OOn the genus two skein algebra
Juliet Cooke and Peter SamuelsonAugust 2020
Abstract
We study the skein algebra of the genus 2 surface and its action on the skein module of the genus2 handlebody. We compute this action explicitly, and we describe how the module decomposes overcertain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally,we show that this algebra is isomorphic to the t = q specialisation of the genus two spherical doubleaffine Hecke algebra recently defined by Arthamonov and Shakirov. The Kauffman bracket skein algebra of a surface Σ is spanned by framed links in the thickened surfaceΣ × [0 , relations imposed:= s + s − = − s − s − Multiplication in the algebra is given by stacking links in the [0 ,
1] direction. Typically, thisalgebra is noncommutative; however, its s = ± s = − s (Σ) is isomorphic tothe ring of functions on the SL character variety of Σ. Bullock, Frohman, and Kania-Bartoszy´nska[BFK99] strengthened this statement by showing that the skein algebra is a quantization of the SL character variety of Σ with respect to Atiyah–Bott–Goldman Poisson bracket.The skein module of a 3-manifold M is defined in the same way as the skein algebra (using linksin M instead of in Σ × [0 , ∂M .The action is given by ‘pushing links from a neighbourhood of the boundary into M ’. At s = − M determines a Lagrangian subvariety of the character variety, which consists ofthe representations π ( ∂M ) → SL ( C ) that extend to representations of π ( M ). The fact that theskein module of M is a module over Sk s ( ∂M ) is an illustration of the general principle that ‘thequantization of a coisotropic subvariety is a module’.From now on we write Σ g,n for the genus g surface with n punctures. For the torus Σ , , the skeinalgebra and its action on the skein module of the solid torus were described explicitly by Frohmanand Gelca [FG00]. This description led to interesting connections to Double Affine Hecke Algebras(DAHAs); for example, it follows from their results that the skein algebra of the torus is isomorphicto the t = q specialisation of the A spherical DAHA, the SL ( Z ) actions on both algebras agree, andthe polynomial representation of the DAHA is isomorphic (again at t = q ) to the skein module of thesolid torus. The main goal of the present paper is to find relationships between various versions ofDAHAs and the skein algebra and module of genus 2 surface and handlebody. By local relation we mean the following. The first picture represents 3 links which are identical outside of an embeddedball in Σ × [0 ,
1] and inside the ball are as pictured. Similarly, the second says a trivially framed unknot in an embeddedball can be removed at the expense of a scalar. a r X i v : . [ m a t h . QA ] A ug ince DAHAs tend to be defined by explicit formulas, our first task is to find concrete descriptionsof the skein module Sk s ( H ) of the handlebody, which we do as follows. The skein algebra Sk s (Σ , )of the closed genus two surface is generated by the five curves A , A , A , B , and B depicted inFigure 1. There are two natural bases to use for the skein module Sk s ( H ) of the handlebody, the thetabasis n ( i, j, k ) and the dumbbell basis m ( i, j, k ). We compute the action of generators of Sk s (Σ , )on Sk s ( H ) in both of these bases in the following theorem (see Theorem 3.2 and Theorem 3.4). Theorem 1.
The operators A , A , A act diagonally in the theta basis n ( i, j, k ) of Sk s ( H ) , andmatrix coefficients for B , B , B with respect to this basis are in equation (11) . The dumbbellbasis m ( i, j, k ) diagonalises the operators A and A , and matrix coefficients for the other operatorsare in Theorem 3.4. There is a third basis of Sk s ( H ) which comes from the fact that the genus 2 handlebody isdiffeomorphic to a thickening of a 2-punctured disc. This means Sk s ( H ) is actually an algebra,and it turns out to be the polynomial algebra in the elements B , B , and B . We provide somecomments about the resulting monomial basis in Section 2 following Lemma 2.13. Double affine Hecke algebras were introduced by Cherednik [Che95] for his proof of Macdonald’sconjectures, and have since been related to a wide variety of areas (see, for example [Che05] andreferences therein). There are two versions that will be relevant for our purposes: the 2-parameterspherical DAHA SH q,t of type A , and the 5-parameter spherical DAHA S H q,t ,t ,t ,t of type( C ∨ , C ). These algebras each have a polynomial representation , which is an analogue of Vermamodules in Lie theory.Terwilliger has given presentations of both spherical DAHAs, and combining these with re-sults of Bullock and Przytycki [BP00] one can construct algebra maps Sk s (Σ , ) → SH q,t andSk s (Σ , ) → S H q, { t i } . This implies the skein algebras of these punctured surfaces act on the polyno-mial representations of the corresponding spherical DAHAs. These skein algebras also act on Sk s ( H )since they map to the skein algebra of the genus 2 surface via the surface maps in Figure 1. A A a a a a x x y y y y a a a a x x B B A B Figure 1: Loops and surface embeddings
Using the structure constants from Theorem 1, we compute how the skein module decomposes overthe subalgebras corresponding to these subsurfaces in the following (see Corollary 3.3, Theorem 4.4,and Corollary 4.8):
Theorem 2.
As a module over Sk s (Σ , ) , the skein module Sk s ( H ) is irreducible.1. As a module over Sk s (Σ , ) ⊗ Sk s (Σ , ) , the skein module decomposes as a direct sum Sk s ( H ) = (cid:77) j ≥ P j ⊗ P j where P j is the specialisation of the A polynomial representation at t = − q − j − . The spherical DAHA is a subalgebra which is analogous to a subalgebra of invariants of a group action. Note that there are two copies of Σ , embedded in Σ , , which means we have two commuting actions of the skeinalgebra Sk s (Σ , ) on Sk s ( H ). . As a module over Sk s (Σ , ) , the skein module decomposes as a direct sum Sk s ( H ) = (cid:77) i,k ≥ P (cid:48) i,k where P (cid:48) i,k is the (unique) finite dimensional quotient of the spherical ( C ∨ , C ) polynomial rep-resentation with parameter specialisations given in equation (26) . We aren’t aware of an explanation or construction of this module using classical DAHA theory, sowe briefly summarise some of its unusual features here. Terwilliger’s universal Askey-Wilson algebra∆ q maps to the skein algebra of the punctured torus and to the skein alegbra of the 4-puncturedsphere. Therefore, there are 3 copies of ∆ q acting simultaneously on the space R [ B , B , B ] ofpolynomials in 3 variables. These algebras have nontrivial intersections which corresponds to theintersections of the respective subsurfaces. For example, the ‘Casimir’ element for both the left andright copies of ∆ q correspond to the curve A above; however, this curve corresponds to the Askey-Wilson operator in the middle copy of ∆ q . Conversely, central elements in the middle copy of ∆ q act as the loops labelled A and A , which correspond to the Macdonald operators in the left andright copies of ∆ q . We also point out that these three subalgebras generate the skein algebra of thegenus 2 surface (see Corollary 2.19), and that over this algebra, the skein module of the handlebodyis irreducible (see Corollary 3.3). Finally, we note that the left and right copies of ∆ q contain themultiplication operators B and B respectively. However, the B multiplication operator isn’t inany of the 3 subalgebras in question; instead, it has to be written as a fairly complicated expressioninvolving generators of all three copies of ∆ q (see Corollary 2.21).Finally, we recall that a Leonard pair is a finite dimensional vector space V with two diagonalizableendomorphisms A, B such that A has a tridiagonal matrix with respect to an eigenbasis of B , andvica-versa (see [Ter01]). There is an extensive literature on Leonard pairs, and they have arisenin representation theory, combinatorics, orthogonal polynomials, and more (see, e.g. [Ter03] andreferences therein). We show Leonard pairs also appear in skein theory; in particular, in Corollary 4.11we show that each P (cid:48) i,k is a Leonard pair with respect to the operators x and x . It seems likely thatthese particular Leonard pairs have appeared before, e.g. in [NT17]. However, using the topologicalpoint of view in the present paper, it is evident that ⊕ i,k ≥ P (cid:48) i,k is a module for ∆ q ⊗ ∆ q , and it isnot clear if this observation has appeared in the literature. If we combine the Terwilliger presentation of the A spherical DAHA SH q,t with the Frohman-Gelcadescription of the skein algebra of the torus, it follows almost immediately that the skein algebraSk s (Σ , ) is isomorphic to the t = q = s specialisation SH s,s of the A spherical DAHA. Furthermore,the SL ( Z ) actions on both algebras agree, and the polynomial representation of the spherical DAHAin this specialisation is isomorphic to the skein module of the solid torus. These results are surprisingsince the objects on the DAHA side are defined in terms of explicit formulas, while the skein-theoreticdefinitions are purely topological. Our second main motivation for the present paper was to generalisethese results to genus 2.In genus 2, this comparison was not possible until the work of Arthamonov and Shakirov [AS19],who recently gave a very interesting proposal for a definition of the genus 2 spherical DAHA . Theydefine their algebra in terms of its action on a space with basis Ψ i,j,k where ( i, j, k ) ranges overthe set of admissible triples . Their algebra depends on two parameters, q and t , and we use ourskein-theoretic results to prove the following (see Theorem 5.10): Theorem 3.
The q = t specialisation of the Arthamonov-Shakirov algebra is isomorphic to the imageof the skein algebra Sk s (Σ , ) in the endomorphism ring of Sk s ( H ) . Thang Le has informed us of work in progress [Le20] where the author(s) show faithfulness of theaction of the skein algebra of a (closed) surface on the skein module of the corresponding handlebody.Le’s result combined with Theorem 3 immediately imply the following corollary (see Corollary 5.11). A tridiagonal matrix only has nonzero entries which are on or adjacent to the diagonal. See Definition 2.8; this definition also comes up in the bases we use for the skein module. orollary. Assuming the faithfulness result in [Le20], the skein algebra Sk s (Σ , ) is isomorphic tothe t = q specialisation of the Arthamonov–Shakirov genus 2 DAHA. Arthamonov and Shakirov raised a number of questions about their algebra; in particular, theyask [AS19, Pg. 17] whether their algebra is a flat deformation of the skein algebra. Our corollaryabove proves that the t = q specialisation is isomorphic to the skein algebra; however, to the best ofour knowledge, it is still unknown whether their algebra is a flat deformation of the skein algebra. The results described above lead or contribute to some interesting questions in both representationtheory and knot theory. On the representation theory side, in [Ter13] Terwilliger has defined a universal Askey-Wilson algebra ∆ q , and showed that this algebra surjects onto the spherical DAHA S H q, { t i } (and hence, onto the A spherical DAHA also). In fact, these surjections factor throughthe skein algebras (see Remark 2.30), so we have maps ∆ q → Sk s (Σ , ) and ∆ q → Sk s (Σ , ). Thisimplies we have three maps from ∆ q to the skein algebra Sk s (Σ , ) of the genus 2 surface. Question 1.
Do the three maps ∆ q → Sk s (Σ , ) deform to maps to the Arthamonov-Shakirov alge-bra? How does their module decompose over the subalgebras given by the images of these maps? In knot theory, there have been quite a number of papers conjecturing and/or proving a relation-ship between DAHAs and knot invariants for torus knots (beginning with [AS15, Che13]), iteratedtorus knots (beginning with [Sam19, CD16]), and iterated torus links [CD17]. One of the recent suc-cesses in this direction was the work of Mellit, Hogancamp and Mellit [Mel17, HM19] which provedthat the Khovanov–Roszansky homology for positive torus knots/links can be computed using theelliptic Hall algebra. Roughly, the (Euler characteristic of) Khovanov–Roszansky homology and theelliptic Hall algebra are the ‘ gl ∞ ’ analogues of the Kauffman bracket knot polynomial and A sphericalDAHA, respectively, both of which correspond to sl .From our point of view, the heuristic for these conjectures and results is that the torus knot T m,n is embedded in the torus, and can therefore be viewed as an operator on the skein module of thesolid torus. Using the SL ( Z ) action on the spherical DAHA, an analogous operator ˆ T m,n can bedefined as an element of the DAHA. The real surprise is that the operator ˆ T m,n and its action in thepolynomial representation can compute the Poincare polynomial of the knot homology of the torusknot (instead of the Euler characteristic, which is computed by the skein algebra action). This leadsto the following question, which was asked slightly differently in the last sentence of [AS19]:
Question 2 ([AS19]) . Can the Poincare polynomial of Khovanov homology of genus 2 knots (suchas the figure eight knot) be computed using the Arthamonov-Shakirov algebra?
The present paper provides some evidence that this question has a positive answer: our Theorem 3above shows that the answer is yes at the level of Euler characteristics. In other words, we have thefollowing corollary, which is stated precisely in Corollary 5.13.
Corollary. If α is a simple closed curve on Σ , , then the Jones polynomial of α can be computedusing the Arthamonov-Shakirov algebra. Finally, we mention that Hikami has also given a proposal for a genus 2 DAHA by gluing togetherthe A and ( C ∨ C ) spherical DAHAs, and has conjectured [Hik19, Conj. 5.5] that his algebra canalso be used to compute (coloured) Jones polynomials of knots embedded on the genus 2 surface.However, the relation between his construction and the Arthamonov-Shakirov construction is notclear, so the results of the present paper do not seem to be immediately applicable to this conjecture.An outline of the paper is as follows. In Section 2, we recall background material on skein theoryand double affine Hecke algebras, and perform some initial computations. In Section 3, we statetheorems giving matrix coefficients of actions of certain loops on the skein module. In Section 4 weuse DAHAs to describe how the skein module decomposes over certain subsurfaces, and we brieflydiscuss Leonard pairs. In Section 5 we show how our skein-theoretic calculations are related to thegenus 2 DAHA defined by Arthamanov and Shakirov. Finally, the appendices contain diagrammaticproofs of the matrix coefficient computations. In this paragraph, citations grouped together within square brackets have been sorted by chronological order withrespect to arXiv posting, since some articles experienced significant publication delays. The standard embedding of Σ , into S allows us to interpret a curve on Σ , as a knot in S . cknowledgements: We would like to thank Paul Terwilliger for his interest and his guidancethrough the literature regarding Leonard pairs. We would also like to thank Semeon Arthamonov,Matt Durham, David Jordan, Thang Le, Gregor Masbaum, Shamil Shakirov for many helpful dis-cussions over the past several years and Thomas Wright for careful proofreading. Both authors werepartially supported by the ERC grant 637618, the second author was partially supported by a SimonsFoundation Collaboration Grant, and the first author was funded by a EPSRC studentship and theF.R.S.-FNRS.
Kauffman bracket skein modules are based on the Kauffman bracket:
Definition 2.1.
Let L be a link without contractible components (but including the empty link).The Kauffman bracket polynomial (cid:104) L (cid:105) in the variable s is defined by the following local skein relations := s + s − , (1)= − s − s − . (2)(These diagrams represent three links which are identical outside of the dotted circles and are aspictured inside the dotted circles.) It is an invariant of framed links and it can be ‘renormalised’to give the Jones polynomial. The Kauffman bracket can also be used to define an invariant of3-manifolds: Definition 2.2.
Let M be a 3-manifold, R be a commutative ring with identity and s be an invertibleelement of R . The Kauffman bracket Skein module Sk s ( M ; R ) is the R -module of all formal linearcombinations of links, modulo the Kauffman bracket skein relations pictured above. Remark . For the remainder of the paper we will use the coefficient ring R := Q ( s ).We now define the Jones-Wenzl idempotents which one can use to construct a diagrammaticcalculus for skein modules. This diagrammatic calculus is heavily used in Appendix A and Appendix Bto calculate the loop actions used throughout this paper. Definition 2.4.
The
Jones-Wenzl idempotent is depicted using a box and is defined recursively asfollows: n 1 = n 1 − [ n ] s [ n + 1] s n-1 11n-1n-1 1 1 = where a strand labelled by an integer n > n parallel strands and by an integer n ≤ Definition 2.5.
Let n be an integer. The quantum integer [ n ] s is[ n ] s = s n − s − n s − s − Remark . Note that [0] s = 1, [1] s = 1 and [ − n ] s = − [ n ] s . efinition 2.7. For any non-negative integer n the quantum factorial [ n ] s ! is defined as[ n ] s ! = [ n ] s [ n − s . . . [0] s , so in particular [0] s ! = 1.Jones-Wenzl idempotents are also used to define trivalent vertices which are used to describegenerating sets of the skein module of a handlebody. Definition 2.8.
A triple ( a, b, c ) is admissible if a, b, c ≥ a + b + c is even and | a − b | ≤ c ≤ a + b .We denote by Ad ⊆ N the set of all admissible triples. Definition 2.9 ([MV94]) . Given an admissible triple ( a, b, c ) one can define the : a bc a bij kc = where i = ( a + b − c )2 , j = ( a + c − b )2 , k = ( b + c − a )2 are integers as ( a, b, c ) is admissible. Using the trivalent vertices described in the previous section one can describe a generating set forthe skein module of a handlebody.
Definition 2.10.
Let H g denote the solid g -handlebody. Theorem 2.11 ([Lic93, Zho04]) . A generating set of Sk s ( H g ) is given by ...a a a a a n a n-1 a n-3 a n-2 where a , . . . , a n are non-negative integers which form admissible triples around every 3-valent node. Other generating sets of Sk s ( H g ) can be obtained by modifying the diagram in Theorem 2.11 attwo adjacent nodes using quantum 6 j -symbols [KL94] as the change of base matrix. This change ofbasis is useful for checking calculations, and it will also been needed in Section 5 as a different basisis used by Arthamonov and Shakirov. Theorem 2.12 (Change of Basis [KL94]) . If ( r, t, j ) and ( s, u, j ) are admissible triples then jr st u = (cid:88) a (cid:8) t r as u j (cid:9) r st ua where the sum is over all a such that ( r, s, a ) and ( t, u, a ) are admissible. Summarising the discussion so far, we have two bases of skein module of the solid 2-torus whichare depicted in Figure 2 and are related via the change of basis formula given in Theorem 2.12. Thefact that these are bases (and not just generating sets) has been well-known to experts for a longtime; we have, however, not found a precise reference so we sketch a proof in Lemma 2.14 below. ji k (a) the basis element n ( i, j, k ) i j k (b) the basis element m ( i, j, k ) Figure 2: Bases for the skein module of the handlebody6 n fact, the skein of the solid 2-torus is an algebra, since the handlebody is diffeomorphic to aninterval crossed with a twice-punctured disc. We recall its algebra structure in the following.
Lemma 2.13 ([BP00, Prop. 1(6)]) . As an algebra, Sk s ( H ) is isomorphic to R [ B , B , B ] (usingthe notation of Figure 4a), a polynomial algebra in 3 variables. In the present notation, we have B = n (1 , ,
0) = m (1 , , ,B = n (0 , ,
1) = m (0 , , ,B = n (1 , ,
1) = m (1 , ,
1) + 1[2] s m (1 , , m ( i, j, k ), the expression for the loop B can be computed using theJones-Wenzl recursion.) Since Sk s ( H ) is isomorphic to a polynomial algebra, this gives us a naturalmonomial basis. We can partially describe the change of basis matrix between the monomial basisand our bases m ( i, j, k ) and n ( i, j, k ) as follows. Let S n ( x ) be the Chebyshev polynomials, which aredetermined uniquely by the condition S n ( X + X − ) = X n +1 − X n − X − X − We then have the following identities: m ( i, ,
0) = n ( i, ,
0) = S i ( B ) , n ( k, , k ) = S k ( B ) , m (0 , , k ) = n (0 , , k ) = S k ( B )In general the change of basis matrix between monomials in B (cid:96)(cid:96) (cid:48) and either m ( i, j, k ) or n ( i, j, k ) isdetermined uniquely by the formulas in Theorem 3.2, which express the action of B (cid:96)(cid:96) (cid:48) as a matrix inthe n ( i, j, k ) basis. However, the change of basis matrices are not so easy to write explicitly outsideof the cases above.As mentioned above, the following lemma is well-known to experts, but for the convenience of thereader we provide a sketch of a proof. Lemma 2.14.
The set n ( i, j, k ) is a basis for Sk s ( H ) .Proof. By Lemma 2.13, the set { B a B b B c } (for a, b, c ∈ N ) is a basis for Sk s ( H ). Using therecursion defining the Jones-Wenzl idempotents, one can see that B a B b B c = n ( a + c, a + b, b + c )plus lower order terms (with respect to the lexicographic order on triples of nonegative integers).This shows the matrix converting from the monomial basis to the theta basis is upper-triangular,with ones on the diagonal. (The fact that the diagonal entries are 1 follows from the fact that therecursion defining the Jones-Wenzl idempotents has a 1 as the first coefficient.) If Σ is a surface then the skein algebra Sk s (Σ × [0 , R ) forms an algebra with multiplication givenby stacking the links on top of each other to obtain a link in Σ × [0 , × [0 ,
1] again.
Definition 2.15.
Let Σ g,n denote the surface with genus g and n punctures.Presentations are known for skein algebras of a small number of surfaces. We shall use thepresentations for the four-punctured sphere Σ , and 1-punctured torus Σ , , which we recall below.These presentations all use the q -Lie bracket, defined as follows:[ a, b ] q := qab − q − ba Let a i denote the loops around the four punctures of Σ , , and let x i denote the loops aroundpunctures 1 and 2, 2 and 3, 1 and 3 respectively (see Figure 3). If curve x i separates a i , a j from a k , a (cid:96) , let p i = a i a j + a k a (cid:96) . Explicitly, p = a a + a a , p = a a + a a , p = a a + a a e now recall the following theorem of Bullock and Przytycki. Theorem 2.16 ([BP00]) . As an algebra over the polynomial ring R [ a , a , a , a ] , the Kauffmanbracket skein algebra Sk s (Σ , ) has a presentation with generators x , x , x and relations [ x i , x i +1 ] s = ( s − s − ) x i +2 + ( s − s − ) p i +2 (indices taken modulo 3) ;Ω K = ( s + s − ) − ( p p p p + p + p + p + p ); where we have used the following ‘Casimir element’: Ω K := − s x x x + s x + s − x + s x + s p x + s − p x + s p x . y y a a a a x x x Figure 3: The generating loops for Sk s (Σ , ) and Sk s (Σ , ) Let y and y denote the loops around the meridian and longitude respectively of a torus. Theseloops cross once and resolving this crossing crossing gives y y = sy + s − z . Theorem 2.17 ([BP00]) . As an algebra over R , the Kauffman bracket skein algebra Sk s (Σ , ) hasa presentation with generators y , y , y and relations [ y i , y i +1 ] s = ( s − s − ) y i +2 (indices taken modulo 3) . The loop p around the puncture of Σ , is obtained from the resolution of the crossing of z and y using the identity zy = s y + s − y − s − s − + p . In this subsection we briefly discuss sets of curves which generate the skein algebra Sk s (Σ , ). In[San18], Santharoubane gave the following very useful criteria for showing a set of curves generatesthe skein algebra of a surface. Theorem 2.18 ([San18]) . Let { γ j } j ∈ I be a finite set of non-separating simple closed curves suchthat the following conditions hold:1. For any i, j ∈ I , the curves γ i and γ j intersect at most once,2. The set of Dehn twists around the curves γ i generate the mapping class group of Σ .Then the curves γ i generate the skein algebra of Σ . Using this theorem we can prove:
Corollary 2.19.
The skein algebra Sk s (Σ , ) is generated by each of the following:1. The set of curves I := { A , A , A , B , B } of Figure 4,2. The subalgebras Sk s (Σ L , ) , Sk s (Σ R , ) , and Sk s (Σ , ) of Figure 5.Proof. In [Hum79], Humphries constructed a finite generating set of the mapping class group forclosed surfaces, and in the genus 2 case it is exactly the set of curves in the first claim. The secondclaim follows since each of the curves in I is contained in one of the subalgebras mentioned.Santharoubane proves his generation result by using some theory that has been developed formapping class groups. These groups are related to skein theory using Lemma 2.20, which is astandard lemma relating Dehn twists to the resolutions of the crossing that are given by the skeinrelation. If α is a simple closed curve, let D α be the right-handed Dehn twist along α . In particular, We have corrected a sign error in the first relation which appears in the published version of the paper [BP00]. f β is a simple closed curve intersecting α once, then D α ( β ) is the simple closed curve which can bedescribed in words as ‘follow β , turn right at the intersection and follow α , then turn right at theintersection and continue following β .’ Lemma 2.20.
Let α and β be two simple closed curves in Σ that intersect once. We have thefollowing relation in the skein algebra of Σ : D (cid:15)α ( β ) = (cid:15) s (cid:15) αβ − s − (cid:15) βα ( s − s − ) where (cid:15) ∈ {± } . We now give an explicit computation of the loop B in terms of the other loops in Figure 4a.(We note that a computation for B also appeared in [Hik19, Fig. 7], which used an identity in themapping class group that is similar to (3).) Corollary 2.21.
We have the following identity in the skein algebra of Σ , : B = − δ − [ A , [ B , [ A , [ A , B ] s − ] s ] s − ] s − where δ = s − s − .Proof. Elementary computations with Dehn twists show the following identity: B = ( D − A ◦ D − B ◦ D A ◦ D A − )( B ) (3)Then the claim follows from Lemma 2.20 Double Affine Hecke Algebras (DAHAs) were introduced by Cherednik [Che95], who used themto prove Macdonald’s constant term conjecture for Macdonald polynomials. These algebras havesince found wider ranging applications particularly in representation theory [Che05]. DAHAs wereassociated to different root systems, and we recall the type A case here. Definition 2.22.
The A double affine Hecke algebra H q,t is the algebra generated by X ± , Y ± and T subject to the relations T XT = X − , T Y − T = Y, XY = q T − Y X, ( T − t )( T + t − ) = 0 Remark . Our presentation here varies slightly from the standard presentation in [Che05], andin particular replaces his t with t − . See [BS18, Rmk. 2.19] for the precise relation.The element e = ( T + t − ) / ( t + t − ) is an idempotent of H q,t , and is used to define the sphericalsubalgebra SH q,t := eH q,t e .We shall also consider Sahi’s [Sah99] DAHA associated to the ( C ∨ , C ) root system (see also[NS04] for the rank 1 case that we use in this paper). The double affine Hecke algebra H q, t of type( C ∨ , C ) is a 5-parameter universal deformation of the affine Weyl group C [ X ± , Y ± ] (cid:111) Z with thedeformation parameters q ∈ C ∗ and t = ( t , t , t , t ) ∈ ( C ∗ ) and it is a generalisation of Cherednik’sdouble affine Hecke algebras of rank 1, since there is an isomorphism H q ; t = H q, , ,t, . Definition 2.24.
The ( C ∨ , C ) double affine Hecke algebra is the algebra with generators T , T , T ∨ , T ∨ and the following relations: ( T − t )( T + t − ) = 0( T − t )( T + t − ) = 0( T ∨ − t )( T ∨ + t − ) = 0( T ∨ − t )( T ∨ + t − ) = 0 T ∨ T T T ∨ = q he spherical subalgebra S H q, t is defined in terms of an idempotent e ∈ H q,t as follows: e := ( T + t − ) / ( t + t − ) , S H q, t := e H q, t e (4)Terwilliger gave presentations of the spherical DAHAs which will be useful for us (for the conver-sion between our notation and Terwilliger’s notation, see [Sam19] and [BS16]). Define the followingelements in S H q, t: x := ( q − T T ∨ + q ( T T ∨ ) − ) e, y := ( T T + T − T − ) e, z := ( T T ∨ + ( T T ∨ ) − ) e In what follows it will be helpful to use the following notation: t := t − t − , qt = qt − q − t − , etc . Theorem 2.25 ([Ter13]) . The spherical double affine Hecke algebra S H q, t has a presentation withgenerators x, y, z defined above and relations [ x, y ] q = ( q − q − ) z − ( q − q − ) γ [ y, z ] q = ( q − q − ) x − ( q − q − ) α [ z, x ] q = ( q − q − ) y − ( q − q − ) β Ω = ( t ) + ( t ) + ( qt ) + ( t ) − t t ( qt ) t + ( q + q − ) where α := t t + qt t , β := t t + qt t , γ := t t + qt t , and the ‘Casimir’ Ω := − qxyz + q x + q − y + q z − qαx − q − βy − qγz. Using Terwilliger’s presentation of the spherical DAHA and the Bullock-Przytycki presentationof the skein algebra of the 4-punctured sphere (Theorem 2.16), we obtain the following result.
Proposition 2.26.
There is an algebra map ϕ : Sk s (Σ , ) → S H q,t given by ϕ ( x ) = x ϕ ( x ) = y ϕ ( x ) = z ϕ ( s ) = q ,ϕ ( a ) = it ϕ ( a ) = it ϕ ( a ) = it ϕ ( a ) = i ( qt ) . where i = − . Under this map, p (cid:55)→ − α, p (cid:55)→ − β, p (cid:55)→ − γ Remark . To the best of our knowledge, Proposition 2.26 first appeared in notes of Terwilligerwhich have not been published. It was also stated in [BS18], but the notational conventions there areslightly different. In particular, the images of a and a are different here and in [BS18], but becauseof the underlying differences in notation, both statements are correct. This statement also was proveddirectly in [Hik19, Thm. 3.8], without relying on Terwilliger’s presentation of the spherical DAHA.Using [BS18, Rmk. 2.19], it isn’t too hard to see that (in our current conventions) the A DAHAis the specialisation of the ( C ∨ , C ) DAHA at t = (1 , , t, x = ( X + X − ) e, y = ( Y + Y − ) e, z = ( qY X + q − X − Y − ) e These generators lead to the following presentation.
Theorem 2.28 ([Ter13]) . The spherical double affine Hecke algebra SH q,t has a presentation withgenerators x, y, z and relations [ x, y ] q = ( q − q − ) z, [ z, x ] q = ( q − q − ) y, [ y, z ] q = ( q − q − ) x The spherical subalgebra is not a unital subalgebra; instead, the unit in the spherical subalgebra is the idempotent e . x + q − y + q z − qxyz = (cid:18) tq − qt (cid:19) + (cid:18) q + 1 q (cid:19) Note that in the following statement we identify s = q instead of s = q . Proposition 2.29 ([Sam19]) . There is an algebra map Sk s (Σ , ) → SH q,t uniquely determined bythe following assignments: y (cid:55)→ x, y (cid:55)→ y, y (cid:55)→ z, p (cid:55)→ − q t − − q − t , s (cid:55)→ q Remark . Terwilliger [Ter13] has introduced a universal Askey-Wilson algebra ∆ q , which mapsto the spherical ( C ∨ , C ) DAHA. The algebra ∆ q is generated by elements A, B, C , and the relationsstate that A + [ B, C ] q / ( q − q − ) (along with its two cyclic permutations A (cid:55)→ B (cid:55)→ C ) are central.He showed that this algebra maps to the spherical ( C ∨ , C ) DAHA (where A, B, C map to x, y, z ),and it follows that it also maps to the spherical A DAHA. Using the presentations of the skeinalgebras of Σ , and Σ , , it is clear that these maps factor through the maps from the skein algebrasin Proposition 2.26 and Proposition 2.29. The skein algebras of Σ , and Σ , act on the skein of the genus 2 handlebody using the maps ofsurfaces in Figure 5. We would like to use Propositions 2.26 and 2.29 to identify these modules interms of representations of DAHAs. In this section we recall the polynomial representation of H q, ton R [ X ± ] from [NS04] and compute some structure constants of this action.First we define two auxiliary operators ˆ σ, ˆ y ∈ End R ( R [ X ± ]):ˆ σ ( f ( X )) := f ( X − ) , ˆ y ( f ( X )) := f ( q − X ) (5)We write the actions of the operators T and T in terms of ˆ σ , ˆ y , and the multiplication operator X : T := t ˆ σ ˆ y − q t X + qt X − q X (1 − ˆ σ ˆ y ) (6) T := t ˆ σ + t + t X − X (1 − ˆ σ )(Here we have used the notation ¯ t i = t i − t − i ). We note that a priori these operators act onrational functions R ( X ), but in fact they preserve the subspace R [ X ± ] of Laurent polynomials,since (1 − ˆ σ ˆ y ) f ( X ) is always divisible by 1 − q X , and similar for T . Since T , T , and X generatethe DAHA, these definitions completely determine the action, and it is shown in [NS04] that theseoperators satisfy the relations of the DAHA H q,t .The spherical subalgebra S H q, { t i } acts on eR [ X ± ] and it isn’t hard to see that the idempotent e projects onto the subspace R [ X + X − ] = R [ x ] of symmetric Laurent polynomials. Lemma 2.31.
We have the following identities: y · t t + t − t − (7) yx · q t − t − + q − t t ) x + (cid:2) − t t + t ( q − t − q t − ) + ( q − − q ) t ( t + t − ) (cid:3) (8) Proof.
The first equation is immediate, and the second is straightforward but somewhat tedious.Using the quadratic relations for T and T , we need to compute( T T + ( T T ) − ) · ( X + X − ) = ( T T + T T − t T − t T + t t ) · ( X + X − ) (9)Using equations (6), we can compute T · X = t q − X − + t X + t q − T · X − = q t − X − qt T · X = t − X − − t T · X − = t X + t X − + t he claimed identity is the result of substituting these four equations into (9) and simplifying.Since the A DAHA is the t = (1 , , t,
1) specialisation of the ( C ∨ , C ) DAHA, we have thefollowing corollary to Lemma 2.31: Corollary 2.32.
The A spherical DAHA acts on R [ x ] , and under this action we have y · t + t − (10) yx · q t − + q − t ) x In this section we shall determine the actions of the loops depicted in Figure 4a and Figure 4b onSk s ( H ), the skein module of the solid 2-handlebody. These loop actions define the actions of the skeinalgebras Sk s (Σ , ) and Sk s (Σ , ) on Sk s ( H ) which we will use in Section 4 to decompose Sk s ( H ).We shall also relate these loop actions to the operators generating the Arthamonov–Shakirov, genus2, spherical DAHA in Section 5. For the loops in Figure 4a, we shall use the theta basis for Sk s ( H ),which is the set { n ( i, j, k ) } for all admissible triples ( i, j, k ). For the loops in Figure 4b, we use thedumbbell basis for Sk s ( H ), which is the set { m ( i, j, k ) } for all i, j, k such that ( i, i, j ) and ( k, k, j )are admissible . A A A B B B (a) A set of generators of Sk s (Σ , ) which weact on Sk s ( H ) with basis n ( i, j, k ). XA A B B A (b) Another set of generators of Sk s (Σ , )which we act on Sk s ( H ) with basis m ( i, j, k ). Figure 4: Generating loops
Definition 3.1.
Let ( i, j, k ) be an admissible triple. We define the coefficients D a,b ( i, j, k ) for a, b = ± D , − ( i, j, k ) = − (cid:2) j + k − i (cid:3) s [ j ] s [ j + 1] s D − , − ( i, j, k ) = (cid:2) i + j + k +22 (cid:3) s (cid:2) i + j − k (cid:3) s [ i ] s [ i + 1] s [ j ] s [ j + 1] s D − , ( i, j, k ) = − (cid:2) i + k − j (cid:3) s [ i ] s [ i + 1] s D , ( i, j, k ) = 1where the coefficient is defined to be 0 if the denominator is 0. Theorem 3.2.
Let ( i, j, k ) be an admissible triple. The A -loops act on the theta basis of Sk s ( H ) byscalars A · n ( i, j, k ) = ( − s i +2 − s − i − ) n ( i, j, k ) A · n ( i, j, k ) = ( − s j +2 − s − j − ) n ( i, j, k ) A · n ( i, j, k ) = ( − s k +2 − s − k − ) n ( i, j, k ) whilst the B -loops act as follows B · n ( i, j, k ) = (cid:88) a,b ∈{− , } D a,b ( i, j, k ) n ( i + a, j + b, k ) (11) Note that the restrictions on i, j, k are exactly those required for the edges entering each trivalent vertex to form anadmissible triple, and thus for the trivalent vertex to be well defined. · n ( i, j, k ) = (cid:88) a,b ∈{− , } D a,b ( i, k, j ) n ( i + a, j, k + b ) B · n ( i, j, k ) = (cid:88) a,b ∈{− , } D a,b ( j, k, i ) n ( i, j + a, k + b ) Proof.
See Proposition A.3 and Proposition A.6.Before computing actions in the dumbbell basis m ( i, j, k ), we use the previous theorem to showthe following corollary. Corollary 3.3.
When the coefficient ring is Q ( s ) , the skein module Sk s ( H ) is irreducible as a moduleover the skein algebra Sk s (Σ , ) .Proof. First, we note that by Lemma 2.13, the skein module Sk s ( H ) is actually an algebra, and as analgebra it is isomorphic to the polynomial algebra in variables B , B , B . Furthermore, the loopsin Σ , with these labels act by multiplication operators, which shows that Sk s ( H ) is generated as amodule by n (0 , ,
0) (which corresponds to 1 in the polynomial algebra). It therefore suffices to showthat if x ∈ Sk s ( H ) is an arbitrary element, then n (0 , ,
0) is contained in the subspace Sk s (Σ , ) · x .Second, we note that the operators A i act diagonally on n ( i, j, k ), and that the joint eigenspacesof the A i are 1-dimensional when s is not a root of unity. This implies that if n ( i, j, k ) appears with anonzero coefficient in the expansion of x in the theta basis, then n ( i, j, k ) ∈ Sk s (Σ , ) · x . Therefore,it suffices to show that if n ( i, j, k ) is arbitrary, then n (0 , ,
0) appears with a nonzero coefficient insome element in Sk s (Σ , ) · n ( i, j, k ).Finally, Lemma 5.4 shows that any admissible triple ( i, j, k ) can be written as ( x + d, y + d, x + y )for some x, y, d ≥
0. If we start with an admissible triple in that form, we first apply B d to n ( i, j, k )to obtain a sum of basis elements which has a nonzero coefficient of n ( x, y, x + y ). We then apply B x to this sum to obtain a nonzero coefficient of n (0 , y, y ), and finally apply B y to obtain a nonzerocoefficient of n (0 , ,
0) as desired.
Theorem 3.4.
Let i, j, k be integers such that the triples ( i, i, j ) and ( k, k, j ) are admissible. The X -loop and two of the A -loops act on the dumbbell basis of Sk s ( H ) by scalars X · m ( i, j, k ) = ( − s j +2 − s − j − ) m ( i, j, k ) A · m ( i, j, k ) = ( − s i +2 − s − i − ) m ( i, j, k ) A · m ( i, j, k ) = ( − s k +2 − s − k − ) m ( i, j, k ) whilst the B -loops B and B act as follows B · m ( i, j, k ) = (cid:40) m ( i + 1 , j, k ) for i = j m ( i + 1 , j, k ) + [2 i + j +1] s [ i − j/ s [ i ] s [ i +1] s m ( i − , j, k ) for i > j/ B · m ( i, j, k ) = (cid:40) m ( i, j, k + 1) for k = j m ( i, j, k + 1) + [2 k + j +1] s [ k − j/ s [ k ] s [ k +1] s m ( i, j, k − for k > j/ The action of the middle A -loop A on the m -basis of Sk s ( H ) is more complex: A · m ( i, j, k ) = − (cid:18) ( s − s − ) [ j/ s [ i + j/ s [ k + j/ s [ j − s [ j ] s [ j + 1] s (cid:19) m ( i, j − , k ) − (cid:18) ( s − s − ) [ i − j/ s [ k − j/ s (cid:19) m ( i, j + 2 , k )+ (cid:18)(cid:0) − s − i + k +1) − s i + k +1) (cid:1) + K (cid:19) m ( i, j, k ) where K = ( s − s − ) (cid:32) [ j/ s [ i − j/ s [ k − j/ s [ j + 1] s [ j + 2] s + [ j/ s [ i + j/ s [ k + j/ s [ j ] s [ j + 1] s (cid:33) roof. See Proposition A.4, Proposition A.7 and Theorem B.2.
Remark . In the above expressions, we have used the convention that if j = 0, then any coefficientterm with [ j ] in the denominator is 0. In fact, this follows by inspection of these coefficients, sincethe term [ j/ s / [ j ] s (and hence each coefficient itself) is equal to zero when j = 0. There are two inclusions of Σ , into Σ , (which we call left and right respectively) and an inclusionof Σ , into Σ , which are depicted in Figure 5. a a a a x x y y y y y y y y a a a a x x Figure 5: Surface embeddings
These inclusions induce actions of Sk s (Σ , ) and Sk s (Σ , ) on Sk s ( H ). The action of the gen-erators of Sk s (Σ , ) and Sk s (Σ , ) on the m ( i, j, k ) basis of Sk s ( H ) has already been computed inSection 3. In this section we use this to describe how the skein module of the genus 2 handlebodydecomposes as a module in terms of the polynomial representations of the A and ( C ∨ , C ) sphericalDAHAs. In this subsection we provide a useful technical lemma for constructing maps between certain modulesover spherical DAHAs. Suppose B is an algebra over a commutative ring R (cid:48) . Suppose B is generatedby elements b , b , b that satisfy three relations of the following form:[ b i , b i +1 ] q = c i +2 b i +2 + z i +2 (12)where the indices are taken modulo 3, and where c i and z i are elements of R (cid:48) , with the c i invertible(Note that we are not assuming that this is a presentation of B as an algebra, only that these threerelations hold). Lemma 4.1.
Suppose M and N are modules over B , and that as modules over the subalgebra R (cid:48) [ b ] ,they are generated by m ∈ M and n ∈ N , respectively. Suppose furthermore that M is free as amodule over R (cid:48) [ b ] . Then the assignment m (cid:55)→ n extends uniquely to a R (cid:48) [ b ] -linear surjection ϕ : M (cid:16) N Furthermore, suppose that for some α, β, c ∈ R (cid:48) we have the following identities: b · m = α m b b · m = βa m + c (13) b · n = α n b b · n = βa n + c Then ϕ is a map of B -modules.Remark . Before we prove this technical lemma, let us explain how it will be used. The skeinalgebras of the 4-punctured sphere and once-punctured torus map to spherical double affine Heckealgebras, and this allows us to restrict the ‘polynomial representations’ of the spherical DAHAsto modules over these skein algebras. Both these skein algebras are generated by 3 elements that atisfy relations of the form (12). We will use this lemma to construct maps from the polynomialrepresentations coming from spherical DAHAs to the skein module of the genus 2 handlebody. Proof.
Since M is free of rank 1 over R (cid:48) [ b ], the claimed surjection exists and is uniquely defined by ϕ ( f ( b ) m ) = f ( b ) n . By construction ϕ is a map of R (cid:48) [ b ]-modules. What remains to be shown isthat this map commutes with the action of b and b . We will prove this by induction on the degreeof f ( b ). For the base case f ( b ) = 1, the required commutativity for b follows from the first twoequations in (13). For b , the relation (12) shows b = c − [ b , b ] q − c − z , and then commutativityfollows from the second two equations of (13).For the inductive step, assume ϕ ( b i f ( b ) m ) = b i f ( b ) n for all f ( b ) of degree at most n . Wewant to show that ϕ ( b i b n +11 m ) = b i b n +11 n for i = 2 ,
3. We have ϕ ( b b n +11 m ) = ϕ (( b b ) b n )= ϕ (( q b b − qc − qz ) b n m )= ( q b b − qc − qz ) b n n = b b n +11 n where the step from the second to third line follows from the induction hypothesis and R (cid:48) [ b ]-linearityof ϕ . This proves commutativity for b , and the proof for b is similar. Sk s (Σ , ) on Sk s ( H ) In this section we first consider the action · = · L as · R is analogous. From Theorem 2.17 we knowthat the skein algebra Sk s (Σ , ) is generated by the loops y , y , y , but as y = [ y , y ] s ( s − s − )it is sufficient to determine the action of y and y . These actions were computed in previous sections,and we recall the results in the present notation. y · m ( i, j, k ) = ( − s i +2 − s − i − ) m ( i, j, k ) (14) y · m ( i, j, k ) = m ( i + 1 , j, k ) for i = j m ( i + 1 , j, k ) + [2 i + j +1] s [ i − j ] s [ i ] s [ i +1] s m ( i − , j, k ) for i > j/ V = Sk s ( H ) with basis { m ( i, j, k ) } , soSk s ( H ) =: V = (cid:77) j is even ∞ (cid:77) i,k = j/ span R { m ( i, j, k ) } Let V j,k denote the subspace with fixed j and k , and V j denote the subspace with fixed j , so V j,k = ∞ (cid:77) i = j/ span R { m ( i, j, k ) } , V j = ∞ (cid:77) i = j/ ∞ (cid:77) k = j/ span R { m ( i, j, k ) } The subspace V j,k is invariant under the · L action of y and y , and V j is invariant under both the · L and · R actions. Therefore there are representations ρ j,k : Sk s (Σ , ) → End R ( V j,k ) ρ j : Sk s (Σ , ) ⊗ R Sk s (Σ , ) → End R ( V j )defined by this action. We first study the representation ρ j,k and then use this to study the repre-sentation ρ j .The submodule V j,k is free of rank 1 as a module over R [ y ] with generator m := m ( j/ , j, k ). urthermore, using equation (14), we see that y · m = ( − s j +2 − s − j − ) m y y · m = ( − s j/ − s − j/ − ) y · m (15) X · m ( i, j, k ) = ( − s j +2 − s − j − ) m ( i, j, k )Recall that the A spherical DAHA acts on the polynomial representation P ( q, t ) = R [ x ]. We canrestrict this action to the skein algebra using the algebra map Sk s (Σ , ) → SH q,t of Proposition 2.29.We recall that in the current notation, this map is uniquely determined by the following assignments: y (cid:55)→ x, y (cid:55)→ y, X (cid:55)→ − q t − − q − t . (16) Theorem 4.3.
The Sk s (Σ , ) -module V is isomorphic to the direct sum (cid:77) j,k ≥ ,j even P ( q = s, t = − s − j − ) (17) Proof.
By equation (10), in the polynomial representation P ( q, t ) we have y · t + t − yx · q t − + q − t ) x When we specialise q = s and t = − q − j − , we see that these formulas are compatible with theformulas (15) under the assignments (16), when we send 1 ∈ P ( q, t ) = R [ x ] to m ∈ V j,k . ThenLemma 4.1 shows P ( q = s, t = − q − j − ) ∼ = V j,k In the previous theorem we only decomposed V as a module over the action of one copy of theDAHA, which corresponds to the embedded punctured torus on the left of Figure 5. This is thereason that the submodules in the decomposition in (17) doesn’t depend on k (Conversely, under the(left) action of the right copy of Sk s (Σ , ), the k -indices vary and the i indices do not). The actionsof the skein algebras of the ‘left’ and ‘right’ punctured tori commute with each other, and we candecompose V as a module over the tensor product of these two algebras as follows.Let B = Sk s (Σ , ) ⊗ R Sk s (Σ , ), which acts on V as described in the previous paragraph. If M and N are modules over the A DAHA, then B acts on M ⊗ R N via restriction of the (tensor squareof) the algebra map Sk s (Σ , ) → SH q,t . Theorem 4.4.
The B -module V has the following decomposition V ∼ = (cid:77) j ≥ ,j even P ( q = s, t = − s − j − ) ⊗ R P ( q = s, t = − s − j − ) (18) Proof.
The proof is essentially the same as the proof of Theorem 4.3, and it proceeds by showing V j ∼ = P ( q = s, t = − s − j − ) ⊗ R P ( q = s, t = − s − j − )This follows from Theorem 4.3 and from the ‘tensor square’ of Lemma 4.1. Sk s (Σ , ) on Sk s ( H ) In this section we study the action of the skein algebra of the 4-punctured sphere on the skein moduleSk s ( H ) of the genus 2 handlebody. To shorten notation, we write V := Sk s ( H ) . rom Theorem 2.16 we know that the skein algebra Sk s (Σ , ) is generated by x , x , x , a , a , a , a where x i is the curve which separates the punctures a i , a i +1 from the punctures a i +2 , a i +3 (with in-dices taken modulo 4). The element x can be written in terms of the other generators, so it sufficesto compute the actions of x , x , a , a , a and a . These computations were done in Theorem 3.4,and we recall the results here in the present notation. For all i, j, k such that the triples ( i, i, j ) and( k, k, j ) are admissible, a · m ( i, j, k ) = a · m ( i, j, k ) = ( − s i +2 − s − i − ) m ( i, j, k ) , (19) a · m ( i, j, k ) = a · m ( i, j, k ) = ( − s k +2 − s − k − ) m ( i, j, k ) , (20) x · m ( i, j, k ) = ( − s j +2 − s − j − ) m ( i, j, k ) . (21)Finally, x · m ( i, j, k ) = − (cid:18) ( s − s − ) [ j/ s [ i + j/ s [ k + j/ s [ j − s [ j ] s [ j + 1] s (cid:19) m ( i, j − , k ) (22) − (cid:18) ( s − s − ) [ i − j/ s [ k − j/ s (cid:19) m ( i, j + 2 , k )+ (cid:18)(cid:0) − s − i + k +1) − s i + k +1) (cid:1) + K (cid:19) m ( i, j, k )where K is a certain constant. Recall that in these formulas, if a coefficient has [ j ] s in the denominator,then that coefficient is equal to 0 when j = 0 (see Remark 3.5).Now define the following R -submodules of V : V i,k := span R { m ( i, j, k ) | ≤ j ≤ i, k } (23)We note that these R modules have R -dimension 1 + min ( i, k ). Lemma 4.5.
Over the skein algebra Sk s (Σ , ) , the module V has the following decomposition: V = (cid:77) i,k ≥ V i,k Proof. If A ⊂ N × is the set of triples ( i, j, k ) with ( i, i, j ) and ( j, k, k ) admissible, then we alreadyknow the set { m ( i, j, k ) | ( i, j, k ) ∈ A} is an R -basis for V . From the definition of admissibility(Definition 2.8), it is clear that each m ( i, j, k ) is an element of exactly one of the V i,k , which showsthe claimed decomposition as R -modules. All that is left is to show that each V i,k is preserved underthe action of Sk s (Σ , ). The a i and x preserve this decomposition since they act diagonally on the m ( i, j, k ), and since the skein algebra is generated by these elements and x , all that remains is toshow that x preserves this decomposition. This follows from the formula for the action of x above,since the operator x doesn’t change i or k , and the coefficient of m ( i, j + 2 , k ) is equal to 0 whenever( i, i, j ) or ( j, k, k ) is not admissible (since [0] q = 0).We now identify the pieces V i,k as finite dimensional quotients of polynomial representations ofDAHAs. More precisely, by Proposition 2.26 (which we recall below), there is a map from the skeinalgebra of the 4-punctured sphere to the ( C ∨ , C ) spherical DAHA, and this algebra acts on C [ x ] asdescribed in Section 2.6. By the computations in that section, this module is generated by 1 and isfree over the subalgebra C [ x ] (where x acts by multiplication). We recall from Lemma 2.31 that wehave the following identities: y · t t + t − t − (24) yx · q t − t − + q − t t ) x + − t t + t ( q − t − q t − ) + ( q − − q ) t ( t + t − ) (25)Let P i,k be this polynomial representation with parameters specialised as follows: q = s , t = ιs − i − , t = ιs (cid:15) (2 k +2) , t = ιs i , t = ιs (cid:15) (2 k +2) (26) ere ι = −
1, and (cid:15) and (cid:15) are signs which can be chosen arbitrarily (the choice does not affectthe computations below). We view P i,k as a module over the skein algebra Sk s (Σ , ) by restrictionalong the algebra map φ : Sk s (Σ , ) → S H q,t described in Proposition 2.26. This algebra map φ isdetermined by the assignments φ ( x ) = x, φ ( a ) = ιt ,φ ( x ) = y, φ ( a ) = ιt ,φ ( x ) = z, φ ( a ) = ιt ,φ ( s ) = q , φ ( a ) = ι ( qt ) . Let R (cid:48) = R [ a , a , a , a ]. Now we construct the following map of Sk s (Σ , )-modules: ϕ : P i,k (cid:16) V i,k (27)This map is uniquely determined by the choice 1 (cid:55)→ m ( i, , k ) and by the requirement that it is amodule map over R [ x ] (since P i,k is freely generated by 1 over R [ x ]). To see that it is a map of R (cid:48) [ x ]-modules, we need to show that ϕ ( a i · m ) = a i · ϕ ( m ) for any m ∈ P i,k . The a i act as scalarsin P i,k by definition, and we can see that they act as scalars in V i,k from the formulas (19) and (20)since these actions just depend on i and k , and these indices are the same for all basis vectors inside V i,k . All that remains is to show that these scalars agree, and this follows immediately from thespecializations in (26). Lemma 4.6.
The map ϕ of equation (27) is a map of Sk s (Σ , ) -modules.Proof. By Lemma 4.1, it suffices to check the following identities: ϕ ( x ·
1) = x · m ( i, , k ) (28) ϕ ( x x ·
1) = x x · m ( i, , k ) (29)By equation (24) and the specializations in (26), we see that x · t t + t − t − = ι ( s − i − s i + s i +2 s − i ) = − s − s − Combining this with (21) proves (28).Let m := m ( i, , k ) and m := m ( i, , k ). To prove equation (29), we first note that equations(22) and (21) show the identity x x m = c cm + c dm = c ( cm + dm ) + ( − c d + c d ) m = c ( x m ) + ( − c + c ) dm where we have used the constants c i := − s i +2 − s − i − and c = − ( s − s − ) [ i ] s [ k ] s , d = − s − i − k − − s i +2 k +2 + ( s − s − ) [ i ] s [ k ] s [2] s Similarly, equation (25) shows that x x · ax + b , where a = q t − t − + q − t t , b = − t t + t ( q − t − q t − ) + ( q − − q ) t ( t + t − )All that remains is to show that with the specialisations of (26) we have a = c , b = ( − c + c ) d and these are straightforward identities of Laurent polynomials in the parameter s . Remark . Oblomkov and Stoica completely classified finite dimensional representations of the( C ∨ , C ) DAHA in [OS09] and showed that they are all quotients of the polynomial representation(possibly with some different signs) at certain special parameter values. They also show that at these arameter values, the polynomial representation has a unique nontrivial quotient, which leads to thefollowing corollary. Corollary 4.8.
The skein module of the genus 2 handlebody is a direct sum of finite dimensional rep-resentations of the spherical ( C ∨ , C ) DAHA. Each summand is the unique quotient of the polynomialrepresentation with parameters specialised as in (26) , with i, k ≥ . Several years ago Terwilliger asked the second author whether Leonard pairs appeared in skein theory,and in this section we show that the answer is yes.
Definition 4.9.
A matrix with entries a ij is irreducible tridiagonal when the only nonzero entriesare a ij with | i − j | ≤
1, and a ij (cid:54) = 0 whenever | i − j | = 1. Definition 4.10. A Leonard pair is a finite dimensional vector space V equipped with endomorphisms A and B which satisfy the following conditions:1. There exists an A -eigenbasis of V for which the matrix representing B is irreducible tridiagonal.2. There exists a B -eigenbasis of V for which the matrix representing A is irreducible tridiagonal.Recall from (23) that V i,k is a finite-dimensional subspace of the skein module of the handlebody,and it is a submodule over the skein algebra Sk s (Σ , ). In particular, x and x act on V i,k . Corollary 4.11.
The vector space V i,k with the endomorphisms x and x is a Leonard pair.Proof. First, by Theorem 2.16 there is a map from Terwilliger’s universal Askey-Wilson algebra ∆ q to Sk s (Σ , ) which takes the generators A, B to x , x . By [Ter18], there is a map from the q -Onsageralgebra O q to ∆ q , and hence to Sk s (Σ , ). This implies that V i,k is a module over O q , and byLemma 4.13, it is irreducible over O q .The q -Onsager algebra is a specialization of the tridiagonal algebra of [Ter01]. Since V i,k isirreducible and finite dimensional, [Ter01, Thm. 3.10] implies the action of x and x on V i,k give atridiagonal pair. Since we know x and x have 1-dimensional eigenspaces, [Ter01, Lem. 2.2] implies x , x is a Leonard pair on V i,k . Remark . Instead of the proof above, we could compute the action of x on the n ( i, j, k ) basis tocheck that it is tridiagonal. This would be similar in difficulty to the computation of the coefficientsof the x action in the m ( i, j, k ) basis.Using our explicit formulas for the action of x and x , we show the following. Lemma 4.13.
With the base ring Q ( s ) , the module V i,k is irreducible as a module over the subalgebraof Sk s (Σ , ) generated by x and x .Proof. Write A for the subalgebra of the skein algebra generated by x and x . We would like toshow that if x ∈ V i,k , then A · x = V i,k . Equation (21) shows that x acts on V i,k with 1-dimensionaleigenspaces. This implies that it is sufficient to show the following implication: for any j , and foreach j (cid:48) with ( i, i, j (cid:48) ) and ( j (cid:48) , k, k ) admissible, the subspace A · m ( i, j, k ) contains an element y whose m ( i, j, k )-expansion has a nonzero coefficient for m ( i, j (cid:48) , k ). Then equation (22) shows that we maytake y = x | j − j (cid:48) | · m ( i, j, k ). In [AS19], Arthamonov and Shakirov propose a definition for a genus 2, type A , spherical DAHA.In this section we show that when q = t this algebra is isomorphic to the skein algebra Sk s (Σ , ) (seeTheorem 5.10).The A spherical DAHA admits a representation on Q ( q, t )[ X ± ] as the algebra generated by twooperators ˆ O B := X + X − , ˆ O A := X − − tXX − − X ˆ δ + X − tX − X − X − ˆ δ − here ˆ δf ( X ) := f ( q / X ) is the shift operator; these two operators are associated with the A-and B-cycles of the torus. In order to generalise the A spherical DAHA to genus 2, Arthamonovand Shakirov generalise this representation: they use six operators ˆ O B , ˆ O B , ˆ O B , ˆ O A , ˆ O A ,ˆ O A , which relate to the six cycles shown in Figure 4a, and these operators act on H , the spaceof Laurent polynomials in variables x , x , x which are symmetric under the Z group of Weylinversions ( x , x , x ) (cid:55)→ ( x u , x v , x w ) where u, v, w ∈ {± } . This space H has a basis given bya family of Laurent polynomials { Ψ i,j,k } where ( i, j, k ) are admissible triples. We shall show that,when q = t = s , the algebra of actions of Sk s (Σ , ) on Sk s ( H ) is isomorphic to this algebra of sixoperators via the mappingloop (cid:55)→ − ˆ O loop and n ( i, j, k ) (cid:55)→ α − ( i, j, k )Ψ i,j,k where α : Ad → Q ( q ± ) is a change of basis map which we define in Definition 5.5 and give a closedform for in Proposition 5.9.We shall first recall the necessary definitions from [AS19]. Definition 5.1.
For any admissible triple ( i, j, k ) ∈ Ad and a, b ∈ {− , } , the Arthamonov andShakirov coefficients are C a,b ( i, j, k ) = ab (cid:2) ai + bj + k , a + b +22 (cid:3) q,t (cid:2) ai + bj − k , a + b (cid:3) q,t [ i − , q,t [ j − , q,t (cid:2) i, a +32 (cid:3) q,t (cid:2) i − , a +32 (cid:3) q,t (cid:2) j, b +32 (cid:3) q,t (cid:2) j − , b +32 (cid:3) q,t where [ n, m ] q,t := q n t m − q − n t − m q − q − . Definition 5.2.
Given a loop in { A , A , A , B , B , B } , the operator ˆ O loop acts on Ψ i,j,k asfollows:ˆ O B Ψ i,j,k = (cid:88) a,b ∈{− , } C a,b ( i, j, k )Ψ i + a,j + b,k ˆ O A Ψ i,j,k = (cid:16) q i/ t / + q − i/ t − / (cid:17) Ψ i,j,k ˆ O B Ψ i,j,k = (cid:88) a,b ∈{− , } C a,b ( i, k, j )Ψ i + a,j,k + b ˆ O A Ψ i,j,k = (cid:16) q j/ t / + q − j/ t − / (cid:17) Ψ i,j,k ˆ O B Ψ i,j,k = (cid:88) a,b ∈{− , } C a,b ( j, k, i )Ψ i,j + a,k + b ˆ O A Ψ i,j,k = (cid:16) q k/ t / + q − k/ t − / (cid:17) Ψ i,j,k where Ψ , , = 1 and Ψ i,j,k := 0 if ( i, j, k ) is not an admissible triple. The Arthamonov–Shakirov,genus , spherical DAHA is the subalgebra of endomorphism ring of H generated by these operators.From now on we shall specialise to q = t = s so that [ n, m ] q,t = [ n + m ] s (this makes sensebecause none of the structure constants above have poles at t = q ). This leads to a simplification ofthe Arthamonov and Shakirov coefficients C , ( i, j, k ). Lemma 5.3.
For all ( i, j, k ) ∈ Ad we have that C , ( i, j, k ) = (cid:2) i + j + k +42 (cid:3) s (cid:2) i + j − k +22 (cid:3) s [ i + 2] s [ j + 2] s (30) and the relations: C , ( i, j, k ) = C , ( j, i, k ) (31) C , ( k, j + 1 , i + 1) C , ( i, j, k ) = C , ( i, j + 1 , k + 1) C , ( k, j, i ) (32) Proof.
Substituting a = b = 1 into the definition of C a,b gives (30). As this is symmetric in i and j we have (31) and by substituting (30) for each term in (32) we see that this relation holds.We also note that any admissible triple can be written in the following form: Lemma 5.4.
A triple is admissible if and only if it is of the form ( x + d, y + d, x + y ) for some x, y, d ≥ . roof. First note that the triple ( x + d, y + d, x + y ) where x, y, d ≥ a, b, c ). As c ≤ a + b there exists an (cid:15) ≥ a + b = c + (cid:15) . As a + b + c = 2 c + (cid:15) is even, (cid:15) is even and thus (cid:15) = 2 d for some d ≥
0. We also have that c ≥ | a − b | = ⇒ a + b − d ≥ | a − b | :if a ≥ b then this implies that a + b − d ≥ a − b = ⇒ b ≥ d and hence a ≥ b ≥ d ; and if on theother hand b ≥ a then we have that a + b − d ≥ b − a = ⇒ a ≥ d and hence b ≥ a ≥ d . Define x := a − d and y := b − d as a, b ≥ d these are both positive, and ( a, b, c ) = ( x + d, y + d, x + y ), sothe triple is of the required form.Therefore, in order to define the change of basis map α : Ad → Q [ s , s − ], which relates thetheta basis of the skein module Sk s ( H ) to the basis of H in Theorem 5.10, it is sufficient to define α ( x + d, y + d, x + y ) for x, y, d ≥ Definition 5.5.
Define α : Ad → Q [ s , s − ] recursively as follows: α (0 , ,
0) = 1 (33) α ( i + 1 , j + 1 , k ) = − α ( i, j, k ) C , ( i, j, k ) ; (34) α ( i + 1 , j, k + 1) = − α ( i, j, k ) C , ( i, k, j ) ; (35) α ( i, j + 1 , k + 1) = − α ( i, j, k ) C , ( j, k, i ) . (36) Lemma 5.6.
The function α : Ad → Q [ s , s − ] is well defined.Proof. In order to prove that α ( i, j, k ) is unambiguously defined it is sufficient to prove that if α ( i, j, k )is unambiguously defined then α ( i + 1 , j + 2 , k + 1), α ( i + 2 , j + 1 , k + 1) and α ( i + 1 , j + 1 , k + 2) areunambiguously defined. α ( i + 1 , j + 2 , k + 1) = − α ( i, j + 1 , k + 1) C , ( i, j + 1 , k + 1) using (34) first= α ( i, j, k ) C , ( i, j + 1 , k + 1) C , ( j, k, i ) α ( i + 1 , j + 2 , k + 1) = − α ( i + 1 , j + 1 , k ) C , ( j + 1 , k, i + 1) using (36) first= α ( i, j, k ) C , ( j + 1 , k, i + 1) C , ( i, j, k )= α ( i, j, k ) C , ( k, j + 1 , i + 1) C , ( i, j, k ) by Lemma 5.3 (31)= α ( i, j, k ) C , ( i, j + 1 , k + 1) C , ( k, j, i ) by Lemma 5.3 (32)= α ( i, j, k ) C , ( i, j + 1 , k + 1) C , ( j, k, i ) by Lemma 5.3 (31)these give the same result. Checking α ( i + 2 , j + 1 , k + 1) and α ( i + 1 , j + 1 , k + 2) are unambiguouslydefined is analogous.We shall now find a non-recursive formula for α . Lemma 5.7.
For all x ≥ , α ( x, , x ) = ( − x [ x + 1] s . Proof.
We proceed by induction on x : for x = 0 the result holds by the definition of α and α ( x + 1 , , x + 1) = − α ( x, , x ) C , ( x, x,
0) = ( − x +1 [ x + 1] s [ x + 2] s [ x + 1] s = ( − x +1 [ x + 2] s by the induction assumption and (30). Lemma 5.8.
For all x, y ≥ , α ( x, y, x + y ) = ( − x + y [ x + 1] s [ y + 1] s . roof. We proceed by induction on y : for y = 0 this is Lemma 5.7 and α ( x + 1 , , x + 1) = − α ( x, , x ) C , ( x, x,
0) = ( − x +1 [ x + 1] s [ x + 2] s [ x + 1] s = ( − x +1 [ x + 2] s by the induction assumption and (30). Proposition 5.9.
For all x, y, d ≥ , α ( x + d, y + d, x + y ) = ( − x + y + d [ x + 1] s ... [ x + d + 1] s [ y + 1] s ... [ y + d + 1] s [1] s ... [ d ] s [ x + y + 2] s ... [ x + y + d + 1] s As all admissible triples can be written in this form this means that this is an equivalent definitionfor α .Proof. Note that using Lemma 5.3 we have that C , ( x + d, y + d, x + y ) = [ x + y + d + 2] s [ d + 1] s [ x + d + 2] s [ y + d + 2] s We shall prove this result by induction on d . The base case holds by Lemma 5.8, and as α ( x + d + 1 , y + d + 1 , x + y )= − α ( x + d, y + d, x + y ) C , ( x + d, y + d, x + y )= (cid:18) ( − x + y + d +1 [ x + 1] s ... [ x + d + 1] s [ y + 1] s ... [ y + d + 1] s [1] s ... [ d ] s [ x + y + 2] s ... [ x + y + d + 1] s (cid:19) (cid:18) [ x + d + 2] s [ y + d + 2] s [ x + y + d + 2] s [ d + 1] s (cid:19) = ( − x + y + d +1 [ x + 1] s ... [ x + d + 1] s [ x + d + 2] s [ y + 1] s ... [ y + d + 1] s [ y + d + 2] s [1] s ... [ d ] s [ d + 1] s [ x + y + 2] s ... [ x + y + d + 1] s [ x + y + d + 2] s we have proven the induction step. Theorem 5.10.
For the specialisation q = t = s , the action of ˆ O loop on Ψ( i, j, k ) ∈ H is equivalentto the action of loop on m ( i, j, k ) ∈ Sk s ( H ) for any loop ∈ { A , A , A , B , B , B } with thecorrespondence given by loop (cid:55)→ − ˆ O loop and n ( i, j, k ) (cid:55)→ α − ( i, j, k )Ψ i,j,k . Hence, the Arthamonov–Shakirov, genus , spherical DAHA is isomorphic to the image of the skeinalgebra Sk s (Σ , ) in the endomorphism ring of Sk s ( H ) .Proof. Under the correspondenceˆ O A · Ψ i,j,k = ( q i/ t / + q − i/ t − / )Ψ i,j,k = ( s i +2 + s − i − )Ψ i,j,k becomes − α ( i, j, k ) A · n ( i, j, k ) = ( s i +2 + s − i − ) α ( i, j, k ) n ( i, j, k ) . Dividing by − α ( i, j, k ) gives A · n ( i, j, k ) = ( − s i +2 − s − i − ) n ( i, j, k ) which from Theorem 3.2 isindeed A · n ( i, j, k ). The results for ˆ O A Ψ i,j,k and ˆ O A Ψ i,j,k follows by symmetry.As the triple ( i, j, k ) is admissible, by Lemma 5.4 we can assume the triple is of the form ( i + d, j + d, i + j ). Under the correspondenceˆ O B Ψ i + d,j + d,i + j = (cid:88) a,b ∈{− , } C a,b ( i + d, j + d, i + j )Ψ i + d + a,j + d + b,i + j maps to B · α ( i + d, j + d, i + j ) n ( i + d, j + d, i + j ) − (cid:88) a,b ∈{− , } C a,b ( i + d, j + d, i + j ) α ( i + d + a, j + d + b, i + j ) n ( i + d + a, j + d + b, i + j ) ⇐⇒ B · n ( i + d, j + d, i + j )= (cid:88) a,b ∈{− , } − C a,b ( i + d, j + d, i + j ) α ( i + d + a, j + d + b, i + j ) α ( i + d, j + d, i + j ) n ( i + d + a, j + d + b, i + j )So it suffices to show that coefficient is D , ( i + d, j + d, i + j ). When a = 1 and b = 1: − C , ( i + d, j + d, i + j ) α ( i + d + 1 , j + d + 1 , i + j ) α ( i + d, j + d, i + j )= ( − i +2 j +2 d +2 C , ( i + d, j + d, i + j ) [ i +1] s ... [ i + d +2] s [ j +1] s ... [ j + d +2] s [1] s ... [ d +1] s [ i + j +2] s ... [ i + j + d +2] s [ i +1] s ... [ i + d +1] s [ j +1] s ... [ j + d +1] s [1] s ... [ d ] s [ i + j +2] s ... [ i + j + d +1] s = C , ( i + d, j + d, i + j ) [ i + d + 2] s [ j + d + 2] s [ d + 1] s [ i + j + d + 2] s = [ i + j + d + 2] s [ d + 1] s [ i + d + 2] s [ j + d + 2] s b The other cases when a = − b = −
1, when a = 1 and b = −
1, and when a = − b = 1 aresimilar. The result for ˆ O B Ψ i,j,k and ˆ O B Ψ i,j,k follows by symmetry.Using Le’s theorem [Le20] that the action of the skein algebra of a closed surface on the skeinmodule of a handlebody is faithful, we immediately obtain the following. Corollary 5.11.
The t = q = s specialization of the Arthamonov-Shakirov algebra is isomorphic tothe skein algebra Sk s (Σ , ) . Finally, we give a precise statement relating these algebras to quantum knot invariants. Thestandard embedding of the handlebody H into S induces a map on the corresponding skein modules.Since the skein module of S is just the ground ring R , we can view this map as an evaluation function:ev : Sk s ( H ) → Sk s ( S ) = R This map has been computed explicitly in the n ( i, j, k ) basis in [MV94], and for concreteness we recallthe formula here. Let ( a, b, c ) be an admissible triple, and let ( i, j, k ) be the labels on the internalvertices, which can be written explicitly as i = ( b + c − a ) / j = ( a + c − b ) /
2, and k = ( a + b − c ) / Theorem 5.12 ([MV94, Thm. 1]) . The evaluation formula is ev( n ( a, b, c )) = ( − i + j + k [ i + j + k + 1]![ i ]![ j ]![ k ]![ i + j ]![ j + k ]![ i + k ]! (37) where [ (cid:96) ]! is the q -factorial in the variable s . Now suppose α is a simple closed curve on Σ , . If we embed Σ , into S in the standard way,this induces an evaluation map ev Σ : Sk s (Σ , ) → Sk s ( S ) = R Under this embedding we can view α as a knot in S , and by definition, the Jones polynomial J ( α )of α is given by ev Σ ( α ). We could instead first embed the knot in the solid handlebody, and thenembed this in S , which leads to the following (tautological) identity: J ( α ) = ev Σ ( α ) = ev( α · n (0 , , Corollary 5.13.
Suppose that ev q,t : H → R is a linear map whose t = q specialisation is equalto the evaluation map in (37) . Suppose that α q,t is an element in the Arthamonov-Shakirov algebrawhose t = q specialisation is equal to α ∈ Sk s (Σ , ) . Then the specialisation ev s ,s ( α s ,s · Ψ(0 , , is equal to the Jones polynomial of α , viewed as a knot in S under the standard embedding. roof. By Corollary 5.11, the Arthamonov-Shakirov algebra and its action on H specialise at t = q = s to the skein algebra of Σ , and its action on the skein module of the handlebody. This impliesev s,s ( α s,s · Ψ(0 , , α · n (0 , , α by equation (38). Remark . The ‘correct’ evaluation map ev q,t should depend nontrivially on q and t , and shouldonly be equal to the skein-theoretic evaluation after these parameters are set equal. However, thisevaluation map isn’t defined in [AS19], and finding the ‘correct’ definition is a nontrivial task. Appendices
A Calculation of Loop Actions
In this appendix we calculate almost all the actions of the loops depicted below on Sk s ( H ), the skeinmodule of the solid 2-handlebody. For the loops in the left figure, we shall use the non-dumbbellbasis for Sk s ( H ): m ( i, j, k ) for all admissible ( i, j, k ). For the loops in the right figure, we use thedumbbell basis for Sk s ( H ): m ( i, j, k ) for all i, j, k such that ( i, i, j ) and ( k, k, j ) are admissible (seeSection 2.2). We leave the calculation of the action of A on the basis m ( i, j, k ) to Appendix B asthis calculation is significantly more complex than the others. A A A B B B XA A B B A In Section 2.1 we defined Jones-Wenzl idempotents and used them to define trivalent vertices. Wenow outline a number of results which we use as graphical calculus for skein modules, using [MV94]as a reference.
Lemma A.1. [MV94, KL94] Let m and n be integers. n = n , (39) n 1 = 0 , (40) n m = n m , (41) m = ( − s m +2 − s − m − ) , m (42) m = ( − m s m ( m +2) m , (43) m = ( − m s − m ( m +2) m (44) Lemma A.2 ([MV94]) . i kj1 = [ i ] s [ i + j ] s ji-1 k+1 (45) i-1 k+11 = [ i + j + k + 1] s [ j ] s [ i + j ] s [ k + j ] s j-1i k (46)Calculating the actions of A , A and A on n ( i, j, k ) follows directly from Lemma A.1 (42) Proposition A.3. A · n ( i, j, k ) = ( − s i +2 − s − i − ) n ( i, j, k ) A · n ( i, j, k ) = ( − s j +2 − s − j − ) n ( i, j, k ) A · n ( i, j, k ) = ( − s k +2 − s − k − ) n ( i, j, k ) Proof.
This follows immediately from Lemma A.1.The actions of A , A and X on m ( i, j, k ) also follow directly from Lemma A.1 (42). Proposition A.4.
For all i, j, k such that the triples ( i, i, j ) and ( k, k, j ) are admissible, X · m ( i, j, k ) = ( − s j +2 − s − j − ) m ( i, j, k ) , (47) A · m ( i, j, k ) = ( − s i +2 − s − i − ) m ( i, j, k ) (48) A · m ( i, j, k ) = ( − s k +2 − s − k − ) m ( i, j, k ) (49) Proof.
This follows from Lemma A.1 (42), for example X · m ( i, j, k ) = i kj1 (42) = ( − s j +2 − s − j − ) i kb . We shall now calculate the action of B , B and B on the n ( i, j, k ) basis of Sk s ( H ). Definition A.5.
Let ( i, j, k ) be an admissible triple. We define the coefficients D a,b ( i, j, k ) for a, b = ± D , − ( i, j, k ) = − (cid:2) j + k − i (cid:3) s [ j ] s [ j + 1] s D − , − ( i, j, k ) = (cid:2) i + j + k +22 (cid:3) s (cid:2) i + j − k (cid:3) s [ i ] s [ i + 1] s [ j ] s [ j + 1] s D − , ( i, j, k ) = − (cid:2) i + k − j (cid:3) s [ i ] s [ i + 1] s D , ( i, j, k ) = 1where the coefficient is defined to be 0 if the denominator is 0. Proposition A.6.
Let ( i, j, k ) be an admissible triple. B · n ( i, j, k ) = (cid:88) a,b ∈{− , } D a,b ( i, j, k ) n ( i + a, j + b, k ) B · n ( i, j, k ) = (cid:88) a,b ∈{− , } D a,b ( i, k, j ) n ( i + a, j, k + b ) B · n ( i, j, k ) = (cid:88) a,b ∈{− , } D a,b ( j, k, i ) n ( i, j + a, k + b ) Proof. B · n ( i, j, k ) efn = i kbbaa cc isotopy = baa ccb Wenzl = baa ccb − [ i ] s [ i + 1] s baa ccb − [ j ] s [ j + 1] s aa ccb + [ i ] s [ j ] s [ i + 1] s [ j + 1] s baa ccb (45) , (46) = baa ccb − [ b ] s [ i + 1] s [ i ] s baa ccb − [ c ] s [ j + 1] s [ j ] s aa ccb + [ a ] s [ a + b + c + 1] s [ i + 1] s [ j + 1] s [ i ] s [ j ] s baa ccb = n ( i + 1 , j + 1 , k ) − (cid:2) i + k − j (cid:3) s [ i ] s [ i + 1] s n ( i − , j + 1 , k ) − (cid:2) j + k − i (cid:3) s [ j ] s [ j + 1] s n ( i + 1 , j − , k ) + (cid:2) i + j − k (cid:3) s (cid:2) i + j + k +22 (cid:3) s [ i ] s [ i + 1] s [ j ] s [ j + 1] s n ( i + 1 , j + 1 , k )and the other cases are symmetric.Now we calculate the action of B and B on the m ( i, j, k ) basis of Sk s ( H ). Proposition A.7.
For all i, j, k such that the triples ( i, i, j ) and ( k, k, j ) are admissible, B · m ( i, j, k ) = (cid:40) m ( i + 1 , j, k ) for i = j m ( i + 1 , j, k ) + [2 i + j +1] s [ i − j/ s [ i ] s [ i +1] s m ( i − , j, k ) for i > j/ B · m ( i, j, k ) = (cid:40) m ( i, j, k + 1) for k = j m ( i, j, k + 1) + [2 k + j +1] s [ k − j/ s [ k ] s [ k +1] s m ( i, j, k − for k > j/ Proof.
When i (cid:54) = j (and so a (cid:54) = 0) we have: B · m ( i, j, k ) defn = j kba isotopy = j kcba Wenzl = j kabc − [ i ] s [ i + 1] s j kbcai-1 sotopy = m ( i + 1 , j, k ) − [ i ] s [ i + 1] s j kbcai-1 (46) = m ( i + 1 , j, k ) + [ i ] s [ a + b + c + 1] s [ a ] s [ i + 1] s [ a + b ] s [ a + c ] s j kbcai-1 = m ( i + 1 , j, k ) − (cid:2) i − j (cid:3) s [2 i + j + 1] s [ i ] s [ i + 1] s m ( i − , j, k )When i = j this implies that a = 0, so at the fourth step the term j kbcai-1 = 0by Equation (41) and Equation (40). The result for B is analogous. B Calculation of A Action
Finally we need to find A · m ( i, j, k ). This is possible using a change of basis: Proposition B.1.
For all i, j, k such that the triples ( i, i, j ) and ( k, k, j ) are admissible, x · m ( i, j, k ) = (cid:88) a,b (cid:0) − s a +2 − s − a − (cid:1) (cid:8) i i ak k j (cid:9) { i k bk i a } m ( i, b, k ) where the sum is over all a, b such that { i, k, a } , { i, i, b } , and { k, k, b } are admissible.Proof. i j k1 = (cid:88) a (cid:8) i i ak k j (cid:9) i ka 1 ( T heorem . (cid:88) a (cid:0) − s a +2 − s − a − (cid:1) (cid:8) i i ak k j (cid:9) i ka (42)= (cid:88) a,b (cid:0) − s a +2 − s − a − (cid:1) (cid:8) i i ak k j (cid:9) { i k bk i a } i kb ( T heorem . a, b such that { i, k, a } , { i, i, b } , and { k, k, b } are admissible.However, this result is not very explicit, it does not even allow one to easily see how many termsthere are, and is not sufficient for our purposes, so we shall compute the result directly. Theorem B.2.
Let i, j, k be such that the triples ( i, i, j ) and ( k, k, j ) are admissible The action ofthe middle A -loop A on the m -basis of Sk s ( H ) is given by A · m ( i, j, k ) = − (cid:18) ( s − s − ) [ j/ s [ i + j/ s [ k + j/ s [ j − s [ j ] s [ j + 1] s (cid:19) m ( i, j − , k ) (cid:18) ( s − s − ) [ i − j/ s [ k − j/ s (cid:19) m ( i, j + 2 , k )+ (cid:18)(cid:0) − s − i + k +1) − s i + k +1) (cid:1) + K (cid:19) m ( i, j, k ) where K = ( s − s − ) (cid:32) [ j/ s [ i − j/ s [ k − j/ s [ j + 1] s [ j + 2] s + [ j/ s [ i + j/ s [ k + j/ s [ j ] s [ j + 1] s (cid:33) Remark
B.3 . In the above expressions, we have used the convention that if j = 0, then any coefficientterm with [ j ] in the denominator is 0. In fact, this follows by inspection of these coefficients, sincethe term [ j/ s / [ j ] s (and hence each coefficient itself) is equal to zero when j = 0.For the proof of Theorem B.2 we first need some lemmas. Lemma B.4. m-11 = s m − m , m-1 1 = s − m +1 m Proof. ( Skein ) = s − (cid:124) (cid:123)(cid:122) (cid:125) † ) + s (cid:124) (cid:123)(cid:122) (cid:125) crossing has moved one position over = · · · = s m − m Where ( † ) is ( Skein ) = s −
11 m-31 + s
111 m-3 (cid:124) (cid:123)(cid:122) (cid:125) = · · · = s − m (cid:124) (cid:123)(cid:122) (cid:125) = 0 . Lemma B.5. = s − m + s m m-11 1m-111 = s m + s − m m-1 11 Proof. ( Skein ) = s − + s Skein ) = s − + (cid:124) (cid:123)(cid:122) (cid:125) + (cid:124) (cid:123)(cid:122) (cid:125) + s = · · · = s − m + s m m-11 1 The second result is analogous.
Lemma B.6.
11i j = − s i , i 11j = s − i i 11j
11 kj = − s − − k
11j k , k11j = s k k11j Proof.
11i j (43) = − s −
11i j (B.4) = − s − i = − s − i i 11j (B.4) = s − i i 11j The final two cases are analogous.
Lemma B.7.
1i a bc j de kf = j+1i+1 k+1 when j = 0 j+1i+1 k+1 − [ j +12 ] s [ j ] s [ j +1] s j-1 i+1 k+1 for j > roof. When j (cid:54) = 0 we have
1i a bc j de kf isotopy = j kdefia bc 1 = b 1 ea c df Wenzl = a c dfb+1 e+1 − [ j ] s [ j + 1] s a c dfb e1 1j-1 (45) = j+1i+1 k+1 − [ j − b ] s [ j ] s [ j − e ] s [ j ] s [ j ] s [ j + 1] s dfa c1 1b ej-b-1 = j+1i+1 k+1 − (cid:2) j − (cid:3) s [ j ] s [ j + 1] s j-1 i+1 k+1 as b = e = j − . When j = 0 we must have b = c = 0. We apply the isotopy as before but insteadof applying the Wenzl recurrence relation we just add another box to the red strand and reverse theisotopy to give the result. Lemma B.8.
When j = 0 ji a bc kd e f 11 = j+2i+1 k+1 − s ji+1 k+1 and when j (cid:54) = 0 ji a bc kd e f 11 = (cid:2) i − j + 1 (cid:3) s (cid:2) k − j + 1 (cid:3) s [ i + 1] s [ k + 1] s j+2i+1 k+1 − (cid:32) (cid:2) j +22 (cid:3) s (cid:2) i − j + 1 (cid:3) s (cid:2) k − j + 1 (cid:3) s [ i + 1] s [ j + 1] s [ j + 2] s [ k + 1] s + (cid:2) j (cid:3) s (cid:2) i + j + 2 (cid:3) s (cid:2) k + j + 2 (cid:3) s [ i + 1] s [ j ] s [ j + 1] s [ k + 1] s (cid:33) ji+1 k+1 (cid:2) j (cid:3) s (cid:2) i + j + 2 (cid:3) s (cid:2) k + j + 2 (cid:3) s [ i + 1] s [ j − s [ j ] s [ j + 1] s [ k + 1] s j-2i+1 k+1 Proof.
When j (cid:54) = 0 we have ji a bc kd e f 11 Wenzl = i a bc 11 1 kdef − [ j ] s [ j +1] s c (45) , (46) = [ a ] s [ a + b ] s [ d ] s [ d + e ] s i a-1 b 1kd-1ef+1 − [ j ] s [ j + 1] s [ a + b + c + 1] s [ b ] s [ b + a ] s [ b + c ] s [ d + e + f + 1] s [ e ] s [ e + d ] s [ e + f ] s i a b-1 1kde-1 c f = (cid:2) i − j + 1 (cid:3) s (cid:2) k − j + 1 (cid:3) s [ i + 1] s [ k + 1] s i a-1 b 1kd-1ef+1 − (cid:2) j (cid:3) s (cid:2) i + j + 2 (cid:3) s (cid:2) k + j + 2 (cid:3) s [ i + 1] s [ j ] s [ j + 1] s [ k + 1] s i a b-1 1kde-1 c f as a = i +2 − j , d = k +2 − j , b = c = e = f = j Lemma B. = (cid:2) i − j + 1 (cid:3) s (cid:2) k − j + 1 (cid:3) s [ i + 1] s [ k + 1] s j+2i+1 k+1 − (cid:32) (cid:2) j +22 (cid:3) s (cid:2) i − j + 1 (cid:3) s (cid:2) k − j + 1 (cid:3) s [ i + 1] s [ j + 1] s [ j + 2] s [ k + 1] s + (cid:2) j (cid:3) s (cid:2) i + j + 2 (cid:3) s (cid:2) k + j + 2 (cid:3) s [ i + 1] s [ j ] s [ j + 1] s [ k + 1] s (cid:33) ji+1 k+1 + (cid:2) j (cid:3) s (cid:2) i + j + 2 (cid:3) s (cid:2) k + j + 2 (cid:3) s [ i + 1] s [ j − s [ j ] s [ j + 1] s [ k + 1] s j-2i+1 k+1 The case when j = 0 is similar except at the first step one does not have to apply the Wenzl relationso there is no second term. e can now use these lemmas to prove Theorem B.2 Proof. i j k1 ( Lemma B. = s i − k + s − i − k
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