty diagram.Let D be a diagram for an oriented link L ⊂ R . We recall the construction [[ D ]] from[BN2], which is a chain complex over the additive category BN . One first forms the cube QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 5
Figure 2.
The two smoothings at a crossing of resolutions as follows. Label the crossings of D by 1 , . . . , n . Every crossing may beresolved in two ways, called the and , shown in Figure 2. For each u = ( u , . . . , u n ) ∈ { , } n , perform the u i -smoothing at the i -th crossing. The resultingdiagram is a collection of disjoint simple closed curves in the plane R , and we denote itby D u . Thinking of elements of { , } n as vertices of an n -dimensional cube, decorate thevertex u by the smoothing D u .Next, let u = ( u , . . . , u n ) and v = ( v , . . . , v n ) be vertices which differ only in the i -thentry, where u i = 0 and v i = 1. Then the diagrams D u and D v are the same outside of a smalldisk around the i -th crossing. There is a cobordism from D u to D v , which is the obvioussaddle (1-handle attachment) near the i -th crossing and the identity (product cobordism)elsewhere. Denote this cobordism by d u,v , and decorate each edge of the n -dimensional cubeby these saddle cobordisms. This forms a commutative cube in the category BN . There isa way to assign s u,v ∈ { , } to each edge in the cube so that multiplying the edge map d u,v by ( − s u,v results in an anti-commutative cube (see [BN2, Section 2.7]).For u = ( u , . . . , u n ) ∈ { , } n , set | u | = (cid:80) i u i . Now, form the chain complex [[ D ]] over BN by setting [[ D ]] i = (cid:77) | u | = i + n − D u { n − − n + − i } in homological degree i , where n − , n + denote the number of negative and positive crossingsin D , and {−} is the upwards grading shift in BN . The differential is given on each summandby the edge map ( − s u,v d u,v . Anti-commutativity of the cube ensures that [[ D ]] is a chaincomplex.The relations in Figure 1 imply the S , T , and 4 T u relations from [BN2].
Theorem 2.2. ([BN2, Theorem 1])
If diagrams D and D (cid:48) are related by a Reidemeistermove, then [[ D ]] and [[ D (cid:48) ]] are chain homotopy equivalent. Thus to obtain link homology, it suffices to apply a functor from BN into an abeliancategory, and Theorem 2.2 guarantees that the homotopy class of the resulting chain complexis a link invariant. In particular, the TQFT (3) yields a chain complex CKh ( D ) := F ([[ D ]])of graded abelian groups. After reversing the quantum grading, this is the chain complexappearing in [Kh1, Section 7].2.2. U (2) -equivariant Khovanov homology. This section reviews the so-called U (2)-equivariant Frobenius pair, denoted ( R, A ). Although this extension is of general importance,it is not necessary for our construction in Section 4; in fact, in Section 4.1 we note that ananalogue of annular APS homology using (
R, A ) is not possible.Consider the graded ring R = Z [ E , E ]with deg( E ) = 2, deg( E ) = 4. The R -algebra A = R [ X ] / ( X − E X + E ) R. AKHMECHET equipped with the trace ε : A → R, (cid:55)→ , X (cid:55)→ R . The rings R and A are the U (2)-equivariant cohomology with Z coefficients of a point and CP , respectively [Kh3]. The Frobenius algebra A determinesa link homology theory as in Section 2.1, obtained by applying the corresponding TQFT tothe formal complex [[ D ]].2.3. U (1) × U (1) -equivariant Khovanov homology. In this section we review an exten-sion of the Frobenius pair (
R, A ). This extension was studied in [KR] and is central to ourconstruction in Section 4.Let R α = Z [ α , α ] , and consider the R α -algebra A α = R α [ X ] / (( X − α )( X − α )) . The trace ε α : A α → R α , (cid:55)→ , X (cid:55)→ . makes A α into a Frobenius algebra, with comultiplication∆ : A α → A α ⊗ A α (cid:55)→ ( X − α ) ⊗ ⊗ ( X − α ) X (cid:55)→ X ⊗ X − α α ⊗ . There is an inclusion (
R, A ) (cid:44) → ( R α , A α ) given by identifying E , E ∈ R with the elementarysymmetric polynomials in R α , E (cid:55)→ α + α E (cid:55)→ α α . As noted in [KR], R α and A α are the U (1) × U (1)-equivariant cohomology with Z coefficientsof a point and 2-sphere S , respectively.Let BN α denote the Bar-Natan category subject to relations coming from A α . Objectsof BN α are formal direct sums of formally graded collections of simple closed curves inthe plane R . Morphisms are matrices whose entries are formal R α -linear combinations ofdotted cobordisms properly embedded in R × I , modulo isotopy relative to the boundary,and subject to the local relations shown in Figure 3. As outlined in Section 2.1, thesetopological relations correspond to algebraic relations in the Frobenius algebra A α . Remark . We note that BN α is induced from the corresponding U (2)-equivariant cobor-dism category with relations dictated by ( R, A ), since the relations involve only symmetricpolynomials in α , α .For a cobordism S ⊂ R × I , define the degree of S as in (1), and put α , α ∈ R α indegree 2. Note that the relations in Figure 3 are homogeneous. The algebra A α is a free R α -module with basis { , X } . Using the same notation as in (2), define a grading qdeg on A α by setting(4) qdeg(1) = − X ) = 1 . QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 7 (a)
Sphere (b)
Neck-cutting (c)
Dotted sphere (d)
Two dots
Figure 3.
Relations in BN α Remark . Viewing A α as an R α -algebra, it is more natural to set 1 and X in degrees 0and 2, respectively, so that multiplication in A α is grading-preserving. When viewing A α as an R α -module with homogeneous basis { , X } according to the grading (4), the elements X − α and X − α should be interpreted as X − α · X − α ·
1, which are homogeneousof degree 1. In either case, multiplication by X is a degree 2 endomorphism of A α .The Frobenius algebra A α defines a two-dimensional TQFT, and it descends to a graded,additive functor(5) F α : BN α → R α − gmodwhich is R α -linear on each morphism space. Moreover, the following diagram commutes(6) BN α R α − gmod BN Z − gmod F α F where the vertical maps are obtained by setting α = α = 0.Given a diagram D for an oriented link L ⊂ R , form the chain complex [[ D ]] as describedin Section 2.1. We may view [[ D ]] as a chain complex over BN α . The relations in Figure 3imply the S , T , and 4 T u relations from [BN2], so by [BN2, Theorem 1], the homotopy classof [[ D ]] is an invariant of L . It follows that the chain complex obtained by applying F α to[[ D ]] is an invariant of L up to chain homotopy equivalence.2.4. Inverting the discriminant and Lee homology.
We recall from [KR] a furtherextension of the Frobenius pair ( R α , A α ). Let(7) D = ( α − α ) denote the discriminant of the quadratic polynomial ( X − α )( X − α ) ∈ R α [ X ]. Let R α D = R α [ D − ]denote the ring obtained by inverting D (equivalently, one may invert α − α ) and let A α D = A α ⊗ R α R α D R. AKHMECHET be the extension of A α to an R α D -algebra. Let F α D denote the composition BN α F α −→ R α − gmod → R α D − gmodwhere the second functor is extension of scalars, ( − ) ⊗ R α R α D . For a link L ⊂ R withdiagram D , let CKh α D ( D ) := F α D ([[ D ]])denote the resulting chain complex. It is an invariant of L up to chain homotopy equivalence,and we will denote its homology by Kh α D ( L ).The elements(8) e = X − α α − α , e = X − α α − α ∈ A α D . form a basis for A α D and satisfy e + e = 1 , e = e , e = e , e e = 0 , so that the algebra A α D decomposes as a product, A α D = R α D e × R α D e . With respect tothe basis { e , e } , comultiplication in A α D is simply given by∆( e ) = ( α − α ) e ⊗ e ∆( e ) = ( α − α ) e ⊗ e . (9)As noted in [KR, Section 1.2], the TQFT F α D is essentially the Lee deformation [Le]. By[Le, Theorem 4.2], the Lee homology of a k -component link is free (over Q ) of rank 2 k . Aquick alternate proof can be found in the final remark in [We]. The following is stated in[KR] without proof, but the arguments in [We] apply without modification. Proposition 2.5.
For a link L ⊂ R with k components, the homology Kh α D ( L ) is a free R α D -module of rank k . Annular Khovanov homology
We give an overview of annular Khovanov homology, also known as annular Asaeda-Przytycki-Sikora (APS) homology. It was originally defined in [APS] as part of a broadercategorification of the Kauffman bracket skein module of I -bundles over surfaces. A conve-nient reference for the annular setting is [GLW].Let A = S × I denote the annulus. An annular link is a link in the thickened annulus A × I , and its diagram is a projection onto the first factor of A × I . Embed A standardly in R as A = { x ∈ R | ≤ | x | ≤ } , so that an annular link diagram and all of its smoothings are drawn in the punctured plane R \ (0 , × . Figure 4 illustrates an example of an annular link diagram. By a circle in A we mean a smoothly and properly embedded S in A . There are two kinds of circles in A : trivial circles, which are contractible in A , and essential ones, which are not contractible.Let BN ( A ) denote the Bar-Natan category of the annulus. Its objects are formal directsums of formally bigraded collections of simple closed curves in A . Morphisms are matriceswhose entries are formal Z -linear combinations of dotted cobordisms properly embedded in QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 9
Figure 4.
An annular link diagram qdeg − adeg − v v X Figure 5.
Bigradings, where { , X } corresponds to trivial circles, and { v , v } correspond to essential ones. See also Remark 2.1. A × I , modulo isotopy relative to the boundary, and subject to the local relations shown inFigure 1. The bidegree of a cobordism S ⊂ A × I is defined to be(10) ( − χ ( S ) + d ( S ) , , where d ( S ) is the number of dots on S .For a ring k , denote by k − ggmod the category of Z × Z -graded k -modules and gradedmaps (of any bidegree) between them. We now describe the annular TQFT G : BN ( A ) → Z − ggmod , which will be additive, graded, and Z -linear on each morphism space.Let C ⊂ A be a collection of n trivial and m essential circles. Embed A × I standardlyinto R × I , and apply the TQFT F from Section 2.1, F ( C ) = A ⊗ n ⊗ A ⊗ m . Define a second grading, called the annular grading and denoted adeg, on F ( C ) in thefollowing way. A tensor factor A corresponding to a trivial circle is concentrated in annulardegree 0. For a factor A corresponding to an essential circle, let v = 1 , v = X. denote a basis for this copy of A , and setadeg( v ) = − v ) = 1 . Bigradings are summarized in Figure 5.The underlying abelian group of G ( C ) is F ( C ), and the bigrading is given by (qdeg , adeg).For a cobordism S ⊂ A × I , first view S as a surface in R × I and consider the map F ( S ).It is shown in [Ro, Section 2] that F ( S ) splits as a sum(11) F ( S ) = F ( S ) + F ( S ) + where F ( S ) preserves adeg and F ( S ) + increases adeg. Set G ( S ) := F ( S ) to be the adeg-preserving part. It follows from (11) that G is functorial with respect tocomposition of cobordisms. By construction, G ( S ) is a map of bidegree( − χ ( S ) + 2 d ( S ) , G is degree preserving on morphism spaces. We will refer to G as the annularTQFT .To distinguish the bigraded modules assigned to trivial and essential circles, write V = G ( C )if C is an essential circle, with basis written as { v , v } , and keep the notation A = G ( C )when C is trivial. Then if C ⊂ A consists of n trivial and m essential circles, the moduleassigned to C is G ( C ) = A ⊗ n ⊗ V ⊗ m . Given a diagram D for an oriented annular link L , form the chain complex [[ D ]] as de-scribed in Section 2.1. The construction is completely local and crossings are away from thepuncture × . Thus we may view [[ D ]] as a chain complex over BN ( A ), with Z -grading shifts {−} in BN rewritten as a Z × Z -grading shifts {− , } in BN ( A ). Isotopies of annular linksare described by Reidemeister moves away from the puncture, and it follows that the homo-topy class of [[ D ]], viewed as a chain complex over BN ( A ), is an invariant of L . Thereforethe chain complex(12) CKh A ( D ) := G ([[ D ]])is an invariant of L up to chain homotopy equivalence.An elementary cobordism is one that has a single non-degenerate critical point with respectto the height function A × I → I . It consists of a union of a product cobordism and a singlecup, cap, or saddle. An elementary cobordism S with ∂S consisting of trivial circles in A isassigned the same map by F and G . We record the maps assigned to the four elementarysaddles involving at least one essential circle, Figure 6. The vertical red arc is the centralaxis of A × I ⊂ R × I . V ⊗ A −→ Vv ⊗ (cid:55)→ v v ⊗ (cid:55)→ v v ⊗ X (cid:55)→ v ⊗ X (cid:55)→ V ⊗ V (II) −−→ A v ⊗ v (cid:55)→ v ⊗ v (cid:55)→ Xv ⊗ v (cid:55)→ Xv ⊗ v (cid:55)→ V (III) −−→ V ⊗ A v (cid:55)→ v ⊗ Xv (cid:55)→ v ⊗ X (15) A −−→ V ⊗ V (cid:55)→ v ⊗ v + v ⊗ v X (cid:55)→ X acts trivially on any essential circle. It follows that a cobordismwith a component that carries a dot and a closed curve which is nonzero in π ( A × I ) is as-signed the zero map. Thus G factors through the relation shown in Figure 7, called Boerner’srelation [Bo]. Indeed, for an essential circle C ⊂ A , there are no nonzero endomorphisms of G ( C ) with bidegree (2 , QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 11 (a)
Type (I) (b)
Type (II) (c)
Type (III) (d)
Type (IV)
Figure 6.
Saddles involving essential circles
Figure 7.
Boerner’s relation
The category BN ( A ) is monoidal, with monoidal product given by taking two copies A , A of A and gluing the boundary component S ×{ } of A to the boundary component S ×{ } of A . The annular TQFT G is evidently monoidal.4. Equivariant annular Khovanov homology
We are interested in an annular version of the theory outlined in Section 2.3. Precisely,the goal is to fill in the dashed arrow in the diagram BN α ( A ) R α − ggmod BN ( A ) Z − ggmod G α G where the vertical arrows are obtained by setting α = α = 0. Section 4.1 justifies workingwith the extension ( R α , A α ) rather than ( R, A ). The desired functor G α is defined in Section4.2. Maps assigned to saddle cobordisms can be found in (20)–(23). In Section 4.3 we invert D in the annular theory and show that the rank of the resulting homology depends only onthe number of components.4.1. A preliminary observation.
Before defining our equivariant annular TQFT, we notethat the U (2)-equivariant Frobenius pair ( R, A ) from Section 2.2 does not admit such a lift,under the minor assumption that modules assigned to circles are free.The ring R = Z [ E , E ] can be made bigraded, with bidegrees of E and E given by(2 ,
0) and (4 , M be a free Z × Z -graded R -module with basis m − , m + in bidegrees ( − , −
1) and (1 , g : M → M is an R -linear map ofbidegree (2 , g ( m − ) = nE m − for some n ∈ Z . In particular, if M is the module assigned to a single essential circle and g is the map assigned to the cobordism in Figure 8, then the relation X − E X + E = 0 in A implies(18) g ( m − ) − E g ( m − ) + E m − = 0 . Figure 8.
Dotted product cobordism on an essential circle qdeg − − adeg − v , v (cid:48) v , v (cid:48) X α , α Figure 9.
Bigradings, where { , X } corresponds to trivial circles, and { v , v } , { v (cid:48) , v (cid:48) } correspond to essential ones. See also Remark 2.4. However, (17) and (18) are incompatible.4.2.
The equivariant annular TQFT G α . Let BN α ( A ) denote the Bar-Natan categoryof the annulus subject to the relations determined by A α . Its objects are formal direct sumsof formally bigraded collections of simple closed curves in A . Morphisms are matrices whoseentries are formal R α -linear combinations of dotted cobordisms properly embedded in A × I ,modulo isotopy relative to the boundary, and subject to the local relations shown in Figure3. The bidegree of a cobordism S ⊂ A × I is given by (10). For an oriented annular link L with diagram D , the formal complex [[ D ]] over BN α ( A ) is an invariant of L up to chainhomotopy equivalence.Let C ⊂ A be a collection of circles, and view C as embedded in R . Consider F α ( C ) withthe following additional annular grading, denoted adeg as in Section 2.3. Define elements of A α , v = 1 , v = X − α ,v (cid:48) = 1 , v (cid:48) = X − α , with the annular gradings(19) adeg( v ) = adeg( v (cid:48) ) = − , adeg( v ) = adeg( v (cid:48) ) = 1 . Remark . The notation v , v was also used in Section 3. Setting α = α = 0 in theabove expressions recovers v , v in the non-equivariant setting.Both { v , v } = { , X − α } and { v (cid:48) , v (cid:48) } = { , X − α } is an R α -basis for A α . Togetherwith the quantum grading, these equip A α with two (isomorphic) structures of a bigraded R α -module, with the bigrading given by (qdeg , adeg). The ground ring R α lies in annulardegree 0.Let C ⊂ A consist of n trivial and m essential circles, with the essential circles orderedfrom innermost (closest to the puncture × ) to outermost. Define the annular grading on F α ( C ) = A ⊗ nα ⊗ A ⊗ mα QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 13 by declaring that every copy of A α corresponding to a trivial circle is concentrated in annulardegree 0 and that the copy of A α corresponding to the i -th essential circle (1 ≤ i ≤ m ) isgiven the homogeneous basis { v , v } = { , X − α } if i is odd and { v (cid:48) , v (cid:48) } = { , X − α } if i is even. In other words, the essential circles are assigned the homogeneous bases { , X − α } or { , X − α } in an alternating manner, with the innermost circle assigned { , X − α } .Bigradings are summarized in Figure 9As in Section 3, it is convenient to distinguish the modules assigned to essential and trivialcircles. Let V α and V (cid:48) α denote the module A α with homogeneous bases { v , v } and { v (cid:48) , v (cid:48) } ,respectively. Then for a collection of circles C ⊂ A , the i -th essential circle in C is assigned V α if i is odd and V (cid:48) α if i is even. We reserve the notation A α for the module assigned to atrivial circle. Note that interchanging α ↔ α also interchanges v ↔ v (cid:48) and v ↔ v (cid:48) . Lemma 4.2.
Let S ⊂ A × I be an elementary cobordism. Viewing S as a cobordism in R × I , the map F α ( S ) splits as a sum F α ( S ) = F α ( S ) + F α ( S ) where F α ( S ) preserves adeg and F α ( S ) increases adeg by .Proof. If the saddle component of S involves only trivial circles then the claim is immediate,since F α ( S ) = F α ( S ) in this case. We verify the claim for the four elementary cobordismsin Figure 6 by rewriting F α ( S ) in terms of the bases for the circles involved. Terms whereadeg is increased by 2 are boxed. V α ⊗ A α (I) −→ V α v ⊗ (cid:55)→ v v ⊗ (cid:55)→ v v ⊗ X (cid:55)→ α v + v v ⊗ X (cid:55)→ α v V α ⊗ V (cid:48) α (II) −−→ A α v ⊗ v (cid:48) (cid:55)→ v ⊗ v (cid:48) (cid:55)→ X − α v ⊗ v (cid:48) (cid:55)→ X − α v ⊗ v (cid:48) (cid:55)→ V α (III) −−→ V α ⊗ A α v (cid:55)→ v ⊗ X − α v ⊗ v ⊗ v (cid:55)→ v ⊗ X − α v ⊗ A α (IV) −−→ V α ⊗ V (cid:48) α (cid:55)→ v ⊗ v (cid:48) + v ⊗ v (cid:48) X (cid:55)→ α v ⊗ v (cid:48) + α v ⊗ v (cid:48) + v ⊗ v (cid:48) Our assignment for essential circles depends on nesting, so strictly speaking the abovecalculations do not handle all cases. However, note that for types (I) and (II), the positionof the essential circle does not change, and for types (III) and (IV), the two essential circlesinvolved in the saddle must be consecutive in the ordering. Thus a full verification amountsto interchanging v ↔ v (cid:48) , v ↔ v (cid:48) in the input of above maps. One may check that thisamounts to interchanging v ↔ v (cid:48) , v ↔ v (cid:48) , and α ↔ α in the output. (cid:3) Corollary 4.3. (1)
Let S ⊂ A × I be a cobordism. Viewing S as a cobordism in R × I ,the map F α ( S ) splits as a sum F α ( S ) = F α ( S ) + F α ( S ) + where F α ( S ) preserves adeg and F α ( S ) + increases adeg . (2) Let S , S ⊂ A × I be composable cobordisms. Then F α ( S S ) = F α ( S ) F α ( S ) . Proof.
For (1), write S as a composition S = S n · · · S where each S i is an elementarycobordism. Functoriality of F α and Lemma 4.2 yield F α ( S ) = F α ( S n ) · · · F α ( S )= ( F α ( S n ) + F α ( S n ) ) · · · ( F α ( S ) + F α ( S ) )= F α ( S n ) · · · F α ( S ) + terms that increase adeg . Therefore F α ( S ) = F α ( S n ) · · · F α ( S ) is the desired adeg-preserving part, and the remaining terms constitute F α ( S ) + . Statement(2) follows from (1) in a similar fashion. (cid:3) We are now ready for the main theorem.
Theorem 4.4.
There exists a functor G α : BN α ( A ) → R α − ggmod such that the followingdiagram commutes BN α ( A ) R α − ggmod BN ( A ) Z − ggmod G α G where the vertical arrows are obtained by setting α = α = 0 .Proof. For a collection of circles C ⊂ A , set G α ( S ) := F α ( C ) , with the bigrading (qdeg , adeg) as defined earlier in this section. For a cobordism S ⊂ A × I ,set G α ( S ) := F α ( S ) as in Corollary 4.3 (1). That G α is well-defined on cobordisms and factors through therelations in BN α ( A ) follows from the analogous statements for F α . Corollary 4.3 (2) impliesfunctoriality of G α . Finally, commutativity of the diagram follows from deleting the boxedterms and setting α = α = 0 in the maps appearing in the proof of Lemma 4.2, andcomparing the result with the maps (13)–(16). (cid:3) Maps assigned to the four elementary saddles in Figure 6 are recorded below. The full setof maps – that is, if other essential circles are present – can be obtained by interchanging α ↔ α . QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 15
Figure 10.
Product cobordism on m > i -th componentdotted V α ⊗ A α (I) −→ V α v ⊗ (cid:55)→ v v ⊗ (cid:55)→ v v ⊗ X (cid:55)→ α v v ⊗ X (cid:55)→ α v (20) V α ⊗ V (cid:48) α (II) −−→ A α v ⊗ v (cid:48) (cid:55)→ v ⊗ v (cid:48) (cid:55)→ X − α v ⊗ v (cid:48) (cid:55)→ X − α v ⊗ v (cid:48) (cid:55)→ V α (III) −−→ V α ⊗ A α v (cid:55)→ v ⊗ X − α v ⊗ v (cid:55)→ v ⊗ X − α v ⊗ A α (IV) −−→ V α ⊗ V (cid:48) α (cid:55)→ v ⊗ v (cid:48) + v ⊗ v (cid:48) X (cid:55)→ α v ⊗ v (cid:48) + α v ⊗ v (cid:48) (23)Let C ⊂ A consist of m > C be the i -th essential circle in C .Consider the cobordism S whose underlying surface is the identity cobordism C × I , with asingle dot on the component C × I , as shown in Figure 10. Then G α ( S ) is the identity onall tensor factors except the one corresponding to C , and on C it is given by the left-handside of (24) if i is odd, and the right-hand side if i is even. V α → V α v (cid:55)→ α v v (cid:55)→ α v V (cid:48) α → V (cid:48) α v (cid:48) (cid:55)→ α v (cid:48) v (cid:48) (cid:55)→ α v (cid:48) (24)Observe that the functor G α is not monoidal, since the action of X on an essential circledepends on its nestedness.Let L ⊂ A × I be an oriented link with diagram D . Let CKh A α ( D ) := G α ([[ D ]])denote the chain complex obtained by applying G α to the formal complex [[ D ]]. The dif-ferential preserves bidegree, and the complex is an invariant of L up to bidegree-preservingchain homotopy equivalence.The remainder of this section discusses variants of G α . Instead of setting both α = α = 0,it is possible to set only α = 0 and rename the remaining parameter α to α = h . Denote the resulting Frobenius pair by ( R h , A h ). Explicitly, R h = Z [ h ] , A h = R h [ X ] / ( X − hX ) . It may also be obtained from (
R, A ) by setting E = h , E = 0; note that the obstructionin Section 4.1 disappears when E = 0. Collapsing ( R h , A h ) further to characteristic 2 (thatis, applying ( − ) ⊗ R h Z [ h ]) recovers Bar-Natan’s theory [BN2, Section 9.3]. We expect thatthe resulting annular homology is related to [TW].Let L ⊂ A × I be an oriented link with diagram D . Viewing D as a diagram in R andapplying F α to [[ D ]] yields a chain complex CKh α ( D ) of bigraded R α -modules. Letting ∂ denote the differential, Lemma 4.2 implies that ∂ splits as ∂ = ∂ + ∂ where ∂ is of bidegree (0 ,
0) and ∂ is of bidegree (0 , β to account for ∂ . Let R αβ = R α [ β ] with β in bidegree (0 , − CKh αβ ( D ) be the chain complex over R αβ with CKh iαβ ( D ) := CKh iα ( D ) ⊗ R α R αβ in homological degree i and differential ∂ β given by ∂ β := ∂ + β∂ . Note that ∂ β preserves bidegree. Maps assigned to the four elementary saddles in Figure 6are given below. V α ⊗ A α (I) −→ V α v ⊗ (cid:55)→ v v ⊗ (cid:55)→ v v ⊗ X (cid:55)→ α v + βv v ⊗ X (cid:55)→ α v V α ⊗ V (cid:48) α (II) −−→ A α v ⊗ v (cid:48) (cid:55)→ βv ⊗ v (cid:48) (cid:55)→ X − α v ⊗ v (cid:48) (cid:55)→ X − α v ⊗ v (cid:48) (cid:55)→ V α (III) −−→ V α ⊗ A α v (cid:55)→ v ⊗ X − α v ⊗ βv ⊗ v (cid:55)→ v ⊗ X − α v ⊗ A α (IV) −−→ V α ⊗ V (cid:48) α (cid:55)→ v ⊗ v (cid:48) + v ⊗ v (cid:48) X (cid:55)→ α v ⊗ v (cid:48) + α v ⊗ v (cid:48) + βv ⊗ v (cid:48) Inverting D in equivariant annular homology. Recall the Frobenius pair ( R α D , A α D )from [KR], which was reviewed in Section 2.4. Let G α D denote the composition BN α ( A ) G α −→ R α − ggmod → R α D − ggmodwhere the second functor is extension of scalars. Consider the following elements of A α D , v := v = 1 , v := v α − α = X − α α − α ,v (cid:48) := v (cid:48) = 1 , v (cid:48) := v (cid:48) α − α = X − α α − α . As in Section 4.2, let V α D and V (cid:48) α D denote the module A α D with distinguished homogeneousbases { v , v } and { v (cid:48) , v (cid:48) } , respectively. For a collection of circles C ⊂ A , the i -th essential QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 17 qdeg − adeg − v , v (cid:48) v , v (cid:48) e , e Figure 11.
Bigradings circle in C is assigned V α D if i is odd and V (cid:48) α D if i is even. The notation A α D is reservedfor trivial circles, with distinguished basis { e , e } , see (8). Bigradings are summarized inFigure 11.With respect to these bases, the maps assigned to the four elementary saddles in Figure6 are recorded below. V α D ⊗ A α D (I) −→ V α D v ⊗ e (cid:55)→ v ⊗ e (cid:55)→ v v ⊗ e (cid:55)→ v v ⊗ e (cid:55)→ V α D ⊗ V (cid:48) α D (II) −−→ A α D v ⊗ v (cid:48) (cid:55)→ v ⊗ v (cid:48) (cid:55)→ e v ⊗ v (cid:48) (cid:55)→ e v ⊗ v (cid:48) (cid:55)→ V α D (III) −−→ V α D ⊗ A α D v (cid:55)→ ( α − α ) v ⊗ e v (cid:55)→ ( α − α ) v ⊗ e (27) A α D (IV) −−→ V α D ⊗ V (cid:48) α D e (cid:55)→ ( α − α ) v ⊗ v (cid:48) e (cid:55)→ ( α − α ) v ⊗ v (cid:48) (28)To obtain the full set of maps – that is, if other essential circles are present – one interchanges α ↔ α , which has the effect of interchanging v ↔ v (cid:48) , v ↔ v (cid:48) , and e ↔ e . They arerecorded below for convenience. V (cid:48) α D ⊗ A α D → V (cid:48) α D v (cid:48) ⊗ e (cid:55)→ v (cid:48) v (cid:48) ⊗ e (cid:55)→ v (cid:48) ⊗ e (cid:55)→ v (cid:48) ⊗ e (cid:55)→ v (cid:48) (29) V (cid:48) α D ⊗ V α D → A α D v (cid:48) ⊗ v (cid:55)→ v (cid:48) ⊗ v (cid:55)→ e v (cid:48) ⊗ v (cid:55)→ e v (cid:48) ⊗ v (cid:55)→ V (cid:48) α D → V (cid:48) α D ⊗ A α D v (cid:48) (cid:55)→ ( α − α ) v (cid:48) ⊗ e v (cid:48) (cid:55)→ ( α − α ) v (cid:48) ⊗ e (31) A α D → V (cid:48) α D ⊗ V α D e (cid:55)→ ( α − α ) v (cid:48) ⊗ v e (cid:55)→ ( α − α ) v (cid:48) ⊗ v (32)These maps may be written uniformly in the following way. Let C ⊂ A be a collectionof circles, and label each circle in C by one of the letters a or b . From such a labeling we obtain a distinguished basis element of G α D ( C ) by using the correspondence(33) a ↔ e , b ↔ e for a trivial circle, and(34) a ↔ (cid:40) v i is odd v (cid:48) i is even , b ↔ (cid:40) v i is odd v (cid:48) i is evenon the i -th essential circle. Then the saddle maps are V α D ⊗ A α D (I) −→ V α D b ⊗ a (cid:55)→ a ⊗ a (cid:55)→ ab ⊗ b (cid:55)→ ba ⊗ b (cid:55)→ V α D ⊗ V (cid:48) α D (II) −−→ A α D b ⊗ a (cid:55)→ a ⊗ a (cid:55)→ ab ⊗ b (cid:55)→ ba ⊗ b (cid:55)→ V α D (III) −−→ V α D ⊗ A α D b (cid:55)→ ( α − α ) b ⊗ ba (cid:55)→ ( α − α ) a ⊗ a (37) A α D (IV) −−→ V α D ⊗ V (cid:48) α D a (cid:55)→ ( α − α ) a ⊗ ab (cid:55)→ ( α − α ) b ⊗ b (38)Moreover, the same formulas hold with V α D and V (cid:48) α D interchanged.For an annular link L with diagram D , let CKh A α D ( D ) := G α D ([[ D ]])denote the chain complex obtained by applying G α D to [[ D ]]. It is an invariant of L up to chainhomotopy equivalence, so we may write Kh A α D ( L ) to denote the homology of CKh A α D ( D ),for any diagram D of L . Theorem 4.5.
Let L ⊂ A × I be a link with diagram D . Viewing L as a link in R , thereis a qdeg -preserving isomorphism ϕ : CKh A α D ( D ) ∼ −→ CKh α D ( D ) . Proof.
For a smoothing D u , the inclusion A (cid:44) → R induces an isomorphism ϕ u : G α D ( D u ) → F α D ( D u ) , defined in terms of the basis elements labeled by a and b by a (cid:55)→ e , b (cid:55)→ e . Comparing the formulas (35)–(38) with multiplication and comultiplication in A α D , we seethat each of the maps ϕ u commute with cobordism maps and thus assemble into an isomor-phism ϕ : CKh A α D ( D ) → CKh α D ( D ). It is evident from Figure 11 that each ϕ u preservesqdeg. Quantum grading shifts in both chain complexes are the same, so the isomorphism ϕ preserves qdeg as well. (cid:3) The following is immediate from Proposition 2.5.
Corollary 4.6.
For a link L ⊂ A × I with k components, the homology Kh A α D ( L ) is a free R α D -module of rank k . QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 19
We recall the canonical generators for Lee homology, following [Le] and [Ra]. Let L ⊂ A × I be a link with diagram D . Given an orientation o on L , let D o ⊂ A denote the result ofperforming the oriented resolution at each crossing, . Each of the resulting circles is naturally oriented. Assign a mod 2 invariant to each circle C as follows. First, consider the number of circles in D o separating C from infinity, mod 2.Add 1 if C is standardly (counterclockwise) oriented, and add 0 otherwise. Now that eachcircle in D o is labeled by 0 or 1, use the correspondence 0 ↔ a , 1 ↔ b to label each circleby a or b , and finally use (33) and (34) to obtain a generator s o in CKh A α D ( D ).For a collection of oriented circles C ⊂ A , let w ( C ) denote the winding number of C .That is, w ( C ) equals the number of counterclockwise essential circles minus the number ofclockwise ones. If C , . . . , C m are the essential circles in C , then w ( C ) = m (cid:88) i =1 w ( C i ) . Proposition 4.7.
Let L ⊂ A × I be a link with diagram D , and let o be an orientation of L . Let m be the number of essential circles in the oriented resolution D o . Then adeg( s o ) = ( − m w ( L, o ) where w ( L, o ) is the winding number of L with respect to the orientation o .Proof. First note that w ( L, o ) = w ( D o ). It is straightforward to verify that each essentialcircle C in D o contributes ( − m w ( C ) to the annular degree of s o . The claim follows, sincetrivial circles do not contribute to the annular degree or the winding number. (cid:3) Dotted Temperley-Lieb category
This section reviews the Temperley-Lieb category TL and its relation to annular Kho-vanov homology, following observations in [BPW]. We then introduce a natural equivariantanalogue.For each n ≥
0, fix a collection n ⊂ (0 ,
1) of n points in the interior of the unit interval.A planar ( n, m ) -tangle is a smooth embedding of a compact 1-manifold M into I × I , suchthat the boundary of M maps to I × { } ∪ I × { } , with n points in ∂M mapping to n ⊂ I × { } , and the remaining m points in ∂M mapping to m ⊂ ×{ } . Planar tangles arealways considered up to isotopy of I × I fixing the boundary.Let TL denote the Z [ q, q − ]-linear category whose objects are nonnegative integers. Themorphism space TL ( n, m ) is freely generated over Z [ q, q − ] by planar ( n, m )-tangles, modulothe local relation that an innermost circle can be removed at the cost of multiplying theremaining tangle by q + q − ,(39) . Composition is defined by vertically stacking planar tangles. Denote the space of morphismsfrom n to m by TL ( n, m ) . (a) Undotted circle (b)
Dotted circle (c)
Two dots relation
Figure 12.
Relations in TL • . The Temperley-Lieb algebra
T L n can then be identified with the endomorphism space TL ( n, n ). Let TL q =1 denote the category obtained by setting q = 1.It was observed in [BPW, Section 6.1] that TL is closely tied to annular Khovanov homol-ogy. By spinning planar tangles in the S direction, one obtains a functor(40) S × ( − ) : TL q =1 → BN ( A ) . Explicitly, n is sent to the essential circles S × n ⊂ A , and a planar tangle T is sent to thecobordism S × T . That S × ( − ) factors through the relation (39) when q = 1 follows fromthe fact that a torus in A × I evaluates to 2 in BN ( A ). Let BBN ( A )denote the category obtained from BN ( A ) by imposing Boerner’s relation, Figure 7. Recallthat the non-equivariant annular TQFT G factors through this relation, so no informationis lost from the point of view of link homology.It is shown in [GLW, Section 4.2] that G can be made to take values in the representationcategory of sl . On the other hand, it is well-known that Temperley-Lieb diagrams (planartangles modulo relation (39)) can be interpreted as U q ( sl )-linear maps between tensor powersof the fundamental representation of U q ( sl ); a convenient reference is [BPW, Appendix A.1].Thus for q = 1, both TL q =1 and BBN ( A ) admit functors F TL and G to U ( sl ) − mod. It wasobserved in [BPW, (6.1)] that the following diagram commutes.(41) TL q =1 BBN ( A ) U ( sl ) − mod S × ( − ) F TL G Moreover, if TL q =1 is made additive and graded by introducing formal direct sums [BN2,Definition 3.2] and formal grading shifts [BN2, Section 6], then the horizontal functor in (41)becomes an equivalence of categories, [BPW, Proposition 6.1].Thus TL q =1 characterizes the skein category BBN ( A ) and the functor F TL characterizesthe annular TQFT G . On the other hand, BN ( A ) was similarly described using planardiagrams in [Ru], which we restate now. A dotted planar ( n, m )-tangle is a planar ( n, m )tangle whose components may be decorated by some number of dots, which are allowed tofloat freely along a component. Let TL • denote the category whose objects are nonnegative integers and whose morphisms are Z -linear combinations of dotted planar tangles, modulo the additional local relations in Figure12. QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 21
Figure 13.
Russel relations. (a) (b) (c)Figure 14.
Two compression disks.
For a dotted planar tangle T , let T u denote the tangle obtained by removing all dots ( u stands for undotted). Consider the functor(42) S × ( − ) : TL • → BN ( A )which sends a dotted planar tangle T to the cobordism whose underlying surface is S × T u ,and which carries k dots on a component if T carried k dots on the corresponding component.The relations in Figure 12 are planar analogues of relations in BN ( A ), Figure 1. Figures 12aand 12b correspond to an undotted torus and a once-dotted torus evaluating to 2 and 0 in BN ( A ), respectively. Figure 12c corresponds to the two dots relation in Figure 1d.Upon introducing formal direct sums and formal grading shifts, the argument in [BPW,Proposition 6.1] shows that the functor (42) is essentially surjective and full. It is not faithful,but by [Ru, Theorem 3.1], its kernel is generated by the local relations shown in Figure 13.Note that the second follows from the first by adding a dot near one of the endpoints ofthe strands and simplifying using the two dots relation. To see that the relations hold,consider two annuli embedded in A × I with a tube joining them, Figure 14a, and performneck-cutting along the two disks shown in Figure 14b and Figure 14c. Denote by (cid:102) TL • the category obtained by imposing the Russel relations. It follows from [Ru, Theorem 3.1]that the induced functor S × ( − ) : (cid:102) TL • → BN ( A )becomes an equivalence of categories after introducing formal direct sums and formal gradingshifts to (cid:102) TL • .An equivariant version of TL • follows from considering the skein category BN α ( A ). Ar-guing as in [BPW, Proposition 6.1], any object of BN α ( A ) is isomorphic to a direct sumof grading-shifted essential circles. Any cobordism in BN α ( A ) can be expressed, in a non-unique way, as an R α -linear combination of cobordisms of the form S × T , where T is adotted planar tangle. It follows that any additive functor out of BN α ( A ) is determined by (a) Undotted circle (b)
Dotted circle (c)
Two dots relation
Figure 15.
Relations in TL α . its value on each collection of n ≥ S × T .This naturally leads to the following definition. Definition 5.1.
Let TL α denote the category whose objects are nonnegative integers, andwhose morphisms are formal R α -linear combinations of dotted planar tangles, modulo thelocal relations shown in Figure 15.For a dotted planar tangle T with d ( T ) dots, define its degree to bedeg( T ) = 2 d ( T ) . Note that the relations in Figure 15 are homogeneous.To motivate the relations, consider the functor(43) S × ( − ) : TL α → BN α ( A )defined as in (42). An undotted torus and a once-dotted torus in A × I evaluate to 2 and α + α in BN α ( A ), respectively, which explains the relations in Figure 15a and Figure 15b.The relation in Figure 15c is a planar analogue of the two dots relation in BN α ( A ), seeFigure 3d. A straightforward induction argument also shows that an innermost circle with k ≥ α k + α k in TL α ,(44) . By composing (43) with the equivariant annular TQFT G α ,(45) TL α S × ( − ) −−−−→ BN α ( A ) G α −→ R α − ggmod , one can view dotted planar tangles as R α -linear maps between tensor powers of A α . Remark . As is the case for BN α ( A ), the relations in TL α involve only symmetric poly-nomials in α and α , so one may consider the U (2)-equivariant analogue instead; see alsoRemark 2.3. However, the functor (45) is not present in the U (2)-equivariant setting.The functor (43) is of course not faithful. For example, it factors through the local relationsshown in Figure 16, which are equivariant analogues of the Russel relations, Figure 13. Let (cid:102) TL α denote the category obtained from TL α by imposing the relations in Figure 16 (it suffices toimpose only the first).We end the section with two questions. The first is motivated by [Ru, Theorem 3.1]. Question 1.
Is the induced spinning functor S × ( − ) : (cid:102) TL α → BN α ( A ) faithful? QUIVARIANT ANNULAR KHOVANOV HOMOLOGY 23
Figure 16.
Equivariant Russel relations. (a) TL (cid:63) ( n, m ) → TL (cid:63) ( n + m, (b) TL (cid:63) ( n + m, → TL (cid:63) ( n, m ) Figure 17.
An isomorphism TL (cid:63) ( n, m ) ∼ = TL (cid:63) ( n + m, By [Ru, Main Theorem] and results in [Kh2], the abelian group (cid:102) TL • (2 n,
0) is free of rank (cid:18) nn (cid:19) . Note that for a symbol (cid:63) ∈ { ∅ , • , α } , the modules TL (cid:63) ( n, m ) and TL (cid:63) ( k, (cid:96) ) are isomorphicwhenever n + m = k + (cid:96) . An isomorphism TL (cid:63) ( n, m ) → TL (cid:63) ( n + m,
0) and its inverse aredepicted in Figure 17. It clearly factors through the Russel relations, so (cid:102) TL • ( n, m ) is free ofrank(46) (cid:18) n + m ( n + m ) / (cid:19) whenever n and m have the same parity, and otherwise it is the zero module. Question 2. Is (cid:102) TL α ( n, m ) free over R α , and, if so, of what rank?Note that TL α ( n, m ) = 0 if n and m have different parities, and otherwise TL α ( n, m ) isfree of rank 2 (cid:96) C ( (cid:96) ) , where (cid:96) = n + m and C ( (cid:96) ) is the (cid:96) -th Catalan number. To see this, consider the collection ofdotted planar ( n, m )-tangles in which every component carries at most one dot and which hasno closed components. They evidently form a basis for TL α ( n, m ). There are C ( (cid:96) ) undottedplanar ( n, m )-tangles with no closed components. A fixed such tangle has (cid:96) components,hence 2 (cid:96) ways to put at most one dot on each component, which yields the count.We can find bases for small values. A basis for (cid:102) TL α (1 ,
1) is given by an undotted and once-dotted vertical strand, and a basis for (cid:102) TL α (2 ,
2) is depicted in (47). That these elementsgenerate the module follows from Figure 16, and linear independence can be verified using S × ( − ) and the TQFT G α . The ranks agree with (46), but it is not clear if this is acoincidence for small examples. For (cid:102) TL α (3 , C (3) = 40 generators and manyrelations, making direct computation difficult. (47) References [APS] M. Asaeda, J. H. Przytycki, and A. Sikora.
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