DDAHA AND SKEIN ALGEBRA OF SURFACE: DOUBLE-TORUS KNOTS
KAZUHIRO HIKAMIAbstract. We study a topological aspect of rank-1 double affine Hecke algebra (DAHA).Clarified is a relationship between the DAHA of A -type (resp. C ∨ C -type) and the skein al-gebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL ( (cid:90) ) actions ofDAHAs are identified with the Dehn twists on the surfaces. Combining these two types ofDAHA, we construct the DAHA representation for the skein algebra on a genus-two sur-face, and we propose a DAHA polynomial for a double-torus knot, which is a simple closedcurve on a genus two Heegaard surface in S . Discussed is a relationship between the DAHApolynomial and the colored Jones polynomial.
1. IntroductionThe double affine Hecke algebra (DAHA) was introduced by Cherednik, and it is a powerfultool in studies of the Macdonald polynomials associated with root systems (see, e.g. , [13, 35]).The Macdonald polynomial is ubiquitous in mathematics and physics, and an interpretationas a q -deformation of a wave-function of the quantum Hall effect suggest an importance ofa topological structure of the DAHA in studies of topological orders. The DAHA was re-cently applied to quantum topology. Proposed was a DAHA polynomial invariant [14, 15],and discussed was a relationship with the refined Chern–Simons invariant and the Kho-vanov homology [2]. The construction of the DAHA polynomial [14] is purely algebraic,but the DAHA polynomial is limited only to torus knots and their descendants, i.e. , all arenon-hyperbolic. An attempt [3] was made towards DAHA for a genus-two surface general-izing DAHA of A -type, but a relationship with the known quantum polynomial invariantsis unclear.A purpose of this article is to combine two rank-1 DAHAs of A -type and C ∨ C -type toconstruct the DAHA representation for double-torus knots. The double-torus knot [28, 29] isa simple closed curve on a genus two Heegaard surface in S , and a large family of knots suchas twist knots belong to this type. Originally the DAHAs of A -type and C ∨ C -type are for theRogers polynomial (or the q -ultraspherical polynomial) and the Askey–Wilson polynomialrespectively. The Askey–Wilson polynomial [4] is on top of the Askey scheme of classifica-tion of orthogonal polynomials of hypergeometric-type, and its algebraic structure receivesrecent active interests (see [32, 33, 45, 46]). Here we pay attentions to a relationship betweenthe DAHA and the Kauffman bracket skein algebra on surfaces. It is known that the DAHAof C ∨ C -type represents a quantization of the affine cubic surface which is the character va-riety of a 4-punctured sphere [41], while the DAHA of A -type is related to the character Date : December 27, 2018.2000
Mathematics Subject Classification.Key words and phrases. knot, colored Jones polynomial, double affine Hecke algebra, skein algebra, Macdon-ald polynomial, Askey–Wilson polynomial. a r X i v : . [ m a t h - ph ] J a n K. HIKAMI variety of a once-punctured torus. Based on the fact [9, 42] that the coordinate ring of thecharacter varieties is a specialization of the Kauffman bracket skein algebra, discussed alsois a relationship with the skein algebra on the 4-punctured sphere and the once-puncturedtorus [6, 7].For each simple closed curve on the genus-two surface, we assign a DAHA operator whichrepresents the skein algebra on the surface. A benefit of our method combining two types ofrank-1 DAHAs is that we can make use of their well-known automorphisms. Due to the rela-tionship between the DAHA and the skein algebra, the DAHA automorphisms are regardedas the mapping class group [8, 19], the group of isotopy classes of orientation-preservingdiffeomorphisms of surface. As the mapping class group is generated by the Dehn twists, wecan clarify the SL ( (cid:90) ) actions of the DAHA of A -type and C ∨ C -type as the Dehn twistsabout curves on each surface. The q -difference DAHA operator is indeed constructed by useof the automorphisms of DAHA as in the case of torus knots by Cherednik [14]. Using theDAHA operator assigned to a simple closed curve (cid:99) on the surface, we propose a DAHApolynomial for (cid:99) . We compute explicitly the DAHA polynomial for double-twist knots, anddiscuss a relationship with the colored Jones polynomial.This paper is organized as follows. In Section 2, we study the once-punctured torus Σ , . Werecall properties of the DAHA of A -type, and establish a relationship with the skein algebraon Σ , . The DAHA polynomial proposed by Cherednik is also reviewed. Section 3 is for the4-punctured sphere Σ , . We recall both the DAHA of C ∨ C -type and the skein algebra on Σ , .Based on the correspondence, we associate a DAHA operator for a simple closed curve with arational slope. In these sections, the SL ( (cid:90) ) -actions on the rank-1 DAHAs are interpreted asthe Dehn twists about certain curves on the surfaces Σ , and Σ , . In Section 4, as a prototypetoward the genus-two surface, we study the skein algebra on a twice-punctured torus Σ , . We“glue” two types of the rank-1 DAHAs, A -type and C ∨ C -type, using a quantum dilogarithmfunction, and give the DAHA representation of the skein algebra on Σ , . Section 5 is forthe genus-two surface. We propose the DAHA polynomial for a simple closed curve on thesurface, and study a relationship with the colored Jones polynomial. In the rest of this section,we collect our notations such as special functions.1.1. Preliminaries.
The Kauffman bracket skein module KBS A ( M ) of a 3-manifold M is de-fined by = A + A − , (1.1) = − A − A − . When M = Σ × [ , ] with an oriented surface Σ , we write KBS A ( Σ ) . Here, a multiplication (cid:120) (cid:121) of curves (cid:120) and (cid:121) means that (cid:120) is vertically above (cid:121) , (cid:120) (cid:121) = (cid:120)(cid:121) (1.2) AHA AND SKEIN ALGEBRA 3
Throughout this paper, we use for simplicity a variant of hyperbolic functionssh ( x ) = x − x − , ch ( x ) = x + x − . (1.3)We recall the standard notations of q -calculus. We use the q -Pochhammer symbol defined by ( x ; q ) n = n (cid:214) j = (cid:0) − x q j − (cid:1) , (1.4) ( x , x , · · · ; q ) n = ( x ; q ) n ( x ; q ) n · · · . Here we mean ( x ; q ) =
1, and for negative integers ( x ; q ) − n = ( x ; q ) ∞ ( x q − n ; q ) ∞ = ( x q − n ; q ) n . (1.5)We also use the q -hypergeometric series r ϕ s (cid:20) a , · · · , a r b , · · · , b s ; q , z (cid:21) = ∞ (cid:213) n = ( a , · · · , a r ; q ) n ( q , b , · · · , b s ; q ) n (cid:16) (− ) n q n ( n − ) (cid:17) + s − r z n . (1.6)See, e.g. , [22] for properties of the hypergeometric functions.AcknowledgmentsThe author would like to thank H. Fuji, A.N. Kirillov, and H. Murakami for communica-tions and comments on a draft of the manuscript. A part of this work was presented at theworkshop “Volume Conjecture in Tokyo” on August 2018, celebrating the 60th birthday ofJun Murakami and Hitoshi Murakami. Thanks to the organizers and the participants. Thiswork is supported in part by KAKENHI JP16H03927, JP17K05239, JP17K18781.2. Once-Punctured Torus2.1. Skein Algebra.
We study the skein module on a once-punctured torus Σ , . We set sim-ple closed curves (cid:120) , (cid:121) , (cid:122) , and (cid:98) as in Fig. 1. See that (cid:98) denotes the boundary circle of thepuncture. b x yz Figure 1.
Depicted are simple closed curves on the once-puncture torus Σ , . Proposition 2.1 ([10]) . The
KBS A ( Σ , ) is generated by (cid:120) , (cid:121) , and (cid:122) , satisfying A (cid:120) (cid:121) − A − (cid:121) (cid:120) = (cid:0) A − A − (cid:1) (cid:122) , A (cid:121) (cid:122) − A − (cid:122) (cid:121) = (cid:0) A − A − (cid:1) (cid:120) , A (cid:122) (cid:120) − A − (cid:120) (cid:122) = (cid:0) A − A − (cid:1) (cid:121) . (2.1) K. HIKAMI
It is noted that the boundary circle (cid:98) is generated by (cid:98) = A (cid:120) (cid:121) (cid:122) − A (cid:120) − A − (cid:121) − A (cid:122) + A + A − . (2.2)2.2. DAHA of A -type. We collect several properties of DAHA of A -type. Essential refer-ences are [13, 35]. Definition 2.2.
The DAHA of A -type (cid:72) q , t is (cid:67) ( q , t ) -algebra generated by Y ± , X ± , T ± satis-fying ( T + t ) (cid:0) T − t − (cid:1) = , T X T = X − , T − Y T − = Y − , X Y = q − Y X T . (2.3)We use an idempotent e = t + t − ( t + T ) , (2.4)which satisfies e = e , e T = T e = t − e . (2.5) Definition 2.3.
The spherical DAHA SH q , t is SH q , t = e (cid:72) q , t e . The automorphisms of A -DAHA are listed in the following [13]. Lemma 2.4. • An automorphism ϵ : (cid:72) q , t → (cid:72) q − , t − ; ϵ : (cid:169)(cid:173)(cid:171) XYT (cid:170)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:171)
YXT − (cid:170)(cid:174)(cid:172) . (2.6) • An anti-automorphism ϵ (cid:48) : (cid:72) q , t → (cid:72) q , t ; ϵ (cid:48) : (cid:169)(cid:173)(cid:171) XYT (cid:170)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:171) Y − X − T (cid:170)(cid:174)(cid:172) . (2.7) Lemma 2.5 ([13]) . The SL ( (cid:90) ) action on (cid:72) q , t is generated by (cid:18) (cid:19) (cid:55)→ τ R , (cid:18) (cid:19) (cid:55)→ τ L , (2.8) where τ • : (cid:72) q , t → (cid:72) q , t is τ R : (cid:169)(cid:173)(cid:171) TYX (cid:170)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:171) T q X YX (cid:170)(cid:174)(cid:172) , τ L : (cid:169)(cid:173)(cid:171) TYX (cid:170)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:171) TY q − Y X (cid:170)(cid:174)(cid:172) . (2.9) AHA AND SKEIN ALGEBRA 5
We note that, with (2.6), we have τ L = ϵ τ R ϵ . (2.10)2.3. Polynomial Representation and Macdonald Polynomial.
We recall a representation onring of the Laurent polynomials (cid:67) [ x ± ] [13]. We use an involution s and a q -difference oper-ator ð respectively defined by s f ( x ) = f ( x − ) , ð f ( x ) = f ( q x ) , (2.11)where f ∈ (cid:67) [ x ± ] . Proposition 2.6.
A polynomial representation in (cid:67) [ x ± ] is given by T (cid:55)→ t − s + (cid:0) t − − t (cid:1) x − ( s − ) , (2.12) X (cid:55)→ x , Y (cid:55)→ ð s T . Here Y is called the Dunkl–Cherednik operator. We see that T f = t − f for the symmetricLaurent polynomials f ∈ (cid:67) [ x + x − ] , and that e (cid:67) [ x ] = (cid:67) [ x + x − ] . Thus SH q , t preservesa symmetric space (cid:67) [ x + x − ] . As a q -difference operator of SH q , t , we have the followingexpression, Y + Y − (cid:12)(cid:12) sym (cid:55)→ t x − t − x − x − x − ð + t − x − t x − x − x − ð − , (2.13)where h | sym means that h ∈ (cid:72) q , t acts on the symmetric Laurent polynomial space (cid:67) [ x + x − ] .This operator is known as the Macdonald operator (see, e.g. , [34]). One also finds that q X Y + q − Y − X − (cid:12)(cid:12)(cid:12) sym (cid:55)→ q x t x − t − x − x − x − ð + q x − t − x − t x − x − x − ð − . (2.14)We have the non-symmetric Macdonald polynomials E m ( x ; q , t ) as eigenfunctions of Y , Y E − m ( x ; q , t ) = t − q − m E − m ( x ; q , t ) , Y E m ( x ; q , t ) = t q m E m ( x ; q , t ) . (2.15)Here m >
0, and the Laurent polynomials E m ( x ; q , t ) have forms of E − m ( x ; q , t ) = x − m + (cid:0) t − t − (cid:1) q m t q m − t − q − m x m + · · · , E m ( x ; q , t ) = x m + · · · , (2.16)where · · · means Laurent polynomials x k with | k | < m . It is noted that E ( x ; q , t ) = Y E ( x ; q , t ) = t − E ( x ; q , t ) .Symmetric eigenfunctions of (2.13) are the Macdonald polynomials of A -type (also knownas the q -ultraspherical polynomial, or the Rogers polynomial [35]). Explicitly, we have (cid:0) Y + Y − (cid:1) M n ( x ; q , t ) = (cid:0) t q n + t − q − n (cid:1) M n ( x ; q , t ) , (2.17) K. HIKAMI where M n ( x ; q , t ) = x n · ϕ (cid:20) t , q − n t − q − n ; q , q t − x − (cid:21) (2.18) = ( q ; q ) n ( t ; q ) n (cid:213) j , k ≥ j + k = n ( t ; q ) j ( t ; q ) k ( q ; q ) j ( q ; q ) k x j − k . Here the polynomials are normalized to be M ( x ; q , t ) = M n ( x ; q , t ) = ( x n + x − n ) + · · · , for n >
0. The polynomials M n ( x ; q , t ) span the symmetric Laurent polynomial space (cid:67) [ x + x − ] . In terms of the non-symmetric polynomials (2.15), we have M m ( x ; q , t ) = E − m ( x ; q , t ) + q m − q − m t q m − q − m E m ( x ; q , t ) (2.19) = t − ( T + t ) E m ( x ; q , t ) . Some of them are explicitly written as follows; M ( x ; q , t ) = , M ( x ; q , t ) = x + x − , (2.20) M ( x ; q , t ) = x + x − + (cid:0) + q (cid:1) (cid:0) − t (cid:1) − q t , M ( x ; q , t ) = x + x − + (cid:0) − q (cid:1) (cid:0) − t (cid:1) ( − q ) ( − q t ) (cid:0) x + x − (cid:1) . Note that the generating function of the A -type Macdonald polynomials is ∞ (cid:213) n = M n ( x ; q , t ) (cid:0) t ; q (cid:1) n ( q ; q ) n z n = (cid:0) t x z , t x − z ; q (cid:1) ∞ ( x z , x − z ; q ) ∞ . (2.21)One sees that the Macdonald polynomial (2.18) reduces at q = t to M n ( x ; q , q ) = x n + − x − n − x − x − = S n ( x + x − ) . (2.22)Here S n ( z ) is the Chebyshev polynomial of the second kind, which is also defined recursivelyby z S n ( z ) = S n + ( z ) + S n − ( z ) , (2.23)with S ( z ) = S ( z ) = z .We list some identities of the A -type Macdonald polynomials. As a typical property oforthogonal polynomials, we have the three-term recurrence relation, ( X + X − ) M n ( X ; q , t ) = M n + ( X ; q , t ) + (cid:0) − q n (cid:1) (cid:0) − q n − t (cid:1) ( − q n − t ) ( − q n t ) M n − ( X ; q , t ) . (2.24)We also have (cid:16) q − Y X + q X − Y − (cid:17) M n ( X ; q , t ) = t q n + M n + ( X ; q , t ) + t − q − n + (cid:0) − q n (cid:1) (cid:0) − q n − t (cid:1) ( − q n − t ) ( − q n t ) M n − ( X ; q , t ) . (2.25)For our later computations, we introduce the raising and lowering operators of M n ( x ; q , t ) . AHA AND SKEIN ALGEBRA 7
Proposition 2.7.
We have the raising operator with a parameter shift, (cid:26) (cid:0) − t x (cid:1) (cid:0) − q t x (cid:1) q t x ( x − ) ð − (cid:0) t − x (cid:1) (cid:0) t q − x (cid:1) q t x ( x − ) ð − (cid:27) M m ( x ; q , q t ) = (cid:0) q m + t − q − m − t − (cid:1) M m + ( x ; q , t ) . (2.26) The lowering operator with a parameter shift is given by xx − (cid:0) ð − ð − (cid:1) M m ( x ; q , t ) = ( q m − q − m ) M m − ( x ; q , q t ) . (2.27) Proof.
It can be proved by calculating actions on the generating function (2.21). See also [30,31]. (cid:3)
These raising and lowering operators, which preserve the symmetric Laurent polynomialspace (cid:67) [ x + x − ] , can be rewritten using the generators of DAHA. For brevity, we denote theraising and lowering operators in Prop. 2.7 as K ( + ) and K (−) respectively. By use of (2.12), wehave t Y − t − Y − (cid:12)(cid:12) sym = t − (cid:0) t − − T (cid:1) ð (cid:12)(cid:12) sym = t − t − x − tx − ( − s ) ð (cid:12)(cid:12) sym , which proves [13, 35] that the lowering operator (2.27) is written as K (−) (cid:12)(cid:12)(cid:12) sym = tt − X − t X − (cid:0) t Y − t − Y − (cid:1)(cid:12)(cid:12)(cid:12) sym . Note that K (−) does not depend on t as operators on (cid:67) [ x + x − ] .Combining the identities (2.26) and (2.27), we have K ( + ) K (−) M m ( x ; q , t ) = (cid:110) (cid:0) Y + Y − (cid:1) − (cid:0) t + t − (cid:1) (cid:111) M m ( x ; q , t ) . Using the above expression for K (−) and the fact that the Macdonald polynomials M m ( x ; q , t ) are bases of (cid:67) [ x + x − ] , we find K ( + ) (cid:12)(cid:12)(cid:12) sym = t − (cid:0) t − Y − t Y − (cid:1) (cid:0) t − X − t X − (cid:1)(cid:12)(cid:12) sym . To conclude, we have the following. We recall that sh ( x ) is defined in (1.3). Proposition 2.8.
Both the raising and lowering operators preserve the symmetric Laurent poly-nomial space (cid:67) [ x + x − ] , and they are written as t − sh (cid:0) t − Y (cid:1) sh (cid:0) t − X (cid:1) M m ( x ; q , q t ) = (cid:0) q m + t − q − m − t − (cid:1) M m + ( x ; q , t ) , t sh ( t − X ) sh ( t Y ) M m ( x ; q , t ) = ( q m − q − m ) M m − ( x ; q , q t ) . (2.28)2.4. Automorphisms as Conjugation.
We revisit the SL ( (cid:90) ) action (2.9) in the polynomialrepresentation of DAHA. As a completion of DAHA [13], we introduce a function U R = exp (cid:18) ( log X ) q (cid:19) . As U R is symmetric in X ↔ X − and s U R = U R s , it commutes with T . One easily sees that ð U R = q X U R ð , and we obtain Y U R = ð s T U R = q X U R ð s T = U R q X Y . K. HIKAMI
Trivial is a commutativity between U R and X , and thus, the automorphism τ R (2.9) is identifiedwith a conjugation by U R . For τ L (2.9), we recall (2.10) to find the following U L . Proposition 2.9.
The SL ( (cid:90) ) action on SH q , t is given as conjugation. In particular, the auto-morphisms τ • (2.8) are written as conjugations τ • : h (cid:55)→ U − • h U • , (2.29) where U R = exp (cid:18) ( log X ) q (cid:19) , U L = exp (cid:18) − ( log Y ) q (cid:19) . (2.30)See [18] where the operators U • were introduced for DAHA of A n -type.2.5. Shift Operator.
As seen from the raising (2.26) and the lowering operators (2.27), it isuseful to introduce a parameter shift operator ð t in SH q , t satisfying ð t t = q t ð t . (2.31)The SL ( (cid:90) ) actions τ • (2.8) on t are trivial, but we have the following action on ð t . Proposition 2.10.
We have τ R : ð t (cid:55)→ ð t , and τ L : ð t (cid:55)→ (cid:16) t − q − Y X (cid:17) sh ( t − X ) ð t . (2.32) Proof.
The conjugation (2.29) shows an invariance of ð t under τ R . For τ L , we use the factthat the Macdonald polynomials span the symmetric polynomial space (cid:67) [ x + x − ] , and wecompute actions on M m ( x ; q , t ) . Recalling that M m ( x ; , q , t ) is a sum of the non-symmetricpolynomials (2.19) and that the operator U L is symmetric in Y ↔ Y − , we have the followingequalities; U − L ð t U L M m ( x ; q , t ) (2.15) = U − L ð t e − ( log ( t qm ))
22 log q M m ( x ; q , t ) = U − L e − ( log ( t qm + ))
22 log q M m ( x ; q , q t ) (2.28) = e − ( log ( t qm + ))
22 log q U − L t sh ( t − X ) ( t − Y ) (cid:0) q m + t − q − m − t − (cid:1) M m + ( x ; q , t ) (2.9) = e − ( log ( t qm + ))
22 log q t sh (cid:16) t − q − YX (cid:17) ( t − Y ) U − L (cid:0) q m + t − q − m − t − (cid:1) M m + ( x ; q , t ) (2.15) = t sh (cid:16) t − q − Y X (cid:17) ( t − Y ) (cid:0) q m + t − q − m − t − (cid:1) M m + ( x ; q , t ) (2.28) = (cid:16) t − q − Y X (cid:17) sh (cid:0) t − X (cid:1) M m ( x ; q , q t ) . This proves the statement. (cid:3)
AHA AND SKEIN ALGEBRA 9
Algebra Embedding.
We shall give a DAHA representation for KBS A ( Σ , ) definedin (2.1). For a simple closed curve (cid:99) ( r , s ) with slope s / r on Σ , , we assign DAHA operators as (cid:99) ( r , s ) (cid:55)→ (cid:77) ( r , s ) = ch (cid:0) (cid:79) ( r , s ) (cid:1) ∈ SH q , t , (2.33)where we recall that ch ( x ) is defined in (1.3). The curves in Fig. 1 are identified as (cid:99) ( , ) = (cid:120) , (cid:99) ( , ) = (cid:121) , and (cid:99) ( , ) = (cid:122) . The algebra KBS A ( Σ , ) was studied in detail in [20] (see also [5]),and known is a “product-to-sum formula” for (cid:99) ( r , s ) . For example, we can check easily thefollowing skein algebra; (cid:99) ( , ) (cid:99) ( , ) = A (cid:99) ( , ) + A − (cid:99) ( , − ) , (cid:99) ( , ) (cid:99) ( , ) = A (cid:99) ( , ) + A − (cid:99) ( , ) , (cid:99) ( , ) (cid:99) ( , ) = A (cid:99) ( , ) + A − (cid:99) ( , ) . (2.34)Here we put (cid:77) ( , ) = ch ( X ) , (cid:77) ( , ) = ch ( Y ) , (cid:77) ( , ) = ch (cid:16) q X Y (cid:17) = ch (cid:16) q − Y X (cid:17) . (2.35)Recall that (cid:77) ( , ) is the Macdonald operator of SH q , t , and see (2.13) and (2.14) for explicitforms of the q -difference operators on the symmetric Laurent polynomial space (cid:67) [ x + x − ] .By direct computations, we get a representation for (2.34) (cid:77) ( , ) (cid:77) ( , ) = q − (cid:77) ( , ) + q (cid:77) ( , − ) , (cid:77) ( , ) (cid:77) ( , ) = q − (cid:77) ( , ) + q (cid:77) ( , ) , (cid:77) ( , ) (cid:77) ( , ) = q − (cid:77) ( , ) + q (cid:77) ( , ) , (2.36)where we have defined (cid:77) ( , − ) = ch (cid:16) q − X − Y (cid:17) , (cid:77) ( , ) = ch ( Y X Y ) , (cid:77) ( , ) = ch ( X Y X ) . (2.37)Using these relations, we can check (2.1) in Prop. 2.1, and we obtain the following. Theorem 2.11.
We have an algebra embedding
KBS A ( Σ , ) → SH q , t with A = q − by (cid:169)(cid:173)(cid:171) (cid:120)(cid:121)(cid:122) (cid:170)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:171) (cid:77) ( , ) (cid:77) ( , ) (cid:77) ( , ) (cid:170)(cid:174)(cid:172) . (2.38)We can also check that (cid:77) ( , − ) (cid:77) ( , ) = q − (cid:77) ( , ) T + q (cid:77) ( , ) T + q + q − − (cid:0) t q − + t − q (cid:1) , (2.39)where we follow the notation of [20], (cid:77) ( , ) T = T ( (cid:77) ( , ) ) = X + X − , (cid:77) ( , ) T = T ( (cid:77) ( , ) ) = Y + Y − . Here T n ( z ) denotes the Chebyshev polynomial of the first kind defined by T n ( x + x − ) = x n + x − n , (2.40) which satisfies the same recurrence equation (2.23) with the second kind polynomial. Fromthe viewpoint of the skein algebra, the last term in (2.39) denotes the boundary circle (cid:98) inFig. 1, (cid:98) (cid:55)→ − t q − − t − q . (2.41)Indeed the identities (2.36) and (2.39) give q − (cid:77) ( , ) (cid:77) ( , ) (cid:77) ( , ) = q − (cid:77) ( , ) + q (cid:77) ( , ) + q − (cid:77) ( , ) + (cid:0) − q − q − (cid:1) + (cid:0) − t q − − t − q (cid:1) . (2.42)Recalling the skein algebra relation (2.2), we see that the embedding (2.38) induces (2.41).Actions of (cid:77) ( r , s ) on the Macdonald polynomial M n ( x ; q , t ) as a basis of the symmetric Lau-rent polynomials (cid:67) [ x + x − ] can be computed explicitly in principle. The Macdonald poly-nomials are eigenpolynomials of (cid:77) ( , ) = Y + Y − as in (2.17), and the three-term recurrencerelations (2.24) and (2.25) denote respectively the actions of (cid:77) ( , ) and (cid:77) ( , ) . These relationsreduce to results of representation of KBS A ( Σ , ) [16, 36] when q is a root of unity.Other operators (cid:77) ( r , s ) can be given explicitly using the SL ( (cid:90) ) -action of SH q , t . For in-stance, when we apply the automorphisms (2.9) to (2.36), we get (cid:77) ( n , ) (cid:77) ( , ) = q − (cid:77) ( n − , ) + q (cid:77) ( n + , ) , (cid:77) ( , ) (cid:77) ( , n ) = q − (cid:77) ( , n − ) + q (cid:77) ( , n + ) , (2.43)where for n ≥ (cid:77) ( n , ) = ch (cid:16) q n − X n − Y X (cid:17) , (cid:77) ( , n ) = ch (cid:16) q − n + Y n − X Y (cid:17) . (2.44)With our algebra embedding, the SL ( (cid:90) ) actions (2.9) of DAHA naturally induce the map-ping class group Mod ( Σ , ) (cid:27) SL ( (cid:90) ) (see, e.g. , [8, 19]), which is generated by the Dehntwists T (cid:120) (resp. T (cid:121) ) about the curve (cid:120) (resp. (cid:121) ). As seen from the fact that τ R : (cid:99) ( , ) (cid:55)→ (cid:99) ( , ) , τ R : (cid:99) ( , ) (cid:55)→ (cid:99) ( , ) , and that the boundary circle (cid:98) is fixed, the automorphism τ R (2.8) denotesthe right Dehn twist T − (cid:120) about (cid:120) = (cid:99) ( , ) . In the same manner, as we have τ L : (cid:99) ( , ) (cid:55)→ (cid:99) ( , ) and τ L : (cid:99) ( , ) (cid:55)→ (cid:99) ( , ) , we can identify the automorphism τ L (2.8) with the left Dehn twist T (cid:121) about (cid:121) = (cid:99) ( , ) . For our later use, we summarize the actions of the Dehn twists as follows. T (cid:121) (cid:55)→ τ L , τ ± L : (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) TXY ð t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) T q ∓ Y ± XY (cid:16) t − q ∓ Y ± X (cid:17) sh (cid:0) t − X (cid:1) ð t (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , T (cid:120) (cid:55)→ τ − R , τ ± R : (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) TXY ð t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) TX q ± X ± Y ð t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) . (2.45) AHA AND SKEIN ALGEBRA 11
DAHA Polynomial.
The N -colored Jones polynomial of knot K is a linear combinationof the Kauffman bracket polynomials for k parallel copies of knot K (1 ≤ k ≤ N − (− ) N − A N − A − N A − A − . We should recall that this is the Chebyshevpolynomial of the second kind. A case of N = (cid:99) ( r , s ) with coprime integers ( r , s ) on the genus-one Heegaard surface in S denotes atorus knot, the DAHA operator (cid:77) ( r , s ) (2.33) associated to (cid:99) ( r , s ) is expected to be related withthe (colored) Jones polynomial of the torus knot. Following Cherednik [14, 15], we define P n ( x , q , t ; (cid:99) ( r , s ) ) = M n − ( (cid:79) ( r , s ) ; q , t )( ) . (2.46)Here (cid:79) ( r , s ) = τ ( X ) where the SL ( (cid:90) ) -action is T (cid:55)→ τ (2.45) when the curve is (cid:99) ( r , s ) = T ( (cid:99) ( , ) ) . Note that the case of n = P ( x , q , t ; (cid:99) ( r , s ) ) = (cid:77) ( r , s ) ( ) = ch ( (cid:79) ( r , s ) )( ) . (2.47)For instance, we have P n ( x , q , t ; (cid:99) ( , ) ) = M n − ( x ; q , t ) , P n ( x , q , t ; (cid:99) ( , ) ) = M n − ( t − ; q , t ) . (2.48)As the Macdonald polynomial reduces to the Chebyshev polynomial of the second kind (2.22)at a specific setting q = t , both polynomials P n ( x , q , t ; (cid:99) ( r , s ) ) for ( r , s ) = ( , ) , ( , ) are re-garded as a deformation of the n -colored Jones polynomial for unknot.We show an explicit result for the curve (cid:99) ( k + , ) = (cid:16) T − k (cid:120) ◦ T (cid:121) (cid:17) ( (cid:99) ( , ) ) . We have (cid:79) ( k + , ) = (cid:16) τ kR ◦ τ L (cid:17) ( X ) = q k − ( X k Y ) X , (2.49)and the DAHA operator associated to (cid:99) ( k + , ) is given by (cid:77) ( k + , ) = ch (cid:16) q k − ( X k Y ) X (cid:17) . This can be written as the operator on the symmetric polynomials (cid:67) [ x + x − ] as (cid:77) ( k + , ) (cid:55)→ (cid:98) M ( )( k + , ) ( x ; q , t ) = ( q x ) k + (cid:0) − t x (cid:1) (cid:0) − q t x (cid:1) t ( − x ) ( − q x ) ð + (cid:0) q x − (cid:1) k + (cid:0) t − x (cid:1) (cid:0) q t − x (cid:1) t ( − x ) ( q − x ) ð − − q t − (cid:0) q − t (cid:1) (cid:0) − t (cid:1) x (cid:0) + x (cid:1) ( q − x ) ( − q x ) . (2.50)From this expression we obtain P (− q , q , − q ; (cid:99) ( k + , ) ) = (cid:98) M ( )( k + , ) ( x ; q , t )( ) (cid:12)(cid:12)(cid:12) x = t = − q = − q k − q k + − q k + = − q k + (cid:0) q + q − (cid:1) J ( q ; T ( k + , ) ) , (2.51) where J N ( q ; T ( s , t ) ) is the colored Jones polynomial for torus knot [38] normalized to be J N ( q ; unknot ) = J N ( q ; T ( s , t ) ) = q st ( − N ) q N − q − N N − (cid:213) r = − N − (cid:16) q str −( s + t ) r + − q str −( s − t ) r − (cid:17) . (2.52)We note that the three-term recurrence relation (2.24) for the Macdonald polynomial givesa recursion relation for the DAHA polynomial, P n + ( x , q , t ; c ( k + , ) ) = (cid:98) M ( )( k + , ) ( x ; q , t ) (cid:0) P n ( x , q , t ; c ( k + , ) ) (cid:1) − (cid:0) − q n − (cid:1) (cid:0) − q n − t (cid:1) ( − q n − t ) ( − q n − t ) P n − ( x , q , t ; c ( k + , ) ) . (2.53)We find that this agrees with the colored Jones polynomial up to framing factor at a specificpoint, P m (− q , q , − q ; (cid:99) ( k + , ) ) = q ( k + )( m − ) q m − q − m q − q − J m ( q ; T ( k + , ) ) . (2.54)See [14, 15] for further computations of the A n DAHA polynomials.3. 4-Punctured Sphere3.1.
Skein Algebra.
We set simple closed curves (cid:120) , (cid:121) , (cid:122) , and (cid:98) j on a 4-punctured sphere Σ , as in Fig. 2. The boundary circles (cid:98) j of the punctures are central. b b b b xyz Figure 2.
Depicted are simple closed curves on the 4-punctured sphere Σ , . Proposition 3.1 ([10]) . The Kauffman bracket skein module
KBS A ( Σ , ) is generated by (cid:120) , (cid:121) , (cid:122) ,and (cid:98) j satisfying A (cid:120) (cid:121) − A − (cid:121) (cid:120) = (cid:0) A − A − (cid:1) (cid:122) + (cid:0) A − A − (cid:1) ( (cid:98) (cid:98) + (cid:98) (cid:98) ) , A (cid:121) (cid:122) − A − (cid:122) (cid:121) = (cid:0) A − A − (cid:1) (cid:120) + (cid:0) A − A − (cid:1) ( (cid:98) (cid:98) + (cid:98) (cid:98) ) , A (cid:122) (cid:120) − A − (cid:120) (cid:122) = (cid:0) A − A − (cid:1) (cid:121) + (cid:0) A − A − (cid:1) ( (cid:98) (cid:98) + (cid:98) (cid:98) ) , (3.1) with A (cid:120) (cid:121) (cid:122) = A (cid:120) + A − (cid:121) + A (cid:122) + A ( (cid:98) (cid:98) + (cid:98) (cid:98) ) (cid:120) + A − ( (cid:98) (cid:98) + (cid:98) (cid:98) ) (cid:121) + A ( (cid:98) (cid:98) + (cid:98) (cid:98) ) (cid:122) + (cid:98) + (cid:98) + (cid:98) + (cid:98) + (cid:98) (cid:98) (cid:98) (cid:98) − (cid:0) A + A − (cid:1) . (3.2) AHA AND SKEIN ALGEBRA 13
Figure 3.
A simple closed curve with slope 5 / Σ , is given. Here punctures areon corners of the square. Essential simple closed curves on Σ , are parameterized by a slope (cid:81) ∪ {∞} . Curves (cid:120) , (cid:121) ,and (cid:122) are identified respectively with curves (cid:99) ( , ) , (cid:99) ( , ) , and (cid:99) ( , ) , where (cid:99) ( r , s ) with coprimeintegers r and s denotes a simple closed curve with a slope s / r (see Fig. 3). The multiplicativestructure of these curves were investigated in detail [5], and a “product-to-sum formula” [20]was given. For example, one finds the following skein relation; (cid:99) ( , ) (cid:99) ( , ) = A (cid:99) ( , ) + A − (cid:99) ( , − ) + ( (cid:98) (cid:98) + (cid:98) (cid:98) ) , (cid:99) ( , ) (cid:99) ( , ) = A (cid:99) ( , ) + A − (cid:99) ( , ) + ( (cid:98) (cid:98) + (cid:98) (cid:98) ) , (cid:99) ( , ) (cid:99) ( , ) = A (cid:99) ( , ) + A − (cid:99) ( , ) + ( (cid:98) (cid:98) + (cid:98) (cid:98) ) . (3.3)3.2. C ∨ C -DAHA. A generalization of the rank-1 DAHA is known as DAHA of C ∨ C -type(or the Askey–Wilson type), which has four parameters besides q . Definition 3.2.
The DAHA of C ∨ C -type (cid:72) q , t , t , t , t is generated by T ± , T ± , T ∨ ± , and T ∨ ± ,satisfying (cid:0) T − t − (cid:1) ( T + t ) = , (cid:0) T − t − (cid:1) ( T + t ) = , (cid:0) T ∨ − t − (cid:1) (cid:0) T ∨ + t (cid:1) = , (cid:0) T ∨ − t − (cid:1) (cid:0) T ∨ + t (cid:1) = , (3.4) and T ∨ T T T ∨ = q − . (3.5)We use an idempotent e = t + t − ( t + T ) . (3.6)satisfying e = e , e T = T e = t − e . (3.7)Hereafter we denote t = ( t , t , t , t ) . Definition 3.3.
The spherical DAHA SH q , t of C ∨ C -type is defined by SH q , t = e (cid:72) q , t e . We recall some of the known automorphisms of DAHA. See, e.g. , [40, 41].
Lemma 3.4.
We have an involutive anti-automorphism ϵ (cid:48) defined by ϵ (cid:48) : (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) T T T ∨ T ∨ (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) T ∨ T T ∨ T (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) t t t t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) t t t t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) . (3.8) Lemma 3.5.
The SL ( (cid:90) ) action on (cid:72) q , t is generated by (cid:18) (cid:19) (cid:55)→ σ R , (cid:18) (cid:19) (cid:55)→ σ L , (3.9) where the automorphisms are σ R : (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) T T T ∨ T ∨ (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) T T ∨ T − T T T ∨ (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) t t t t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) t t t t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , σ L : (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) T T T ∨ T ∨ (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) T T T ∨ T ∨ − T ∨ T ∨ (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) t t t t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) t t t t (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) . (3.10)We note that σ R = ϵ (cid:48) σ L ϵ (cid:48) , (3.11)where ϵ (cid:48) is an involutive anti-automorphism (3.8).3.3. Polynomial Representation and Askey–Wilson Polynomial.
We review the represen-tation on the Laurent polynomials (cid:67) [ x ± ] . See, e.g. , [39, 40, 43]. Proposition 3.6.
A polynomial representation is given by T (cid:55)→ t − s ð − q − (cid:0) t − − t (cid:1) x + q − (cid:0) t − − t (cid:1) x − q − x ( − s ð ) , T (cid:55)→ t − s + (cid:0) t − − t (cid:1) + (cid:0) t − − t (cid:1) xx − ( s − ) , T ∨ (cid:55)→ q − T − x , T ∨ (cid:55)→ x − T − . (3.12)Based on this representation, we define Y = T T , X = (cid:0) T ∨ T (cid:1) − , (3.13)where Y is the Dunkl–Cherednik operator for the Askey–Wilson polynomial.As in the case of A -type, we have T f = t − f for a symmetric Laurent polynomial f ∈ (cid:67) [ x + x − ] . We see that the projection e is (cid:67) [ x ] → (cid:67) [ x + x − ] , and SH q , t preserves (cid:67) [ x + x − ] . On this symmetric polynomial space, the so-called Askey–Wilson operator is explicitlywritten as Y + Y − (cid:12)(cid:12) sym (cid:55)→ A ( x ; t ) ( ð − ) + A ( x − ; t ) ( ð − − ) + t t + ( t t ) − , (3.14) AHA AND SKEIN ALGEBRA 15 where A ( x ; t ) = t t (cid:16) − t t x (cid:17) (cid:16) + t t x (cid:17) (cid:18) − q t t x (cid:19) (cid:18) + q t t x (cid:19) ( − x ) ( − q x ) . (3.15)Eigenfunctions of Y (3.13) are called the non-symmetric Askey–Wilson polynomial, Y E m ( x ; q , t ) = ( t t ) − q m E m ( x ; q , t ) , Y E − m ( x ; q , t ) = t t q − m E − m ( x ; q , t ) . (3.16)Here m > E m ( x ; q , t ) = x m + (cid:0) t − (cid:1) t + q m (cid:0) t − (cid:1) ( t t ) q − m − q m x − m + · · · , (3.17) E − m ( x ; q , t ) = x − m + · · · , where · · · denote the Laurent polynomials x k with | k | < m . We note that E ( x ; q , t ) = Y E ( x ; q , t ) = ( t t ) − E ( x ; q , t ) .The eigenfunctions of (3.14) are the symmetric Askey–Wilson polynomials [4]. We have (cid:0) Y + Y − (cid:1) P m ( x ; q , t ) = (cid:16) ( t t ) − q m + t t q − m (cid:17) P m ( x ; q , t ) . (3.18)Here we have P m ( x ; q , t ) = ( a b , a c , a d ; q ) m a m ( a b c d q m − ; q ) m ϕ (cid:20) q − m , q m − a b c d , a x , a x − a b , a c , a d ; q , q (cid:21) , (3.19)where a = t t , b = − t t , c = q t t , d = − q t t . (3.20)Note that we have normalized the polynomials so that P m ( x ; q , t ) = ( x m + x − m ) + · · · , and P ( x ; q , t ) =
1. These are written in terms of the non-symmetric polynomials as P m ( x ; q , t ) = E m ( x ; q , t ) + ( q m − ) (cid:0) t + q m (cid:1) t q m − ( t t ) E − m ( x ; q , t ) = t ( T + t ) E − m ( x ; q , t ) . (3.21)Some of them are explicitly written as P ( x ; q , t ) = , P ( x ; q , t ) = x + x − + q t (cid:0) + t (cid:1) (cid:0) − t (cid:1) t + (cid:0) q + t (cid:1) t t (cid:0) − t (cid:1)(cid:0) q − t t (cid:1) t t . (3.22)Higher order polynomials are generated from the three-term recurrence relation. It is readas (see, e.g. , [22]) (cid:0) X + X − (cid:1) P m ( x ; q , t ) = P m + ( x ; q , t ) + B m P m ( x ; q , t ) + C m P m − ( x ; q , t ) . (3.23) Here using (3.20) we have C n = − abcdq n − ( − a b c d q n − ) ( − a b c d q n − ) · − q n ( − a b c d q n − ) ( − a b c d q n − )× (cid:0) − a b q n − (cid:1) (cid:0) − a c q n − (cid:1) (cid:0) − a d q n − (cid:1) (cid:0) − b c q n − (cid:1) (cid:0) − b d q n − (cid:1) (cid:0) − c d q n − (cid:1) , B n = a + a − − − a b c d q n − ( − a b c d q n − ) ( − a b c d q n ) a − ( − a b q n ) ( − a c q n ) ( − a d q n )− C n (cid:0) − a b c d q n − (cid:1) (cid:0) − a b c d q n − (cid:1) − a b c d q n − a ( − a b q n − ) ( − a c q n − ) ( − a d q n − ) . (3.24)It is noted that we also have (cid:16) T T ∨ + (cid:0) T T ∨ (cid:1) − (cid:17) P m ( x ; q , t ) = q m + ( t t ) − P m + ( x ; q , t ) + (cid:169)(cid:173)(cid:173)(cid:171) B m − q m − (cid:16) t − q t (cid:17) (cid:16) + q t t (cid:17) ( t − t ) ( + t t ) (cid:16) q m − + t t (cid:17) (cid:16) q m + + t t (cid:17) t t (cid:170)(cid:174)(cid:174)(cid:172) P m ( x ; q , t ) + t t q − m + C m P m − ( x ; q , t ) . (3.25)3.4. Automorphisms as Conjugation.
We study the SL ( (cid:90) ) action (3.10) of SH q , t under thepolynomial representation (3.12). We introduce V R = exp (cid:18) − ( log X ) q (cid:19) . This function is symmetric in X ↔ X − , thus s V R = V R s , and commutes with T and T ∨ . Aswe have s ð V R = q − X V R s ð , we get V R T = (cid:16) q x − a ( x ) s ð + b ( x ) (cid:17) V R , where we have used T = a ( x ) s ð + b ( x ) in (3.12) for brevity. We see that the expression inthe parenthesis coincides with T ∨ = q − T − x = q x − a ( x ) s ð + q − x (cid:0) b ( x ) + t − t − (cid:1) , when ( t − t ) ( + t t ) = . (3.26)So assuming (3.26), we have V − R T ∨ V R = T , and also T V R T = T T ∨ V R = q − x V R = V R q − x = V R T T ∨ , which proves V − R T V R = T T ∨ T − . The automorphism σ R (3.10) is thus realized by conjuga-tion of V R .For σ L (3.10), we recall (3.11) where the anti-involution ϵ (cid:48) (3.8) sends X (cid:55)→ Y − and Y (cid:55)→ X − .As in the case of A -type, the automorphism σ L is also realized by conjugation of V L . Toconclude, we have the following. Proposition 3.7.
Under the condition (3.26) , the SL ( (cid:90) ) actions (3.10) are conjugations σ • : h (cid:55)→ V − • h V • , AHA AND SKEIN ALGEBRA 17 where V R = exp (cid:18) − ( log X ) q (cid:19) , V L = exp (cid:18) ( log Y ) q (cid:19) . (3.27)3.5. Algebra Embedding.
We shall define (cid:65) ( r , s ) ∈ SH q , t associated to a simple closed curve (cid:99) ( r , s ) on the sphere Σ , with a slope s / r , (cid:99) ( r , s ) (cid:55)→ (cid:65) ( r , s ) = ch (cid:0) (cid:79) ( r , s ) (cid:1) . (3.28)Amongst them, we put (cid:65) ( , ) = X + X − = ch (cid:0) T ∨ T (cid:1) = ch (cid:16) q T T ∨ (cid:17) , (cid:65) ( , ) = Y + Y − = ch ( T T ) , (cid:65) ( , ) = ch (cid:0) T T ∨ (cid:1) = ch (cid:16) q − T T − X (cid:17) . (3.29)A tedious but straightforward computation proves (cid:65) ( , ) (cid:65) ( , ) = q − (cid:65) ( , ) + q (cid:65) ( , − ) − t , , (3.30) (cid:65) ( , ) (cid:65) ( , ) = q − (cid:65) ( , ) + q (cid:65) ( , ) − t , , (cid:65) ( , ) (cid:65) ( , ) = q − (cid:65) ( , ) + q (cid:65) ( , ) − t , . Here we have (cid:65) ( , − ) = ch (cid:16) q T T ∨ (cid:17) , (cid:65) ( , ) = ch (cid:16) T T ∨ T ∨ (cid:0) T ∨ (cid:1) − (cid:17) = σ − L ( (cid:65) ( , ) ) , (cid:65) ( , ) = ch (cid:16) T (cid:0) T ∨ (cid:1) − T T ∨ (cid:17) = σ − R ( (cid:65) ( , ) ) , (3.31)and t , = (cid:16) q t − q − t − (cid:17) (cid:0) t − t − (cid:1) + (cid:0) t − t − (cid:1) (cid:0) t − t − (cid:1) , t , = (cid:16) q t − q − t − (cid:17) (cid:0) t − t − (cid:1) + (cid:0) t − t − (cid:1) (cid:0) t − t − (cid:1) , t , = (cid:0) t − t − (cid:1) (cid:0) t − t − (cid:1) + (cid:0) t − t − (cid:1) (cid:16) q t − q − t − (cid:17) . (3.32)Furthermore, we can check that (cid:65) ( , − ) (cid:65) ( , ) = q − (cid:65) ( , ) T + q (cid:65) ( , ) T − q − t , (cid:65) ( , ) − q t , (cid:65) ( , ) + (cid:16) q − q − (cid:17) − (cid:0) t − t − (cid:1) − (cid:16) q t − q − t − (cid:17) − (cid:0) t − t − (cid:1) − (cid:0) t − t − (cid:1) + (cid:0) t − t − (cid:1) (cid:16) q t − q − t − (cid:17) (cid:0) t − t − (cid:1) (cid:0) t − t − (cid:1) , (3.33)where we have in terms of the Chebyshev polynomial (2.40) (cid:65) ( , ) T = T ( (cid:65) ( , ) ) = (cid:0) T ∨ T (cid:1) + (cid:0) T ∨ T (cid:1) − , (cid:65) ( , ) T = T ( (cid:65) ( , ) ) = ( T T ) + ( T T ) − . (3.34)Combining (3.30) with (3.33), we find that (cid:65) ( , ) , (cid:65) ( , ) , and (cid:65) ( , ) fulfill the cubic rela-tion (3.2). See also [41, 45]. As a result, we have the algebra embedding of the skein al-gebra (3.1) and (3.2) as follows. Theorem 3.8.
We have an algebra embedding
KBS A ( Σ , ) → SH q , t with A = q − by (cid:169)(cid:173)(cid:171) (cid:120)(cid:121)(cid:122) (cid:170)(cid:174)(cid:172) (cid:55)→ (cid:169)(cid:173)(cid:171) (cid:65) ( , ) (cid:65) ( , ) (cid:65) ( , ) (cid:170)(cid:174)(cid:172) , (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) (cid:98) (cid:98) (cid:98) (cid:98) (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (cid:55)→ ± (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) i (cid:0) t − t − (cid:1) i (cid:0) t − t − (cid:1) i (cid:16) q t − q − t − (cid:17) i (cid:0) t − t − (cid:1) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . (3.35)With the embedding, the three term relations (3.23) and (3.25) give the representation of (cid:120) and (cid:122) in terms of the Askey–Wilson polynomials as eigenfunctions of (cid:121) (3.18). As in thecase of the skein algebra on the once-punctured torus [16], we obtain a finite-dimensionalrepresentation when we set q to be a root of unity.It should be noted that a torus is a two-fold branched cover of a sphere over four-points.This may correspond to the fact [25] that the DAHA of C ∨ C -type is constructed using the A -type with the reflection equation [44].The other DAHA operators (cid:65) ( r , s ) associated to (cid:99) ( r , s ) can be given using the SL ( (cid:90) ) actionson SH q , t . For instance, when we apply the automorphisms (3.10) to (3.30), we get (cid:65) ( , n ) (cid:65) ( , ) = q − (cid:65) ( , n + ) + q (cid:65) ( , n − ) − (cid:40) t , , for odd n , t , , for even n , (cid:65) ( , ) (cid:65) ( n , ) = q − (cid:65) ( n + , ) + q (cid:65) ( n − , ) − (cid:40) t , , for odd n , t , , for even n , (3.36)where (cid:65) ( , k ) = ch (cid:16) Y − k X − T − Y k T (cid:17) , (cid:65) ( , k + ) = ch (cid:16) q − Y − k − T XY k T (cid:17) , (3.37) (cid:65) ( k , ) = ch (cid:16) T X − k T − YX k (cid:17) , (cid:65) ( k + , ) = ch (cid:16) q − T X − k Y − T X k + (cid:17) . See [5] for the algorithms to obtain the curve (cid:99) ( r , s ) from the viewpoint of the skein algebraon the sphere Σ , .3.6. Automorphisms and Braiding.
The relationship between DAHA of C ∨ C -type and theskein algebra KBS A ( Σ , ) gives an interpretation of the SL ( (cid:90) ) action (3.10) of DAHA. Themapping class group Mod ( Σ , ) is generated by half Dehn twists, and it is known that the SL ( (cid:90) ) action corresponds to the Artin braid group B , which denotes the subgroup ofMod ( Σ , ) fixing one puncture [8, 19]. As seen from the fact that σ R : (cid:65) ( , ) (cid:55)→ (cid:65) ( , − ) , (cid:65) ( , ) (cid:55)→ (cid:65) ( , ) , σ L : (cid:65) ( , ) (cid:55)→ (cid:65) ( , − ) , (cid:65) ( , ) (cid:55)→ (cid:65) ( , ) , the automorphism σ R (resp. σ L ) is identified with a braiding of punctures (cid:98) and (cid:98) (resp. (cid:98) and (cid:98) ) as in Fig. 4, and it denotes the Dehn twist about (cid:120) (resp. (cid:121) ). AHA AND SKEIN ALGEBRA 19 b b b b σ L σ R Figure 4.
The automorphisms σ R and σ L are braidings of punctures. DAHA polynomials for (cid:99) ( r , s ) . For the simple closed curve (cid:99) ( r , s ) with slope s / r on Σ , ,we have assigned the DAHA operator (cid:65) ( r , s ) ∈ SH q , t (3.28). The DAHA polynomial associatedto the curve (cid:99) ( r , s ) is defined by P n ( x , q , t ; (cid:99) ( r , s ) ) = M n − ( (cid:79) ( r , s ) ; q , q )( ) . (3.38)Especially P ( x , q , t ; (cid:99) ( r , s ) ) = (cid:65) ( r , s ) ( ) . (3.39)This is why we use the A -type Macdonald polynomial in (3.38) rather than the Askey–Wilsonpolynomial, P ( x ; q , t ) (cid:44) x + x − , but we may introduce a new parameter as a t -parameterof the A -Macdonald polynomial (3.38). Indeed a different definition was used in [15] as the C ∨ C DAHA polynomial.We give some explicit forms in the following. We have P n ( x , q , t ; (cid:99) ( , ) ) = M n − ( x ; q , q ) , P n ( x , q , t ; (cid:99) ( , ) ) = M n − (( t t ) − ; q , q ) . Using (3.29) and (3.31) we have P ( x , q , t ; (cid:99) ( , ) ) = q ( t t ) − (cid:0) x + x − (cid:1) − q t − (cid:0) t − t − (cid:1) − t − (cid:0) t − t − (cid:1) , P ( x , q , t ; (cid:99) ( , − ) ) = q − x − A ( x ; t ) + q − x A ( x − ; t ) − A ( x ; ˜ t ) − A ( x − ; ˜ t ) + t t + ( t t ) − , where ˜ t = ( t , t , t , t ) . 4. Twice-punctured Torus4.1. Skein Algebra.
The skein algebra KBS A ( Σ , ) on a twice-punctured torus was studiedin [10]. We define simple closed curves (cid:120) , (cid:121) , (cid:120) u , (cid:121) u , (cid:98) , (cid:98) as in Fig. 5, where (cid:98) and (cid:98) arethe boundary circles. We regard Σ , as the surface constructed by gluing an annulus S ×[ , ] with Σ , in Fig. 2, Σ , = Σ , ∪ S × [ , ] , where both S × { } ≈ (cid:98) and S × { } ≈ (cid:98) areisotopic to (cid:120) u . Then we have the skein algebra of Σ , -type, (3.1) and (3.2) with (cid:98) ≈ (cid:98) ≈ (cid:120) u . Here (cid:122) is generated by (cid:120) and (cid:121) from the first identity of (3.1). On the other hand, Σ , is regarded as the surface given by gluing a once-punctured torus with a thrice-puncturedsphere, Σ , = Σ , ∪ Σ , , where the boundary circle of Σ , is (cid:120) and the three boundary circlesof Σ , are isotopic to (cid:120) , (cid:98) , (cid:98) . Then we have the algebra (2.1) of Σ , for (cid:120) u , (cid:121) u , and (cid:122) u . Here (cid:122) u is generated by (cid:120) u and (cid:121) u by the first identity of (2.1). We see that (cid:120) , which is isotopicto the boundary circle of Σ , , is generated by (2.2). In addition, we need the consistencycondition (4.5) for (cid:121) and (cid:121) u as the skein algebra for Σ , , which can be checked directly. b b y xy u x u Figure 5.
Depicted are simple closed curves on the twice-punctured torus Σ , . Proposition 4.1.
The skein algebra
KBS A ( Σ , ) is as follows; • Σ , -type, A (cid:120) (cid:121) − A − (cid:121) (cid:120) = (cid:0) A − A − (cid:1) (cid:122) + (cid:0) A − A − (cid:1) ( (cid:98) + (cid:98) ) (cid:120) u , A (cid:121) (cid:122) − A − (cid:122) (cid:121) = (cid:0) A − A − (cid:1) (cid:120) + (cid:0) A − A − (cid:1) (cid:0) (cid:120) u + (cid:98) (cid:98) (cid:1) , A (cid:122) (cid:120) − A − (cid:120) (cid:122) = (cid:0) A − A − (cid:1) (cid:121) + (cid:0) A − A − (cid:1) ( (cid:98) + (cid:98) ) (cid:120) u , (4.1) with A (cid:120) (cid:121) (cid:122) = A (cid:120) + A − (cid:121) + A (cid:122) + A (cid:0) (cid:120) u + (cid:98) (cid:98) (cid:1) (cid:120) + A − ( (cid:98) + (cid:98) ) (cid:120) u (cid:121) + A ( (cid:98) + (cid:98) ) (cid:120) u (cid:122) + (cid:120) u + (cid:98) + (cid:98) + (cid:120) u (cid:98) (cid:98) − (cid:0) A + A − (cid:1) , (4.2) • Σ , -type, A (cid:120) u (cid:121) u − A − (cid:121) u (cid:120) u = (cid:0) A − A − (cid:1) (cid:122) u , A (cid:121) u (cid:122) u − A − (cid:122) u (cid:121) u = (cid:0) A − A − (cid:1) (cid:120) u , A (cid:122) u (cid:120) u − A − (cid:120) u (cid:122) u = (cid:0) A − A − (cid:1) (cid:121) u , (4.3) with (cid:120) = A (cid:120) u (cid:121) u (cid:122) u − A (cid:120) u − A − (cid:121) u − A (cid:122) u + A + A − , (4.4) • consistency, − (cid:121) (cid:121) u + (cid:0) A + A − (cid:1) (cid:121) (cid:121) u (cid:121) − (cid:121) u (cid:121) = (cid:0) A − A − (cid:1) (cid:121) u , − (cid:121) u (cid:121) + (cid:0) A + A − (cid:1) (cid:121) u (cid:121) (cid:121) u − (cid:121) (cid:121) u = (cid:0) A − A − (cid:1) (cid:121) , (4.5) with (cid:120) (cid:121) u = (cid:121) u (cid:120) , (cid:120) u (cid:121) = (cid:121) (cid:120) u . (4.6) It is noted that the boundary circles (cid:98) and (cid:98) are central. We note that the Σ , -type relations (4.3) are redundant, and we have − (cid:121) u (cid:120) u + (cid:0) A + A − (cid:1) (cid:121) u (cid:120) u (cid:121) u − (cid:120) u (cid:121) u = (cid:0) A − A − (cid:1) (cid:120) u , − (cid:120) u (cid:121) u + (cid:0) A + A − (cid:1) (cid:120) u (cid:121) u (cid:120) u − (cid:121) u (cid:120) u = (cid:0) A − A − (cid:1) (cid:121) u . (4.7) AHA AND SKEIN ALGEBRA 21
Construction of DAHA.
To construct the DAHA representation for KBS A ( Σ , ) inProp. 4.1, we shall first make use of the DAHA of C ∨ C -type which represents KBS A ( Σ , ) in Fig. 2. Due to that the curves (cid:98) and (cid:98) are set to be isotopic, (cid:98) ≈ (cid:98) ≈ (cid:120) u , and that wehave the embedding (3.35) for KBS A ( Σ , ) , we put t = t = i x u . Namely the parameters of C ∨ C -DAHA (cid:72) q , t (3.14) are set to be ( t , t , t , t ) = (cid:16) i x u , i q − x (cid:96) , i x u , i x r (cid:17) = t (cid:92) . (4.8)In the spherical C ∨ C -DAHA SH q , t (cid:92) with t (cid:92) (4.8), we assign X + X − and Y + Y − for the curves (cid:120) and (cid:121) respectively. Explicitly, we have the following representation on (cid:67) [ x + x − ] satisfyingthe Σ , -type skein relations (4.1), (cid:120) (cid:55)→ x + x − , (cid:121) (cid:55)→ − β ( x , x u , x (cid:96) , x r ) ð − β ( x − , x u , x (cid:96) , x r ) ð − − φ ( x , x u , x (cid:96) , x r ) , (4.9) (cid:120) u (cid:55)→ x u + x − u , (cid:98) (cid:55)→ x (cid:96) + x − (cid:96) , (cid:98) (cid:55)→ x r + x − r , where β ( x , x u , x (cid:96) , x r ) = (cid:16) x (cid:96) + q x x r (cid:17) (cid:16) q x + x (cid:96) x r (cid:17) (cid:16) q x + x u (cid:17) q (cid:16) − q x (cid:17) ( − x ) x (cid:96) x r x u , φ ( x , x u , x (cid:96) , x r ) = − x ( x (cid:96) + x r ) ( + x (cid:96) x r ) (cid:0) + x u (cid:1)(cid:16) − q − x (cid:17) (cid:16) − q x (cid:17) x (cid:96) x r x u . (4.10)To give representations for (cid:120) u and (cid:121) u satisfying (4.3) and (4.4), we use the A -type DAHASH q u , t . The skein algebra embeddings in Theorems 2.11 and 3.8 suggest to set q u = q . (4.11)Recalling that the boundary circle (cid:98) in Fig. 1 is generated by (2.2) and that the DAHA repre-sentation gives (2.41), we see that (4.4) is satisfied by t = i q u x . (4.12)For the consistency conditions (4.5), we may take a conjugation of SH q u , t by use of a “gluingfunction” G ( x , x u ) to be determined as (cid:120) u (cid:55)→ G ( x , x u ) − (cid:0) x u + x − u (cid:1) G ( x , x u ) = x u + x − u , (cid:121) u (cid:55)→ G ( x , x u ) − (cid:0) γ ( x , x u ) ð u + γ ( x , x − u ) ð − u (cid:1) G ( x , x u ) . (4.13)Here ð u is a difference operator for x u , ð u x u = q u x u ð u , (4.14)and γ ( x , x u ) is for the Macdonald operator (2.13) of SH q u , i q u x γ ( x , x u ) = i q u x x u − (cid:16) i q u x (cid:17) − x − u x u − x − u . (4.15) Under this setting, the commutativity (4.6) is trivial, and we have the Σ , -type skein rela-tion (4.3) because of the conjugation of the A DAHA representation. To check the remainingconsistency condition (4.5), we assume that (cid:121) u (cid:55)→ γ ( x u ) ð u + γ ( x , x u ) ð − u , (4.16)By brute force computations, we find that the consistency conditions (4.5) are fulfilled when γ ( x u ) = i q − − − x u , γ ( x , x u ) = i q (cid:16) + q − x − x u (cid:17) (cid:16) + q − x x u (cid:17) − x u . (4.17)The representation (4.16) is indeed the form of (4.13), when the gluing function is defined interms of the quantum dilogarithm function by G ( x , x u ) = e log x log xu log q (− q x x u ; q ) ∞ . (4.18)We should note that the gluing function G ( x , x u ) satisfies the following q -difference equa-tions, G ( q x , x u ) G ( x , x u ) = x u + q x x u , G ( x , q x u ) G ( x , x u ) = x + q x x u . (4.19)As a result, we obtain the following. Theorem 4.2.
We have an algebra embedding (4.9) , (4.13) , (4.18) of KBS A ( Σ , ) → SH q , t (cid:92) with A = q − as operators on symmetric polynomial (cid:67) [ x + x − ] . It should be stressed that the above representation preserves the symmetric Laurent poly-nomial space (cid:67) [ x + x − ] . Due to conjugation by the gluing function G ( x , x u ) (4.18), broken isa symmetry x u ↔ x − u . We can recover the symmetry of x u when we discard the symmetry x ↔ x − , by taking an inverse conjugation, h (cid:55)→ G ( x , x u ) h G ( x , x u ) − , for the representationsin Theorem 4.2. We use t by x = − t q u , (4.20)as in (4.12), and a q -difference operator for t in SH q u , t is ð t t = q u t ð t . (4.21)We obtain the following representation which acts on (cid:67) [ x u + x − u ] . AHA AND SKEIN ALGEBRA 23
Corollary 4.3.
We have an algebra embedding,
KBS A ( Σ , ) → SH q u , t , with A = q − u , (cid:120) u (cid:55)→ X u + X − u , (cid:121) u (cid:55)→ Y u + Y − u , (cid:98) (cid:55)→ x (cid:96) + x − (cid:96) , (cid:98) (cid:55)→ x r + x − r , (cid:121) (cid:55)→ − q u t (cid:0) x (cid:96) − x r t (cid:1) (cid:0) x (cid:96) x r − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) x (cid:96) x r (cid:0) X − u t − − X u t (cid:1) (cid:0) X u t − − X − u t (cid:1) ð t − q u (cid:0) q u x r − x (cid:96) t (cid:1) (cid:0) q u − x (cid:96) x r t (cid:1)(cid:0) t + q u (cid:1) (cid:0) q u − t (cid:1) x (cid:96) x r ð − t − q u t ( x (cid:96) + x r ) ( + x (cid:96) x r )( + t ) (cid:0) t + q u (cid:1) x (cid:96) x r (cid:0) X u + X − u (cid:1) . (4.22) Here X u and Y u defined by (2.12) acting on the Laurent polynomial of x u constitute the DAHA SH q u , t . The representation (4.22) preserves the symmetric polynomials (cid:67) [ x u + x − u ] , and theyare explicitly written as follows; (cid:120) u (cid:55)→ x u + x − u , (cid:121) u (cid:55)→ t x u − t − x − u x u − x − u ð u + t − x u − t x − u x u − x − u ð − u , (cid:98) (cid:55)→ x (cid:96) + x − (cid:96) , (cid:98) (cid:55)→ x r + x − r , (cid:121) (cid:55)→ − q u (cid:0) x (cid:96) − x r t (cid:1) (cid:0) x (cid:96) x r − t (cid:1) (cid:0) − x u t (cid:1) (cid:0) x u − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) x (cid:96) x r x u ð t − q u (cid:0) q u x r − x (cid:96) t (cid:1) (cid:0) q u − x (cid:96) x r t (cid:1)(cid:0) t + q u (cid:1) (cid:0) q u − t (cid:1) x (cid:96) x r ð − t − q u t ( x (cid:96) + x r ) ( + x (cid:96) x r ) (cid:0) + x u (cid:1) ( + t ) (cid:0) t + q u (cid:1) x (cid:96) x r x u . (4.23)4.3. DAHA Polynomial of Simple Closed Curves on Σ , . The representation in Theorem 4.2is on (cid:67) [ x + x − ] , while the representation (4.22) is on (cid:67) [ x u + x − u ] . Using Theorem 4.2, wecan assign an operator (cid:65) acting on (cid:67) [ x + x − ] for a simple closed curve (cid:99) on Σ , , (cid:99) (cid:55)→ (cid:65) = ch ( (cid:79) ) . (4.24)When (cid:99) is given by Dehn twists from (cid:121) , (cid:99) = T ( (cid:121) ) , we have (cid:79) = γ ( Y ) due to that (cid:121) (cid:55)→ Y + Y − ∈ SH q , t (cid:92) . The automorphism γ is induced from T , and it is written as the conjugationby U • and V • . We then define the DAHA polynomial for (cid:99) by Q n ( x , x u , x (cid:96) , x r , q ; (cid:99) ) = M n − ( (cid:79) ; q , q )( ) . (4.25)Especially we have Q ( x , x u , x (cid:96) , x r , q ; (cid:99) ) = (cid:65) ( ) . For example, we have Q ( x , x u , x (cid:96) , x r , q ; (cid:121) u ) = i q − x u − x u (cid:0) x + x − (cid:1) − i q − q ( − q ) − x u − x u , Q ( x , x u , x (cid:96) , x r , q ; (cid:121) ) = − q − x u x (cid:96) − q x − u x − (cid:96) . These are in (cid:67) ( q , x u , x (cid:96) , x r )[ x + x − ] , and we do not know a relationship with the previouslyknown quantum polynomial invariants at this stage. We shall pay attention to the representation (4.22) which acts on (cid:67) [ x u + x − u ] . We assumethat a simple closed curve (cid:99) on Σ , is given from either (cid:120) u or (cid:99) ( r , s ) on subsurface Σ , ⊂ Σ , by the Dehn twist T , where T is generated by T (cid:120) u and T (cid:121) u . Then we have (cid:99) (cid:55)→ (cid:40) ch ( γ ( X u )) , when (cid:99) = T ( (cid:120) u ) ,ch ( γ ( (cid:79) ( r , s ) )) , when (cid:99) = T ( (cid:99) ( r , s ) ) ,where γ denotes automorphisms of DAHA (cid:72) q u , t induced from T . We define P n ( x u , x (cid:96) , x r , q u , t ; (cid:99) ) = (cid:40) M n − ( γ ( X u ) ; q u , t )( ) , when (cid:99) = T ( (cid:120) u ) , M n − ( γ ( (cid:79) ( r , s ) ) ; q u , t )( ) , when (cid:99) = T ( (cid:99) ( r , s ) ) . (4.26)We show some concrete examples. In the following τ •( u ) denotes the automorphisms (2.8)for X u and Y u of A -DAHA, which correspond to the Dehn twists about the curve (cid:120) u and (cid:121) u respectively. The first example is (cid:99) (cid:48)( k + , ) = (cid:16) T − k (cid:120) u ◦ T (cid:121) u (cid:17) ( (cid:120) u ) , In the DAHA representation we have the automorphism τ kR ( u ) ◦ τ L ( u ) on X u (2.49) to obtain thesame results for torus knots (cid:99) ( k + , ) on the once-punctured torus, (cid:99) (cid:48)( k + , ) (cid:55)→ (cid:98) M ( )( k + , ) ( x u ; q u , t ) , (4.27)where (cid:98) M ( )( k + , ) ( x ; q , t ) is defined in (2.50). We thus have P ( x u , x (cid:96) , x r , q u , t ; (cid:99) (cid:48)( k + , ) ) = (cid:98) M ( )( k + , ) ( x u ; q u , t )( ) , which reduces to P (cid:16) x u = − q , x (cid:96) = − q − , x r = − q − , q , t = q ; (cid:99) (cid:48)( k + , ) (cid:17) = − q k − q k + − q k + , (4.28)which is the Jones polynomial for torus knot T ( k + , ) as in (2.51).As the second example, we treat (cid:99) (cid:48)(cid:48)( k + , ) = (cid:16) T − k (cid:120) u ◦ T (cid:121) u (cid:17) ( (cid:121) ) , We need the action τ kR ( u ) ◦ τ L ( u ) on the representation (4.22) of (cid:121) arising from the DAHA (cid:72) q , t (cid:92) of C ∨ C type. Recalling (2.45), we have (cid:99) (cid:48)(cid:48)( k + , ) (cid:55)→ q u t (cid:0) x (cid:96) − x r t (cid:1) (cid:0) x (cid:96) x r − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) x (cid:96) x r sh (cid:16) t q k − u ( X ku Y u ) X u (cid:17) sh ( t − X u ) ð t − q u (cid:0) q u x r − x (cid:96) t (cid:1) (cid:0) q u − x (cid:96) x r t (cid:1)(cid:0) t + q u (cid:1) (cid:0) q u − t (cid:1) x (cid:96) x r ð − t ( t − X u ) sh (cid:16) t − q k − u ( X ku Y u ) X u (cid:17) − q u t ( x (cid:96) + x r ) ( + x (cid:96) x r )( + t ) (cid:0) t + q u (cid:1) x (cid:96) x r ch (cid:16) q k − u ( X ku Y u ) X u (cid:17) . (4.29) AHA AND SKEIN ALGEBRA 25
This can be written as an operator on (cid:67) [ x u + x − u ] as (cid:99) (cid:48)(cid:48)( k + , ) (cid:55)→ − q u t (cid:0) x (cid:96) − x r t (cid:1) (cid:0) x (cid:96) x r − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) x (cid:96) x r (cid:98) M ( + )( k + , ) ( x u ; q u , t ) ð t − q u (cid:0) q u x r − x (cid:96) t (cid:1) (cid:0) q u − x (cid:96) x r t (cid:1)(cid:0) t + q u (cid:1) (cid:0) q u − t (cid:1) x (cid:96) x r ð − t (cid:98) M (−)( k + , ) ( x u ; q u , t )− q u t ( x (cid:96) + x r ) ( + x (cid:96) x r )( + t ) (cid:0) t + q u (cid:1) x (cid:96) x r (cid:98) M ( )( k + , ) ( x u ; q u , t ) , (4.30)where we use (2.50) and the operators (cid:98) M (±)( k + , ) ( x ; q , t ) are given by (cid:98) M ( + )( k + , ) ( x ; q , t ) = (cid:26) ( q x ) k (cid:0) − t x (cid:1) (cid:0) − q t x (cid:1) (cid:0) − q t x (cid:1) q t ( − x ) ( − q x ) (cid:27) ð + (cid:8) x → x − (cid:9) ð − + q (cid:0) + q (cid:1) (cid:0) − t (cid:1) (cid:0) t − x (cid:1) (cid:0) − t x (cid:1) t ( q − x ) ( − q x ) , (cid:98) M (−)( k + , ) ( x ; q , t ) = (cid:26) ( q x ) k q x (cid:0) − + q t x (cid:1) ( − x ) ( − q x ) (cid:27) ð + (cid:8) x → x − (cid:9) ð − + q (cid:0) + q (cid:1) (cid:0) − t (cid:1) x ( q − x ) ( − q x ) . (4.31)Here ð is a q -difference operator for x , and (cid:8) x → x − (cid:9) in the second term denotes the coeffi-cient {· · · } of ð in the first term replacing x by x − so that the operators (cid:98) M (±) ( x ; q , t ) preservethe symmetric Laurent polynomial space (cid:67) [ x + x − ] . Then we obtain the DAHA polynomialfor (cid:99) (cid:48)(cid:48)( k + , ) by acting (4.30) on 1. This is indeed a deformation of the Jones polynomial fortorus knot T ( k + , ) ; it reduces to the Jones polynomial for (cid:99) ( k + , ) up to framing when wetake specific values for deformation parameters, P (cid:16) x u = − q , x (cid:96) = − q − , x r = − q − , q , t = q ; (cid:99) (cid:48)(cid:48)( k + , ) (cid:17) = − q k − q k + − q k + . (4.32)5. Genus-Two Torus5.1. Skein Algebra.
We shall construct the skein algebra KBS A ( Σ , ) based on the previoussections. We define simple closed curves on Σ , as in Fig. 6. We regard the twice-puncturedtorus Σ , in Fig. 5 as a subsurface of Σ , , and we can construct Σ , = Σ , ∪ S × [ , ] where both S × { } ≈ (cid:98) and S × { } ≈ (cid:98) are isotopic to (cid:120) d . As a reduction of the C ∨ C -type DAHA (3.1), the simple closed curves (cid:120) , (cid:121) , (cid:122) constitute the following skein algebra (5.1)of Σ , -type. Here (cid:122) is generated by (cid:120) and (cid:121) as in (5.1). On the other hand, we can regard Σ , as a union of two once-punctured tori, Σ , = Σ u , ∪ Σ d , , where Σ u , ∩ Σ d , ≈ (cid:120) , and thecurves (cid:120) u and (cid:121) u (resp. (cid:120) d and (cid:121) d ) satisfy the skein algebra (5.3) on the once-puncture torus Σ u , (resp. Σ d , ). Two sets of curves { (cid:120) (cid:125) , (cid:121) (cid:125) , (cid:122) (cid:125) } for (cid:125) ∈ { u , d } fulfill the skein algebra of Σ , whose boundary circle is isotopic to (cid:120) . Further we need skein relations for (cid:101) (cid:121) in Fig. 5, andthey can be written in (5.7).In summary, we have the following skein algebra. See also [3]. Proposition 5.1.
The skein algebra
KBS A ( Σ , ) is y xx u x d y u y d e y Figure 6.
Simple closed curves on the genus-two torus Σ , are given. • Σ , -type, A (cid:120) (cid:121) − A − (cid:121) (cid:120) = (cid:0) A − A − (cid:1) (cid:122) + (cid:0) A − A − (cid:1) (cid:120) d (cid:120) u , (5.1) A (cid:121) (cid:122) − A − (cid:122) (cid:121) = (cid:0) A − A − (cid:1) (cid:120) + (cid:0) A − A − (cid:1) (cid:0) (cid:120) u + (cid:120) d (cid:1) , A (cid:122) (cid:120) − A − (cid:120) (cid:122) = (cid:0) A − A − (cid:1) (cid:121) + (cid:0) A − A − (cid:1) (cid:120) d (cid:120) u , with A (cid:120) (cid:121) (cid:122) = A (cid:120) + A − (cid:121) + A (cid:122) + A (cid:0) (cid:120) u + (cid:120) d (cid:1) (cid:120) + A − (cid:120) d (cid:120) u (cid:121) + A (cid:120) d (cid:120) u (cid:122) + (cid:120) u + (cid:120) d + (cid:120) u (cid:120) d − (cid:0) A + A − (cid:1) , (5.2) • Σ , -type, A (cid:120) (cid:125) (cid:121) (cid:125) − A − (cid:121) (cid:125) (cid:120) (cid:125) = (cid:0) A − A − (cid:1) (cid:122) (cid:125) , (5.3) A (cid:121) (cid:125) (cid:122) (cid:125) − A − (cid:122) (cid:125) (cid:121) (cid:125) = (cid:0) A − A − (cid:1) (cid:120) (cid:125) , A (cid:122) (cid:125) (cid:120) (cid:125) − A − (cid:120) (cid:125) (cid:122) (cid:125) = (cid:0) A − A − (cid:1) (cid:121) (cid:125) , with (cid:120) = A (cid:120) (cid:125) (cid:121) (cid:125) (cid:122) (cid:125) − A (cid:120) (cid:125) − A − (cid:121) (cid:125) − A (cid:122) (cid:125) + A + A − , (5.4) where (cid:125) ∈ { u , d } , • consistency, − (cid:121) (cid:121) (cid:125) + (cid:0) A + A − (cid:1) (cid:121) (cid:121) (cid:125) (cid:121) − (cid:121) (cid:125) (cid:121) = (cid:0) A − A − (cid:1) (cid:121) (cid:125) , (5.5) − (cid:121) (cid:125) (cid:121) + (cid:0) A + A − (cid:1) (cid:121) (cid:125) (cid:121) (cid:121) (cid:125) − (cid:121) (cid:121) (cid:125) = (cid:0) A − A − (cid:1) (cid:121) , with (cid:120) (cid:121) (cid:125) = (cid:121) (cid:125) (cid:120) , (cid:120) (cid:125) (cid:121) = (cid:121) (cid:120) (cid:125) , (5.6) where (cid:125) ∈ { u , d } , and (cid:120) u (cid:120) d = (cid:120) d (cid:120) u , (cid:120) u (cid:120) d = (cid:120) d (cid:120) u , (cid:121) u (cid:120) d = (cid:120) d (cid:121) u , (cid:121) u (cid:120) d = (cid:120) d (cid:121) u , • skein relations for (cid:101) (cid:121) , (cid:101) (cid:121) (cid:121) (cid:125) = (cid:121) (cid:125) (cid:101) (cid:121) , (5.7) − (cid:101) (cid:121) (cid:120) (cid:125) + (cid:0) A + A − (cid:1) (cid:101) (cid:121) (cid:120) (cid:125) (cid:101) (cid:121) − (cid:120) (cid:125) (cid:101) (cid:121) = (cid:0) A − A − (cid:1) (cid:120) (cid:125) , − (cid:120) (cid:125) (cid:101) (cid:121) + (cid:0) A + A − (cid:1) (cid:120) (cid:125) (cid:101) (cid:121) (cid:120) (cid:125) − (cid:101) (cid:121) (cid:120) (cid:125) = (cid:0) A − A − (cid:1) (cid:101) (cid:121) . AHA AND SKEIN ALGEBRA 27 where (cid:125) ∈ { u , d } , and (cid:101) (cid:121) (cid:120) u (cid:121) u − (cid:121) u (cid:120) u (cid:101) (cid:121) = (cid:121) (cid:121) d (cid:120) d − (cid:120) d (cid:121) d (cid:121) , (5.8) (cid:101) (cid:121) (cid:121) = (cid:121) (cid:101) (cid:121) , Note that the above Σ , -type relations (5.3) are redundant, and we have − (cid:121) (cid:125) (cid:120) (cid:125) + (cid:0) A + A − (cid:1) (cid:121) (cid:125) (cid:120) (cid:125) (cid:121) (cid:125) − (cid:120) (cid:125) (cid:121) (cid:125) = (cid:0) A − A − (cid:1) (cid:120) (cid:125) , (5.9) − (cid:120) (cid:125) (cid:121) (cid:125) + (cid:0) A + A − (cid:1) (cid:120) (cid:125) (cid:121) (cid:125) (cid:120) (cid:125) − (cid:121) (cid:125) (cid:120) (cid:125) = (cid:0) A − A − (cid:1) (cid:121) (cid:125) . Polynomial Representation.
To give a representation, we make use of the DAHA rep-resentation for the twice-punctured torus in the previous section. We glue the two puncturesof Σ , in Fig. 5 by setting x (cid:96) = x r = x d . (5.10)It should be remarked that four parameters t of DAHA of C ∨ C -type SH q , t (see Fig. 2) arenow set to be ( t , t , t , t ) = (cid:16) i x u , i q − x d , i x u , i x d (cid:17) = t (cid:63) , (5.11)which means that we have glued the punctures, (cid:98) with (cid:98) , and (cid:98) with (cid:98) , together. In gluing (cid:98) with (cid:98) in the previous section, we have employed DAHA of A -type SH q u , i q / u x / so that (cid:120) is generated from (cid:120) u and (cid:121) u as the boundary circle (2.2) of Σ u , . As we have Σ u , ∩ Σ d , ≈ (cid:120) ,another DAHA SH q u , i q / u x / plays the role of Σ d , so that (cid:120) is also generated from (cid:120) d and (cid:121) d as the boundary circle (2.2) of Σ d , . Thus it is natural to use the gluing function (4.18) for (cid:121) d ,and to put as follows; (cid:120) (cid:55)→ x + x − , (cid:121) (cid:55)→ − β ( x , x u , x d ) ð − β ( x − , x u , x d ) ð − − φ ( x , x u , x d ) , (cid:120) u (cid:55)→ x u + x − u , (cid:121) u (cid:55)→ G ( x , x u ) − (cid:0) γ ( x , x u ) ð u + γ ( x , x − u ) ð − u (cid:1) G ( x , x u ) = γ ( x u ) ð u + γ ( x , x u ) ð − u , (cid:120) d (cid:55)→ x d + x − d , (cid:121) d (cid:55)→ G ( x , x d ) − (cid:0) γ ( x , x d ) ð d + γ ( x , x − d ) ð − d (cid:1) G ( x , x d ) = γ ( x d ) ð d + γ ( x , x d ) ð − d , (5.12)where γ ( x , x u ) , γ ( x u ) , and γ ( x , x u ) are defined in (4.15) and (4.16). The functions β ( x , x u , x d ) and φ ( x , x u , x d ) come from the Askey–Wilson operator (3.14) with t (cid:63) (5.11), β ( x , x u , x d ) = q − + x (cid:16) − q x (cid:17) ( − x ) (cid:16) x u + q x x − u (cid:17) (cid:16) x d + q x x − d (cid:17) , φ ( x , x u , x d ) = (cid:16) q − − x − (cid:17) (cid:16) − q x (cid:17) (cid:0) x u + x − u (cid:1) (cid:0) x d + x − d (cid:1) . We should note that (cid:121) is represented by the Askey–Wilson difference operator (3.14), while (cid:121) u and (cid:121) d are the (conjugated) A -type Macdonald difference operators (2.13). The q -difference operators ð u and ð d are for x u and x d respectively ð u ð d = ð d ð u , ð u f ( x , x u , x d ) = f ( x , q u x u , x d ) , ð d f ( x , x u , x d ) = f ( x , x u , q d x d ) , where we mean q u = q d = q (4.11). Note that the representation for (cid:121) is symmetric x u ↔ x d ,and that the associated Askey–Wilson polynomial for t (cid:63) (5.11) is written in a symmetric formby use of the Sears’ transformation formula [22] to (3.19) as P m ( x ; q , x u , x d ) = (− ) m q − m (cid:16) qx u , qx d , q ; q (cid:17) m (cid:16) q m + x d x u ; q (cid:17) m ϕ q − m , q m + x d x u , − q x , − q x − qx d , qx u , q ; q , q . (5.13)By construction, both skein relations of the Σ , -type (5.1) and the Σ , -type (5.3) are ful-filled by the above representation (5.12). Moreover the above Askey–Wilson operator for (cid:121) issymmetric in x u ↔ x d , and the consistency conditions (5.5) are satisfied thanks to the resultsfor the skein algebra on Σ , in the previous section. Hence a non-trivial is for the simpleclosed curve (cid:101) (cid:121) in Fig. 5. To give a representation, we suppose that (cid:101) (cid:121) (cid:55)→ (cid:213) ε ∈{− , , + } (cid:213) ε u , ε d = ± a ε , ε u , ε d ( x , x u , x d ) ð ε ð ε u u ð ε d d , (5.14)and consider a condition for the first identity of (5.8). Equating each coefficient of ð ð u ð − d , ð ð − u ð − d , ð ð − d , we have the following functional equations for a + ±− ( x , x u , x d ) ; a ++ − ( x , q u x u , x d ) a ++ − ( x , x u , x d ) = γ ( q u x u ) γ ( x u ) , a + −− ( x , q − u x u , x d ) a + −− ( x , x u , x d ) = γ ( qx , q − u x u ) γ ( x , x u ) , (cid:0) q − u x u + q u x − u (cid:1) (cid:8) γ ( q − u x u ) a + −− ( x , x u , x d ) − γ ( x , x u ) a ++ − ( x , q − u x u , x d ) (cid:9) − (cid:0) q u x u + q − u x − u (cid:1) { γ ( x u ) a + −− ( x , q u x u , x d ) − γ ( q x , q u x u ) a ++ − ( x , x u , x d )} = i ( − q ) (cid:16) + q x (cid:17) (cid:16) q x + x d (cid:17) (cid:16) q x + x d (cid:17) (cid:16) q x + x u (cid:17) q x ( − x ) (cid:16) − x d (cid:17) x u . The first two are solved to be a + −− ( x , x u , x d ) = ˜ a + −− ( x , x d ) (cid:16) q x + x u (cid:17) (cid:16) q x + x u (cid:17) − x u , a ++ − ( x , x u , x d ) = ˜ a ++ − ( x , x d ) − x u . Due to that functions ˜ a +++ and ˜ a + − + do not depend on x u , we can solve them from the abovethird equations, ˜ a + −− ( x , x d ) = (cid:16) + q x (cid:17) (cid:16) q x + x d (cid:17) (cid:16) q x + x d (cid:17) q x ( − x ) (cid:16) − q x (cid:17) (cid:16) − x d (cid:17) , ˜ a ++ − ( x , x d ) = − q x ˜ a + −− ( x , x d ) . AHA AND SKEIN ALGEBRA 29
In this manner, we get a ± , ε u , ε d ( x , x u , x d ) .For a , ± , ± ( x , x u , x d ) , we see that a sum for ε (cid:44) (cid:121) u and (cid:121) d .Consequently we can suppose that a sum for ε = ψ ( x ) (cid:0) γ ( x d ) ð d + γ ( x , x d ) ð − d (cid:1) (cid:0) γ ( x u ) ð u + γ ( x , x u ) ð − u (cid:1) . A commutativity between (cid:121) and (cid:101) (cid:121) solves ψ ( x ) , and as a result we obtain (cid:101) (cid:121) (cid:55)→ (cid:0) κ ( x d ) ð d + λ ( x , x d ) ð − d (cid:1) (cid:0) κ ( x u ) ð u + λ ( x , x u ) ð − u (cid:1) ω ( x ) ð + (cid:0) κ ( x d ) ð d + λ ( x − , x d ) ð − d (cid:1) (cid:0) κ ( x u ) ð u + λ ( x − , x u ) ð − u (cid:1) ω ( x − ) ð − + ψ ( x ) (cid:0) γ ( x d ) ð d + γ ( x , x d ) ð − d (cid:1) (cid:0) γ ( x u ) ð u + γ ( x , x u ) ð − u (cid:1) , (5.15)where ω ( x ) = x (cid:16) + q x (cid:17) q ( − x ) (cid:16) − q x (cid:17) , ψ ( x ) = x (cid:16) − q − x (cid:17) (cid:16) − q x (cid:17) , (5.16) λ ( x , x u ) = (cid:16) q x + x u (cid:17) (cid:16) q x + x u (cid:17) q x (cid:0) − x u (cid:1) , κ ( x u ) = − − x u . With the setting (5.15), we can check the relations (5.7) by tedious computations.As a result, we have obtained the following theorem.
Theorem 5.2.
We have a representation of
KBS A ( Σ , ) by (5.12) , (5.15) with A = q − . This representation denotes the difference operators on (cid:67) [ x + x − ] . As representation on (cid:67) [ x u + x − u , x d + x − d ] , we take a conjugation h (cid:55)→ G h G − with G = G ( x , x u ) G ( x , x d ) . (5.17)Using (4.20), we have the following representation. Corollary 5.3.
We have a representation of
KBS A ( Σ , ) with A = q − u by (cid:120) (cid:125) (cid:55)→ X (cid:125) + X − (cid:125) , (cid:121) (cid:125) (cid:55)→ Y (cid:125) + Y − (cid:125) , for (cid:125) ∈ { u , d } , (5.18) (cid:121) (cid:55)→ − q u t (cid:0) − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) (cid:214) (cid:125) ∈{ u , d } sh ( t X (cid:125) ) sh (cid:0) t − X (cid:125) (cid:1) ð t − q u (cid:0) q u − t (cid:1)(cid:0) q u + t (cid:1) (cid:0) q u − t (cid:1) ð − t − q u t (cid:0) t + q u (cid:1) ( t + ) (cid:214) (cid:125) ∈{ u , d } ch ( X (cid:125) ) , (5.19) (cid:101) (cid:121) (cid:55)→ q u t (cid:0) − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) (cid:214) (cid:125) ∈{ u , d } t − sh (cid:0) t − Y (cid:125) (cid:1) sh (cid:0) t − X (cid:125) (cid:1) ð t + q u (cid:0) q u − t (cid:1)(cid:0) q u + t (cid:1) (cid:0) q u − t (cid:1) (cid:214) (cid:125) ∈{ u , d } t sh ( t − X (cid:125) ) sh ( t Y (cid:125) ) ð − t − q u t (cid:0) q u + t (cid:1) ( + t ) (cid:214) (cid:125) ∈{ u , d } ch ( Y (cid:125) ) . (5.20) Here { X (cid:125) , Y (cid:125) , T (cid:125) } are generators of (cid:72) q u , t , and these representations preserve symmetric space (cid:67) [ x u + x − u , x d + x − d ] . Recalling (2.28) , these are explicitly written as operators on the symmetricpolynomial space, (cid:120) (cid:125) (cid:55)→ x (cid:125) + x − (cid:125) , (cid:121) (cid:125) (cid:55)→ t x (cid:125) − t − x − (cid:125) x (cid:125) − x − (cid:125) ð (cid:125) + t − x (cid:125) − t x − (cid:125) x (cid:125) − x − (cid:125) ð − (cid:125) , for (cid:125) ∈ { u , d } , (5.21) (cid:120) (cid:55)→ − q u t − − q − u t , (cid:121) (cid:55)→ − q u t (cid:0) − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) (cid:214) (cid:125) ∈{ u , d } (cid:0) t x (cid:125) − t − x − (cid:125) (cid:1) (cid:0) t − x (cid:125) − t x − (cid:125) (cid:1) ð t − q u (cid:0) q u − t (cid:1)(cid:0) q u + t (cid:1) (cid:0) q u − t (cid:1) ð − t − q u t (cid:0) q u + t (cid:1) ( + t ) (cid:214) (cid:125) ∈{ u , d } (cid:0) x (cid:125) + x − (cid:125) (cid:1) , (5.22) (cid:101) (cid:121) (cid:55)→ q u t (cid:0) − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) (cid:214) (cid:125) ∈{ u , d } (cid:18) (cid:0) − t x (cid:125) (cid:1) (cid:0) − q u t x (cid:125) (cid:1) q u t x (cid:125) (cid:0) x (cid:125) − (cid:1) ð (cid:125) − (cid:0) t − x (cid:125) (cid:1) (cid:0) t q u − x (cid:125) (cid:1) q u t x (cid:125) (cid:0) x (cid:125) − (cid:1) ð − (cid:125) (cid:19) ð t + q u (cid:0) q u − t (cid:1)(cid:0) q u + t (cid:1) (cid:0) q u − t (cid:1) (cid:214) (cid:125) ∈{ u , d } x (cid:125) x (cid:125) − (cid:0) ð (cid:125) − ð − (cid:125) (cid:1) ð − t − q u t (cid:0) q u + t (cid:1) ( + t ) (cid:214) (cid:125) ∈{ u , d } (cid:18) t x (cid:125) − t − x − (cid:125) x (cid:125) − x − (cid:125) ð (cid:125) + t − x (cid:125) − t x − (cid:125) x (cid:125) − x − (cid:125) ð − (cid:125) (cid:19) . (5.23)We should note that the representation (5.20) of (cid:101) (cid:121) can be recovered by use of the automor-phisms of (cid:72) q u , t . As shown in Fig. 7, we have (cid:101) (cid:121) = T (cid:120) u T − (cid:120) d T (cid:121) u T − (cid:121) d ( (cid:121) ) , and the DAHA oper-ator for (cid:121) = (cid:99) ( , ) is (cid:99) ( , ) (cid:55)→ (cid:65) ( , ) = ch ( Y ) in (5.19). Using (2.45), an action τ − R ( u ) τ R ( d ) τ L ( u ) τ − L ( d ) is (cid:18) X u X d ð t (cid:19) (cid:55)→ (cid:169)(cid:173)(cid:171) q − X − u Y u X u Y − d ( t − q − X − u Y u X u ) sh ( t − X u ) ( t − Y − d ) sh ( t − X d ) ð t (cid:170)(cid:174)(cid:172) . We then recover (5.20) from (5.19) as operators on SH q u , t × SH q u , t .As depicted in Fig. 7, the simple closed curve (cid:101) (cid:121) is also given from the curve (cid:99) ( , ) by (cid:101) (cid:121) = T (cid:120) u T (cid:120) d T (cid:121) u T (cid:121) d ( (cid:99) ( , ) ) . The associated operator (cid:99) ( , ) (cid:55)→ (cid:65) ( , ) = ch ( T T ∨ ) ∈ SH q , t (cid:63) AHA AND SKEIN ALGEBRA 31 is explicitly written as the operator on (cid:67) [ x + x − ] , (cid:65) ( , ) (cid:12)(cid:12) sym = − x (cid:16) + q x (cid:17) (cid:16) q x x − u + x u (cid:17) (cid:16) q x x − d + x d (cid:17) ( − x ) (cid:16) − q x (cid:17) ð − (cid:16) q + x (cid:17) (cid:16) q x − u + x x u (cid:17) (cid:16) q x − d + x x d (cid:17) x ( x − ) (cid:16) x − q (cid:17) ð − + q x (cid:0) x u + x − u (cid:1) (cid:16) x d + x − d (cid:17)(cid:16) q − x (cid:17) (cid:16) − q x (cid:17) , which follows from (3.29) with t (cid:63) (5.11). We take a conjugation (cid:65) ( , ) (cid:55)→ G (cid:65) ( , ) G − withthe gluing function (5.17), to have an operator on (cid:67) [ x u + x − u , x d + x − d ] . Making a change ofvariables (4.20), we get G (cid:65) ( , ) (cid:12)(cid:12) sym G − = q u t ( − t ) (cid:0) q u − t (cid:1) ( + t ) (cid:214) (cid:125) ∈{ u , d } sh ( t X (cid:125) ) sh (cid:0) t − X (cid:125) (cid:1) ð t + q u (cid:0) q u − t (cid:1) t (cid:0) q u − t (cid:1) (cid:0) q u + t (cid:1) ð − t − q u t (cid:0) q u + t (cid:1) ( + t ) (cid:214) (cid:125) ∈{ u , d } ch ( X (cid:125) ) . (5.24)Applying the automorphism τ − R ( u ) τ − R ( d ) τ L ( u ) τ L ( d ) , we obtain (5.20) as well. T (cid:121) u T − (cid:121) d T (cid:120) u T − (cid:120) d ≈ T (cid:121) u T (cid:121) d T (cid:120) u T (cid:120) d ≈ Figure 7.
A simple closed curve (cid:101) (cid:121) is given from curves with slope-1 / / As a simple closed curve (cid:101) (cid:121) does not intersect with both (cid:121) u and (cid:121) d , the Dehn twists T (cid:121) u and T (cid:121) d have trivial actions on (cid:101) (cid:121) , (cid:101) (cid:121) = T (cid:121) u ( (cid:101) (cid:121) ) = T (cid:121) d ( (cid:101) (cid:121) ) . Our representation for (cid:101) (cid:121) is indeedinvariant under τ L ( (cid:125) ) due to the following proposition. Proposition 5.4.
The operators of SH q , t , t − sh ( t − Y ) sh ( t − X ) ð t , and t sh ( t − X ) sh ( t Y ) ð − t , are invariant under τ L . Proof.
It is straightforward to see the invariance of the first operator using (2.8) and (2.32).For the second, we recall that it denotes the lowering operator K (−) which does not dependon t , and that it commutes with ð t . We then have ð − t t sh ( t − X ) sh ( t Y ) (cid:55)→ ð − t ( t − X ) sh ( t − q − Y X ) t sh ( t − q − Y X ) sh ( t Y ) . which proves the invariance under τ L . (cid:3) Double-Torus Knots.
A simple closed curve on a genus two Heegaard surface in S iscalled a double-torus knot. A construction of a non-trivial knot was studied in [28, 29], but aclassification of the double-torus knots is far from complete.We shall propose a DAHA polynomial for the double-torus knot. We assign a differenceoperator for the simple closed curve (cid:99) as follows. We suppose that (cid:99) is given from (cid:99) ( r , s ) byDehn twists T which is generated by T (cid:120) u , T (cid:121) u , T (cid:120) d , and T (cid:121) d , (cid:99) = T ( (cid:99) ( r , s ) ) , and that a curve (cid:99) ( r , s ) does not intersect with (cid:120) u , (cid:120) d , Here we mean that (cid:99) ( r , s ) is a simpleclosed curve on Σ , ⊂ Σ , with slope s / r , when we regard Σ , = Σ , ∪ S × [ , ] ∪ S × [ , ] and the boundary circles in Fig. 2 are (cid:98) ≈ (cid:98) ≈ (cid:120) u and (cid:98) ≈ (cid:98) ≈ (cid:120) d . Using the C ∨ C -typeDAHA SH q , t (cid:63) with t (cid:63) (5.11), we have the q -difference operator (cid:99) ( r , s ) (cid:55)→ (cid:65) ( r , s ) = ch ( (cid:79) ( r , s ) ) ∈ SH q , t (cid:63) . We then take a conjugation G (cid:65) ( r , s ) G − by the gluing function (5.17), and make a change ofvariables (4.20). Applying the automorphism γ associated to the Dehn twist T , we have thedifference operator (cid:99) (cid:55)→ γ ( G (cid:65) ( r , s ) G − ) . We then define the DAHA polynomial of (cid:99) by P n ( t , q u , x u , x d ; (cid:99) ) = γ (cid:0) G M n − ( (cid:79) ( r , s ) ; q , q ) G − (cid:1) ( ) (5.25) = γ (cid:0) G S n − ( (cid:65) ( r , s ) ) G − (cid:1) ( ) . It is noted that we can further take a conjugation by Φ to be determined, (cid:99) (cid:55)→ Φ ( q u , t ) γ ( G (cid:65) ( r , s ) G − ) Φ ( q u , t ) − , and can modify the definition of the DAHA polynomial as P n ( t , q u , x u , x d , Φ ; (cid:99) ) = γ (cid:0) Φ ( q u , t ) G S n − ( (cid:65) ( r , s ) ) G − Φ ( q u , t ) − (cid:1) ( ) . (5.26)We expect that, by suitably choosing the function Φ , there may exist a relationship betweenour DAHA polynomial and a Poincaré polynomial of knot homology (see, e.g. , [21]), but wedo not know at this stage.To see a relationship with the Jones polynomial, we pay attention to the constant term ð t in the operators S n − ( (cid:65) ( r , s ) ) . This extraction is realized in (5.26) by putting t = q u with acondition Φ ( q u , q u ) = . (5.27) AHA AND SKEIN ALGEBRA 33
For simplicity, we write the reduced DAHA polynomial as the constant term ð t of S n − ( (cid:65) ( r , s ) ) as P n ( q u , x u , x d ; (cid:99) ) = γ (cid:16) Φ ( q u , t ) G M n − ( (cid:79) ( r , s ) ; q , q ) G − Φ ( q u , t ) − (cid:12)(cid:12) t = q u (cid:17) ( ) = γ (cid:16) Const (cid:0) S n − ( (cid:65) ( r , s ) ) (cid:1)(cid:12)(cid:12) t = q u (cid:17) ( ) , (5.28)where Const (•) is the ð t term of • . We do not need to take a conjugation by the gluingfunction G to pick up the constant term. Conjecture 5.5.
The DAHA invariant P n ( q u , x u , x d ; (cid:99) ) for a simple closed curve (cid:99) on Σ , co-incides with the n -colored Jones polynomial for the double-torus knot (cid:99) up to framing when x u = x d = q u . T (cid:121) u T − (cid:121) d Figure 8.
Shown is a curve (cid:99) ( , ) : ( , − ) as an example of a Dehn twist on a simpleclosed curve (cid:99) ( , ) with slope 2. In this case, we get the figure-eight knot4 in S . We give explicit examples. When T is generated by T (cid:120) u , T (cid:121) u , T (cid:120) d , T (cid:121) d , the Dehn twist T ( (cid:99) ( , ) ) is a connected sum of torus knots. To have a hyperbolic knot, we consider a simpleclosed curve (cid:99) = T ( (cid:99) ( , ) ) given from the slope-2 / (cid:99) ( , ) : ( k ,(cid:96) ) be a simple curve on Σ , given by T k (cid:121) u T (cid:96) (cid:121) d ( (cid:99) ( , ) ) . See Fig. 8 for a case of ( k , (cid:96) ) = ( , − ) . For small k and (cid:96) , thesimple closed curve (cid:99) ( , ) : ( k ,(cid:96) ) is identified with a knot of Rolfsen’s notation as in Table 1 [17,28]. The curve (cid:99) ( , ) : ( , p ) is the twist knot K p in S . knot 3 ( k , (cid:96) ) ( , ) ( , − ) ( , ) (− , ) ( , ) (− , − ) (− , ) (− , ) ( , ) ( , ) (− , ) (− , ) Table 1.
Examples of double-torus knots (cid:99) ( , ) : ( k ,(cid:96) ) . The so-called twist knot corre-sponds to the case of k = The operator associated to (cid:99) ( , ) on Σ , , (cid:99) ( , ) (cid:55)→ (cid:65) ( , ) ∈ SH q , t (cid:63) , is given in (3.31), whichpreserves (cid:67) [ x + x − ] . At t (cid:63) (5.11) it is written as (cid:65) ( , ) (cid:12)(cid:12) sym = A [ ]( , ) ( q , x , x u , x d ) + (cid:213) j = (cid:213) ε = ± A [ j ]( , ) ( q , x ε , x u , x d ) ð ε j , (5.29) where A [ ]( , ) ( q , x , x u , x d ) = x (cid:16) + q x (cid:17) (cid:16) + q x (cid:17) ( − x ) ( − q x ) (cid:16) − q x (cid:17) (cid:16) − q x (cid:17) (cid:214) (cid:125) ∈{ u , d } (cid:16) x (cid:125) + q x x − (cid:125) (cid:17) (cid:16) x (cid:125) + q x x − (cid:125) (cid:17) , A [ ]( , ) ( q , x , x u , x d ) = − q x (cid:16) + q x (cid:17) ( − x ) (cid:16) q − x (cid:17) (cid:16) − q x (cid:17) (cid:16) − q x (cid:17) (cid:214) (cid:125) ∈{ u , d } (cid:0) x (cid:125) + x − (cid:125) (cid:1) (cid:16) x (cid:125) + q x x − (cid:125) (cid:17) , A [ ]( , ) ( x , q , x u , x d ) = q x (cid:110) q (cid:0) + x (cid:1) + (cid:0) + q (cid:1) x + q ( + q ) (cid:0) + x (cid:1) x (cid:111)(cid:16) q − x (cid:17) (cid:16) − q x (cid:17) ( q − x ) ( − q x ) (cid:0) x u + x − u (cid:1) (cid:0) x d + x − d (cid:1) + q x (cid:110) q (cid:0) + x (cid:1) + ( + q ) (cid:0) + q (cid:1) x (cid:111) ( q − x ) ( − q x ) (cid:110) (cid:0) x u + x − u (cid:1) + (cid:0) x d + x − d (cid:1) (cid:111) + q ( − q ) x (cid:0) + x (cid:1) ( q − x ) ( − q x ) . We take a conjugation by the gluing function G (5.17), and then replace x with t (4.20) toobtain an operator on SH q u , t × SH q u , t as G (cid:65) ( , ) (cid:12)(cid:12) sym G − = a [ ]( , ) ( q u , t ) (cid:2) sh (cid:0) t − X u (cid:1) sh ( t X u ) sh (cid:0) t − X d (cid:1) sh ( t X d ) ð t (cid:3) + a [ ]( , ) ( q u , t ) ch ( X u ) ch ( X d ) (cid:2) sh (cid:0) t − X u (cid:1) sh ( t X u ) sh (cid:0) t − X d (cid:1) sh ( t X d ) ð t (cid:3) + a [ ]( , ) ( q u , t , X u , X d ) + a [ ]( , ) ( q u , q u t − ) ch ( X u ) ch ( X d ) ð − t + a [ ]( , ) ( q u , q u t − ) ð − t . (5.30)Here we have a [ ]( , ) ( q u , t ) = − q u t (cid:0) − t (cid:1) (cid:0) − q u t (cid:1)(cid:0) q u − t (cid:1) ( + t ) (cid:0) − q u t (cid:1) (cid:0) + q u t (cid:1) , a [ ]( , ) ( q u , t ) = q u t (cid:0) − t (cid:1) ( + t ) (cid:0) q u − t (cid:1) (cid:0) q u + t (cid:1) (cid:0) + q u t (cid:1) , and a [ ]( , ) ( q u , t , X u , X d ) = − q u t (cid:8) t (cid:0) − t + t (cid:1) + q u (cid:0) − t + t (cid:1) + q u t (cid:0) − + t − t (cid:1) (cid:9)(cid:0) q u − t (cid:1) (cid:0) − q u t (cid:1) ( + t ) (cid:0) q u + t (cid:1) ch ( X u ) ch ( X d ) + q u t (cid:8) t ( + q u ) + q u ( t − ) + q u t ( − t ) (cid:9)(cid:0) q u − t (cid:1) (cid:0) − q u t (cid:1) (cid:8) ch ( X u ) + ch ( X d ) (cid:9) + − q u t (cid:0) − q u (cid:1) (cid:0) q u + t (cid:1)(cid:0) q u − t (cid:1) (cid:0) − q u t (cid:1) . (5.31)The reduced DAHA polynomial P for the simple closed curve (cid:99) ( , ) : ( k ,(cid:96) ) is given by apply-ing automorphism τ kR ( u ) τ (cid:96) R ( d ) to (5.30). At t = q u , the constant term ð of (cid:65) ( , ) reduces to a AHA AND SKEIN ALGEBRA 35 symmetric bilinear form, a [ ]( , ) ( q u , q u , X u , X d ) = − q u (cid:0) − q u (cid:1) − q u (cid:169)(cid:173)(cid:171) (cid:0) − q u (cid:1) q u (cid:0) − q u (cid:1) − − q u q u (cid:0) − q u (cid:1) (cid:214) (cid:125) ∈{ u , d } (cid:0) − q u X (cid:125) (cid:1) (cid:0) − q u X − (cid:125) (cid:1)(cid:170)(cid:174)(cid:172) = − q u (cid:0) − q u (cid:1) − q u ( , S ( ch X u )) (cid:32) − q u ( − q u ) − q u (cid:33) (cid:18) S ( ch X d ) (cid:19) , (5.32)where we have used the Chebyshev polynomial of the second kind (2.22). As we have forSH q , t ch (cid:18) (cid:16) q − k Y k X (cid:17) (cid:19) ( ) = ( q t ) k M ( x ; q , t ) − (cid:0) − t (cid:1) (cid:0) + q (cid:1) − q t , we obtain P ( q u , x u , x d ; (cid:99) ( , ) : ( k ,(cid:96) ) ) = a [ ]( , ) ( q u , q u , q − k u Y ku X u , q − (cid:96) u Y (cid:96) u X u )( ) = − q u (cid:0) − q u (cid:1) − q u (cid:16) , q ku S ( ch x u ) (cid:17) (cid:32) − q u ( − q u ) − q u (cid:33) (cid:18) q (cid:96) u S ( ch x d ) (cid:19) . (5.33)We have checked that P ( q u , q u , q u ; (cid:99) ( , ) : ( k ,(cid:96) ) ) for ( k , (cid:96) ) in Table 1 agrees with the Jones poly-nomial up to framing (see, e.g. , [1, 11]).We compute the colored reduced DAHA polynomial P n > (5.28). For n =
3, we need thedifference operator S ( (cid:65) ( , ) ) = (cid:65) ( , ) −
1. Using (5.30), we find that the operator whichsurvives at t = q u is a term ð given by a [ ]( , ) ( q u , q u ) a [ ]( , ) ( q u , q − u ) (cid:214) (cid:125) ∈{ u , d } sh ( q − u X (cid:125) ) sh ( q − u X (cid:125) ) sh ( q u X (cid:125) ) sh ( q u X (cid:125) ) + a [ ]( , ) ( q u , q u ) a [ ]( , ) ( q u , q − u ) (cid:214) (cid:125) ∈{ u , d } ch ( X (cid:125) ) sh ( q − u X (cid:125) ) sh ( q u X (cid:125) ) + (cid:16) a [ ]( , ) ( q u , q u , X u , X d ) (cid:17) − = q ( − q ) − q (cid:40) (cid:0) − q (cid:1) q ( − q ) − − q q ( − q ) (cid:214) (cid:125) ∈{ u , d } (cid:0) − q X (cid:125) (cid:1) (cid:0) − q X − (cid:125) (cid:1) + ( − q ) (cid:0) − q (cid:1) q ( − q ) ( − q ) (cid:214) (cid:125) ∈{ u , d } (cid:0) − q X (cid:125) (cid:1) (cid:0) − q X − (cid:125) (cid:1) (cid:0) − q X (cid:125) (cid:1) (cid:0) − q X − (cid:125) (cid:1) (cid:41) , where q = q u . The 3-colored polynomial for (cid:99) ( , ) : ( k ,(cid:96) ) is given by replacing X u (resp. X d ) by q − k u Y ku X u (resp. q − (cid:96) u Y (cid:96) d X d ). The DAHA SH q , t proves M j ( q − k Y k X ; q , t )( ) = (cid:16) q j t j (cid:17) k M j ( x ; q , t ) , (5.34) which gives P ( q u , x u , x d ; (cid:99) ( , ) : ( k ,(cid:96) ) ) = q u (cid:0) − q u (cid:1) − q u × (cid:16) , q ku S ( ch x u ) , q ku S ( ch x u ) (cid:17) (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ( − q u )( − q u )( − q u )( − q u ) − q u ( − q u ) − q u − q u ( − q u ) − q u q u ( − q u )( − q u )( − q u )( − q u ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) (cid:169)(cid:173)(cid:171) q (cid:96) u S ( ch x d ) q (cid:96) u S ( ch x d ) (cid:170)(cid:174)(cid:172) . (5.35)We have checked that the results for knots in Table 1 coincide with the known N = n = S ( (cid:65) ( , ) ) = (cid:65) ( , ) − (cid:65) ( , ) . Based on these explicit computations, we have the following observation forthe N -colored reduced DAHA polynomial P N . We define s N ( x ) = (cid:0) S j ( x + x − ) (cid:1) ≤ j ≤ N − , (5.36) v N ( q , x ) = (cid:16) (cid:0) q x , q x − ; q (cid:1) j (cid:17) ≤ j ≤ N − . (5.37)These are bases of the symmetric Laurent polynomial space (cid:67) [ x + x − ] of even power, andwe have v N ( q , x ) = s N ( x ) B N ( q ) , (5.38)where the N × N triangular matrix B N ( q ) is defined by ( B N ( q )) j , k = (− ) j q j ( j + ) ( q ; q ) k + ( q ; q ) k − j ( q ; q ) k + j + , for 0 ≤ j ≤ k ≤ N − , otherwise.One sees that these bases were used in [24, 37] for a cyclotomic expansion of the coloredJones polynomial. Conjecture 5.6.
The constant term ð of S N − ( (cid:65) ( , ) ) at t = q u = q is given by Const (cid:16) S N − ( (cid:65) ( , ) ) (cid:12)(cid:12) t = q u (cid:17) = (− ) N − q ( N − ) − q − q N v N ( q , X u ) T N ( q ; (cid:99) ( , ) ) v N ( q , X d ) (cid:62) , (5.39) where the diagonal matrix T N is T N ( q ; (cid:99) ( , ) ) = diag (cid:18) (− ) k − q k ( k + − N ) ( q k ; q ) N + − k ( q k ; q ) N + − k ( q k ; q ) N − k ( q k ; q ) N − k (cid:19) ≤ k ≤ N . (5.40)For computations of the reduced DAHA polynomial of T k (cid:121) u T (cid:96) (cid:121) d ( (cid:99) ( , ) ) , we need v N ( q , τ kL ( X (cid:125) ))( ) in (5.39). By changing bases using (5.38) to the Chebyshev polynomial, wehave s N ( τ kL ( X (cid:125) ))( ) = s N ( q − k u Y k (cid:125) X (cid:125) )( ) at t = q u , which can be computed by SH q , t as (5.34).To conclude, Conjecture 5.5 of the reduced DAHA polynomial for (cid:99) ( , ) : ( k ,(cid:96) ) is read underConjecture 5.6 as follows. Conjecture 5.7.
The N -colored Jones polynomial for (cid:99) ( , ) : ( k ,(cid:96) ) = T k (cid:121) u T (cid:96) (cid:121) d ( (cid:99) ( , ) ) coincides up toframing with the reduced DAHA polynomial P N ( q u , x u , x d ; (cid:99) ) at x u = x d = q u = q , where we AHA AND SKEIN ALGEBRA 37 have P N ( q u , x u , x d ; (cid:99) ( , ) : ( k ,(cid:96) ) ) = (− ) N − q ( N − ) − q − q N × s N ( x u ) diag (cid:16) q j ( j + ) k (cid:17) ≤ j ≤ N − B N ( q ) T N ( q ; (cid:99) ( , ) ) B N ( q ) (cid:62) diag (cid:16) q j ( j + ) (cid:96) (cid:17) ≤ j ≤ N − s N ( x d ) (cid:62) . (5.41)One can check the conjecture for small N and the knots in Table 1. Furthermore, as anexplicit form of the N -colored Jones polynomial for twist knot K p is given in [37], we havechecked the equality for small p and N , P N ( q u , q u , q u ; (cid:99) ( , ) : ( , p ) ) = (− ) N − q N / − q − N / q / − q − / J N ( q ; K p ) . (5.42)Here the colored Jones polynomial is read as J N ( q ; K p ) = ∞ (cid:213) n = q n ( q − N , q + N ; q ) n n (cid:213) j = (− ) j q j ( j + ) p + j ( j − ) (cid:0) − q j + (cid:1) ( q ; q ) n ( q ; q ) n + j + ( q ; q ) n − j , (5.43)which is normalized such that J N ( q ; unknot ) =
1. See also [23] where a similar expressionwith (5.41) was given for the Poincaré polynomial for knot homology (cid:99) ( , ) : ( k ,(cid:96) ) .6. Concluding RemarksWe have clarified topological aspects of rank-1 DAHAs, A -type and C ∨ C -type. Motivatedby the DAHA–Jones polynomial for torus knots by Cherednik [14], we have proposed tocombine these two rank-1 DAHAs as a representation of the skein algebra on the genus-two surface Σ , . We have constructed the reduced DAHA polynomial P n ( q u , x u , x d ; (cid:99) ) fora simple closed curve (cid:99) on Σ , , and have clarified the relationship with the colored Jonespolynomial for double-torus knot. In this paper, we have mainly studied double twist knots T k (cid:121) u T (cid:96) (cid:121) d ( (cid:99) ( , ) ) , and in Appendix A we give some results on other simple closed curves givenfrom (cid:99) ( , ) .In Appendix B, we give some results concerning simple closed curves derived from (cid:99) ( , ) ,and discuss a relationship with the colored Jones polynomial. These results indicate an im-portance of the Askey–Wilson operator at t (cid:63) (5.11) associated to a simple closed curve (cid:99) ( r , s ) instudies of quantum invariants of knot. The symmetric bilinear form (5.41) for those knot fam-ilies seems to be promising for other quantum polynomial invariants of knots. Also studiesof the (non-reduced) DAHA polynomial P n ( t , q u , x u , x d , Φ ; (cid:99) ) , higher rank cases, and skein al-gebra on higher genus surfaces will reveal a fruitful structure [27]. Also the cluster algebraicconstruction of the skein algebra [26] will be useful in DAHA.We have studied the reduced DAHA polynomial for a simple curve (cid:99) = T ( (cid:99) ( r , s ) ) where theDehn twist T is generated by T (cid:120) u , T (cid:121) u , T (cid:120) d , and T (cid:121) d . In this case, we need only to apply the SL ( (cid:90) ) actions of A -type DAHA to the q -difference operator which is defined in terms ofthe C ∨ C -type DAHA. Even in the case that we further have the Dehn twist T (cid:121) about (cid:121) , it ispossible to define the DAHA polynomial using V L corresponding to the Dehn twist about (cid:121) , as we have constructed the conjugation (3.27) for the C ∨ C -type DAHA. Unfortunately suchcomputations are much involved, and it remains for future studies.Appendix A. Other Simple Closed Curves derived from (cid:99) ( , ) The Dehn twist T − k (cid:120) (cid:125) T (cid:121) (cid:125) induces the automorphism of SH q , t , τ kR τ L ( X ) = q ( k − ) X k YX . At t = q , we have S j ( ch ( q ( k − ) X k Y ))( ) = j (cid:213) i = − j S ( k + ) i ( ch x ) q i (( k + ) i + ) . (A.1)Here as we have S n ( x + x − ) = x n + − x − n − x − x − for n ≥
0, the Chebyshev polynomial for negativeintegers means S − n ( ch x ) = − S n − ( ch x ) . The Dehn twist T k (cid:121) (cid:125) T (cid:120) (cid:125) T (cid:121) (cid:125) induces the SH q , t automorphism, τ kL τ R τ L ( X ) = q − k Y k XY k + X .At t = q , we have S j ( ch ( q − k Y k XY k + X ))( ) = j (cid:213) i = (− ) i q ( k + ) i ( i + ) S i ( ch x ) . (A.2)As we have discussed, the reduced DAHA polynomials can be given from the constantterm (5.39) by applying the automorphisms. For instance, when a simple closed curve (cid:99) isgiven from (cid:99) ( , ) as T k (cid:121) u T − (cid:120) u T (cid:121) u T (cid:96) (cid:121) d , the reduced DAHA polynomial is then given by P N ( q u , x u , x d ; T k (cid:121) u T − (cid:120) u T (cid:121) u T (cid:96) (cid:121) d ( (cid:99) ( , ) )) = (− ) N − q ( N − ) − q − q N s N − ( x u ) (cid:16) (− ) i q ( k + ) i ( i + ) (cid:17) ≤ i ≤ j ≤ ( N − ) × B N ( q ) T N ( q ; (cid:99) ( , ) ) B N ( q ) (cid:62) diag (cid:16) q j ( j − ) (cid:96) (cid:17) ≤ j ≤ N s N ( x d ) (cid:62) . (A.3)We have checked the polynomial at x u = x d = q u for ( k , (cid:96) ) = (− , ) (resp. (− , − ) ) coinci-dences with the N -colored Jones polynomial for 6 (resp. 5 ). We can give the formula forthe curves obtained by the above Dehn twists for X (cid:125) by applying (A.1) and (A.2)Appendix B. Double-torus knots from (cid:99) ( , ) We give some results on the reduced DAHA polynomial for double-torus knots, which aregiven from the simple closed curve (cid:99) ( , ) on Σ , . The difference operator (cid:65) ( , ) ∈ SH q , t (cid:63) associated to the curve (cid:99) ( , ) is given in (3.37). At t (cid:63) (5.11), we can compute it explicitly as AHA AND SKEIN ALGEBRA 39 follows. G (cid:65) ( , ) (cid:12)(cid:12) sym G − = a [ ]( , ) ( q u , t ) (cid:2) sh ( t − X u ) sh ( t X u ) sh ( t − X d ) sh ( t X d ) ð t (cid:3) + a [ ]( , ) ( q u , t ) ch ( X u ) ch ( X d ) (cid:2) sh ( t − X u ) sh ( t X u ) sh ( t − X d ) sh ( t X d ) ð t (cid:3) + a [ ]( , ) ( q u , t , X u , X d ) (cid:2) sh ( t − X u ) sh ( t X u ) sh ( t − X d ) sh ( t X d ) ð t (cid:3) + a [ ]( , ) ( q u , t , X u , X d ) + a [ ]( , ) ( q u , q u t − , X u , X d ) ð − t + a [ ]( , ) ( q u , q u t − ) ch ( X u ) ch ( X d ) ð − t + a [ ]( , ) ( q u , q u t − ) ð − t , (B.1)where G is the gluing function (5.17), and a [ ]( , ) ( q u , t ) = q u t ( − t )( − q u t )( − q u t )( + t )( + q u t )( + q u t )( q u − t )( − q u t )( − q u t ) , a [ ]( , ) ( q u , t ) = − q u t ( − t )( − q u t ) (cid:0) − ( + q u ) t (cid:1) ( + t )( q u + t )( + q u t )( + q u t )( q u − t )( − q u t ) , a [ ]( , ) ( q u , t , X u , X d ) = q u t (cid:0) − t (cid:1) ( + t ) (cid:0) q u + t (cid:1) (cid:0) + q u t (cid:1) (cid:0) q u − t (cid:1) (cid:0) q u − t (cid:1) (cid:0) − q u t (cid:1) × (cid:40) − q u (cid:0) t ( + q u )(− + t ) − q u t ( + q u )(− + t − t + t ) + q u (− + t − t + t − t ) (cid:1) ch ( X u ) ch ( X d )− ( q u + t )( + q u t ) (cid:18) t ( + q u ) + q u ( + q u ) t ( − t ) + q u (− + t + t ) (cid:19) (cid:0) ch ( X u ) + ch ( X d ) (cid:1) + ( q u + t )( + q u t )( − q u ) ( t + q u + q u t ) (cid:41) , a [ ]( , ) ( q u , t , X u , X d ) = q u t ( + t ) (cid:0) q u + t (cid:1) (cid:0) q u + t (cid:1) (cid:0) + q u t (cid:1) (cid:0) q u − t (cid:1) (cid:0) − q u t (cid:1) ch ( X u ) ch ( X d )× (cid:40) q u t (cid:18) t ( − t + t ) + q u t (− + t − t + t ) + q u ( − t + t − t ) + q u ( − t + t ) (cid:19) ch ( X u ) ch ( X d )− q u (cid:18) t + q u t ( − t − t + t ) + q u t (− + t − t ) + q u t (− + t − t ) + q u ( − t − t + t ) + q u t (cid:19) (cid:0) ch ( X u ) + ch ( X d ) (cid:1) + (cid:18) t ( + t ) + q u t ( + t + t − t ) − q u t (− − t + t + t + t )− q u t ( + t − t + t + t ) − q u ( + t − t + t + t ) + q u (− − t − t + t + t ) + q u (− + t + t + t ) + q u t ( + t ) (cid:19) (cid:41) . The constant term ð in (cid:65) ( , ) at t = q u is written as a [ ]( , ) ( q u , q u , X u , X d ) = − q u (cid:0) − q u (cid:1) − q u (cid:40) + (cid:0) − q u (cid:1) (cid:0) − q u (cid:1)(cid:0) − q u (cid:1) (cid:0) − q u (cid:1) (cid:214) (cid:125) ∈{ u , d } (cid:0) − q u X (cid:125) (cid:1) (cid:0) − q u X − (cid:125) (cid:1) (cid:41) (cid:214) (cid:125) ∈{ u , d } (cid:0) X (cid:125) + X − (cid:125) (cid:1) . (B.2)Let (cid:99) ( , ) : ( k ,(cid:96) ) be a simple closed curve on Σ , given by T k (cid:121) u T (cid:96) (cid:121) d ( (cid:99) ( , ) ) . Applying the DAHAautomorphisms to (B.2), the reduced DAHA polynomial is given by P ( q u , x u , x d ; (cid:99) ( , ) : ( k ,(cid:96) ) ) = a [ ]( , ) ( q u , q u , q − k u Y ku X u , q − (cid:96) u Y (cid:96) d X d )( ) . (B.3)As X (cid:125) and Y (cid:125) are generators of SH q u , t , we can make use of (5.34) of SH q , t in the computation.As a result, we obtain a symmetric bilinear form for the reduced DAHA polynomial P ( q u , x u , x d ; (cid:99) ( , ) : ( k ,(cid:96) ) ) = − q u (cid:0) − q u (cid:1) − q u × (cid:16) q ku S ( ch x u ) , q ku S ( ch x u ) (cid:17) (cid:169)(cid:173)(cid:173)(cid:171) + ( − q u )( − q u )( − q u )( − q u ) − q u ( − q u ) − q u − q u ( − q u ) − q u q u ( − q u )( − q u )( − q u )( − q u ) (cid:170)(cid:174)(cid:174)(cid:172) (cid:32) q (cid:96) u S ( ch x d ) q (cid:96) u S ( ch x d ) (cid:33) . (B.4)We see that the curve (cid:99) ( , ) : ( k ,(cid:96) ) with ( k , (cid:96) ) = ( , − ) denotes a connected sum of trefoils 3 (the square knot). Also SnapPy [17] tells us that the curves (cid:99) ( , ) : ( k ,(cid:96) ) for ( k , (cid:96) ) = ( , ) and ( , ) are 9 and k respectively. We have checked that the reduced DAHA polynomial P (B.4) for these closed curves coincide with the Jones polynomials for 3 , 9 , and k in [11, 12] at x u = x d = q u .For the n = P , the constant term of S ( (cid:65) ( , ) ) = (cid:65) ( , ) − t = q u = q is computed as a [ ]( , ) ( q u , q u ) a [ ]( , ) ( q u , q − u ) (cid:214) (cid:125) ∈{ u , d } (cid:214) j = sh ( q − ju X (cid:125) ) sh ( q ju X (cid:125) ) + a [ ]( , ) ( q u , q u ) a [ ]( , ) ( q u , q − u ) (cid:214) (cid:125) ∈{ u , d } ch ( X (cid:125) ) (cid:214) j = sh ( q − ju X (cid:125) ) sh ( q ju X (cid:125) ) + a [ ]( , ) ( q u , q u , X u , X d ) a [ ]( , ) ( q u , q − u , X u , X d ) (cid:214) (cid:125) ∈{ u , d } sh ( q − u X (cid:125) ) sh ( q u X (cid:125) ) + (cid:16) a [ ]( , ) ( q u , q u , X u , X d ) (cid:17) − = q ( − q ) (cid:0) − q (cid:1) ( − q ) ( − q ) ( − q ) (cid:214) (cid:125) ∈{ u , d } ( q X (cid:125) , q X − (cid:125) ; q ) (cid:18) ch ( X (cid:125) ) + + q + q (cid:19) + q ( − q ) (cid:0) − q (cid:1) ( − q ) ( − q ) (cid:214) (cid:125) ∈{ u , d } (cid:0) − q X (cid:125) (cid:1) (cid:0) − q X − (cid:125) (cid:1) ch ( X (cid:125) ) + q ( − q ) − q (cid:214) (cid:125) ∈{ u , d } (cid:0) ch ( X (cid:125) ) + (cid:1) . AHA AND SKEIN ALGEBRA 41
Applying the SL ( (cid:90) ) actions on X (cid:125) , we obtain a symmetric bilinear form for the reducedDAHA polynomial of (cid:99) ( , ) : ( k ,(cid:96) ) as P ( q u , x u , x d ; (cid:99) ( , ) : ( k ,(cid:96) ) ) = q ( − q ) − q s ( x u ) diag (cid:16) , q k , q k , q k (cid:17) × (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ( − q ) ( − q ) ( − q )( − q ) − q ( − q ) − q ( − q ) ( − q ) ( − q )( − q ) ( − q ) ( − q )( + q + q + q + q ) ( − q )( − q ) − q ( − q ) ( − q )( + q + q + q ) ( − q )( − q ) q ( − q ) ( − q ) ( − q )( − q )− q ( − q ) − q − q ( − q ) ( − q )( + q + q + q ) ( − q )( − q ) q ( − q ) ( − q )( + q + q + q ) ( − q )( − q )( − q ) − q ( − q ) ( − q ) ( − q )( − q )( − q ) q ( − q ) ( − q ) ( − q )( − q ) − q ( − q ) ( − q ) ( − q )( − q )( − q ) q ( − q ) ( − q )( − q ) ( − q )( − q )( − q ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) × diag (cid:16) , q (cid:96) , q (cid:96) , q (cid:96) (cid:17) s ( x d ) (cid:62) , (B.5)where s N ( x ) is defined in (5.36). We have checked that the results for ( k , (cid:96) ) = ( , − ) and ( , ) agree up to framing factor with the n = and 9 in [11] respectively when x u = x d = q u = q as expected.References[1] The Knot Atlas , http://katlas.org/ , accessed: 2018-11-24.[2] M. Aganagic and S. Shakirov, Refined Chern–Simons theory and knot homology , inJ. Block, J. Distler, R. Donagi, and E. Sharpe, eds.,
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