aa r X i v : . [ m a t h . QA ] M a y DAHAS AND SKEIN THEORY
H. R. MORTON AND P. SAMUELSON
Abstract.
We give a skein-theoretic realization of the gl n double affine Hecke algebra of Cherednikusing braids and tangles in the punctured torus. We use this to provide evidence of a relationshipwe conjecture between the classical skein algebra of the punctured torus and the elliptic Hall algebraof Burban and Schiffmann. Contents
1. Introduction 11.1. The Kauffman bracket 11.2. The elliptic Hall algebra and Homflypt skeins 21.3. DAHAs for gl n
32. Algebraic background 62.1. The Elliptic Hall algebra 62.2. Limits of DAHAs 63. Skeins with a base string 83.1. Isotopies of braids in the punctured torus 83.2. A presentation for the algebra BSk n ( T , ∗ ) 143.3. Isotopies of relations 164. The tangle skein algebra 195. Relations with the elliptic Hall algebra 235.1. Certain closed curves 245.2. Comparison with the algebraic approach 275.3. Without the symmetrizer 295.4. The punctured torus and elliptic Hall algebra 30References 311. Introduction
This paper concerns a relation between double affine Hecke algebras and algebras associated tothe punctured and unpunctured torus that are defined using skein theory. As partial motivation,we first briefly discuss previous results in the sl case of the Kauffman bracket, and then go on todiscuss the conjectures and results of the present paper.1.1. The Kauffman bracket.
The
Kauffman bracket skein algebra K s ( F ) of a surface F isspanned by embedded links in the thickening F × [0 , s ∈ C ∗ , and which are similar to equation(3.13). The product is given by stacking links in the [0 ,
1] direction, and this algebra can be viewedas a quantization of the ring of functions on the SL character variety of F [PS00, BFKB99]. Forthe torus and punctured torus these algebras have been described explicitly by Frohman, Gelca,and by Bullock, Przytycki, respectively. The double affine Hecke algebra was defined by Cherednik (see, e.g. [Che05]) using explicitgenerators and relations, and it depends on two parameters, q, t ∈ C ∗ . In rank 1, its sphericalsubalgebra SH q,t of the DAHA was described explicitly by Koornwinder and later by Terwilliger.Combining these explicit descriptions leads to the following theorem. Theorem 1.1 ([FG00, Ter13, Koo08]) . There is an isomorphism K s ( T ) ∼ = SH s,s between the Kauffman bracket skein algebra of the torus and the t = q = s specialization of therank 1 spherical DAHA. Combining the same algebraic theorems with the description of the skein algebra of the puncturedtorus instead, we obtain the following.
Theorem 1.2 ([BP00, Ter13, Koo08]) . There is a surjective map K s ( T − D ) ։ SH q = s,t from the skein algebra of the punctured torus to the spherical rank 1 DAHA. We note that the source algebra still only depends on one parameter – the second parameter t inthe target appears in the relations describing the kernel of the map. One rough way of thinking ofthese results is that the spherical DAHA can be obtained from the skein algebra of the puncturedtorus using some kind of “decoration” at the puncture.Let us also comment briefly on the importance of two parameters. The Macdonald polynomialsare symmetric polynomials depending on the parameters q and t which have been studied inten-sively, and this had led (at the very least) to very interesting combinatorics, geometry, and algebra.In the t = q specialization, the Macdonald polynomials degenerate to Schur polynomials, whichare much more well-understood and less subtle. It is therefore desirable to be able to “see” bothparameters from topology.1.2. The elliptic Hall algebra and Homflypt skeins.
This paper deals with the “infinite rank”versions of the algebras in the previous section. The Kauffman bracket skein algebras are replacedwith the
Homflypt skein algebra
Sk( T ) of closed links in the thickened torus modulo the Homflyptskein relations. These relations are recalled in equations (1.1) and (1.2), and they depend onparameters s, v ∈ C × . The spherical DAHA is replaced by the elliptic Hall algebra E σ, ¯ σ defined byBurban and Schiffmann [BS12]. In earlier work we proved the analogue of Theorem 1.1: Theorem ([MS17]) . There is an isomorphism
Sk( T ) ∼ = E s,s between the Homflypt skein algebra of the torus and the σ = ¯ σ = s specialization of the ellipticHall algebra. We make the following conjecture which is the analogue of Theorem 1.2. Note that as in theKauffman bracket case, the source algebra only depends on one parameter s – we expect the secondparameter in the elliptic Hall algebra to arise as in the kernel of the map from Sk( T − D ). Also,we point out that Sk( T − D ) is “much bigger” than the elliptic Hall algebra: as a vector spaceit is isomorphic to a polynomial algebra with generators “conjugacy classes in the free group ofrank 2,” while the elliptic Hall algebra is isomorphic as a vector space to a polynomial algebra withgenerators indexed by Z . Technically, Frohman and Gelca showed skein algebra is isomorphic to the t DAHA = 1, q DAHA = s skein special-ization, but the presentations of Koornwinder and Terwilliger show that this spherical subalgebra is isomorphic tothe spherical subalgebra in the t DAHA = q DAHA = s skein specialization, which is a nontrivial statement. To be precise, the presentation of Sk( T ) does not depend on the parameter v , so technically the right hand sideof the isomorphism should be E s,s ⊗ k C [ v ± ]. AHAS AND SKEIN THEORY 3
Conjecture 1.3.
There is a surjective algebra map Sk( T − D ) ։ E σ, ¯ σ .The currently available proofs of all three of the previous theorems involve giving explicit presen-tations of the algebras in the statements, and then using these presentation to construct an algebramap by hand. Giving a presentation of Sk( T − D ) seems difficult, so instead of doing this, we givesome evidence for this conjecture using other techniques, which we describe in the next subsection.These techniques are closely related to some techniques used or mentioned by others – one reasonwe make precise statements of our own version is that it gives evidence for the conjecture above.1.3. DAHAs for gl n . The elliptic Hall algebra E σ, ¯ σ is closely related to the double affine Heckealgebras ¨ H n of type gl n , as detailed in the work of Schiffman and Vasserot, [SV11]. We wereintrigued by the nature of the presentation of the algebras ¨ H n , which involved Homfly type relationsand braids in the torus T . This led us to speculate on the possibility of constructing some formof skein theoretic model which would incorporate both the algebra ¨ H n , in terms of braids, and ouroriginal algebra of closed curves in the thickened torus.As a start we considered the possibility of a direct skein-based model in terms of n -braids in T for the double affine Hecke algebra ¨ H n with parameters t and q .The naive approach of considering n -braids in T modulo the Homfly relations gives a modelthat works for one of the parameters t but only covers the case q = 1 for the other parameter. Asearch of the literature came up with a paper by Burella et al, [BWPV14], suggesting that a modelbased on framed braids could handle the more general case of q = 1, where adding a twist to theframing of a braid was reflected in multiplication by q . Their model depends on the product ofcertain braids with explicit framing resulting in a single twist on the framing of one string. Wetried without success to follow the diagrammatic views of this product, which appears to us tohave the trivial framing on all strings, and not the desired twist. We worked out a uniform way ofspecifying a framing on the strings of a torus braid, noted in Theorem 4.1 below, and we came tothe conclusion that the use of framing alone would not provide a means of incorporating the secondparameter q into a geometric model for ¨ H n .We were still hopeful of making a skein-based geometric model, and we came up instead withone that includes an extra string. Instead of working with n -braids in the torus we use ( n + 1)-braids in which one distinguished string, called the base string , is fixed throughout. Equivalentlyour geometric elements are n -braids in the once-punctured torus, regarding the fixed base string asdetermining the puncture. In our model we use linear combinations of these braids. The regular n -string braids are allowed to interact as braids by the Homfly relations and the parameter q = c is introduced when a regular string is allowed to cross through the base string.These relations can be summarised in diagrammatic form as − = ( s − s − ) . ∗ = c ∗ In Section 3 of this paper we set up our skein model starting from Z [ s ± , c ± ]-linear combinationsof n -braids in the punctured torus, up to equivalence. We give a presentation for this algebra asa quotient of the group algebra of the braid group of n -braids in the punctured torus, using anexplicit presentation by Bellingeri, [Bel04, Theorem 1.1], for this braid group with generators σ , . . . , σ n − , a, b . Our emphasis here is on the use of geometric diagrams to represent the elementsof the algebra. Such an approach is used elsewhere with oriented framed (banded) curves in avariety of manifolds as the basic ingredients subject to the 3-term linear relations above. A newaddition in our current setting is the use of the base string, and the relation introducing the secondparameter c when a string is moved across it. H. R. MORTON AND P. SAMUELSON
We give diagrammatic illustrations of some useful braids and their interrelations, and show howto interpret Bellingeri’s presentation in terms of our braids.The skein relations can then be included by adding the relations σ i − σ − i = s − s − or ( σ i − s )( σ i + s − ) = 0in quadratic form, and P = c , where P is the braid taking string n once round the puncture and fixing the other strings. There isa simple formula for P in terms of the generators of the punctured braid group, which we provide.The outcome is a presentation for the algebra of braids in the punctured torus modulo theskein relations, which is our skein-based model, BSk n ( T , ∗ ), see Definition 3.1. We establish thispresentation in Theorem 3.5, and show how it corresponds exactly to the presentation in [SV11]for the double affine Hecke algebra ¨ H n : Theorem (see Thm. 3.5) . The braid skein algebra
BSk n ( T , ∗ ) is isomorphic to the gl n doubleaffine Hecke algebra ¨ H n ; q,t . Remark 1.4.
While this paper was in preparation, D. Jordan and M. Vazirani proved a very similarstatement, that the DAHA is a quotient of the group algebra of the braid group of the puncturedtorus [JV17, Prop. 4.1]. The difference between their statement and ours is that they imposerelations algebraically, while we impose ours topologically, giving a visually appealing interpretationof elements of the algebra and the underlying relations. It is therefore a-priori possible that weimpose more relations than them, and the content of this theorem is that their relations imply ours.Let us also mention here that Cherednik states in [Che05] that the gl n DAHA is a deformation ofthe Hecke quotient of the braid group of the closed torus. One of our desires was to remove the word“deformation” from this statement so that the deformation parameter q has a direct topologicalmeaning.In Section 5 we make use of framed n -tangles in the full framed Homfly skein of the puncturedtorus, Sk n ( T , ∗ ). In this setting an n -tangle consists of n framed oriented arcs in the thickenedtorus, along with a number of framed oriented closed curves. The arcs are no longer restricted tolying as braids in T × I . We work with linear combinations of framed tangles and impose the localrelation(1.1) − = ( s − s − )between framed tangles, as well as the change of framing relation(1.2) = v − using a second parameter v . In keeping with the first section we include a base string defining thepuncture, and allow a framed string to cross through it at the expense of the scalar c , with localrelation ∗ = c ∗ .There is a homomorphism from BSk n ( T , ∗ ) to Sk n ( T , ∗ ), since the braids in BSk n ( T , ∗ ) canbe given a consistent framing using a nonvanishing vector field on the torus so that the relationsin BSk n ( T , ∗ ) continue to hold in the wider tangle skein. It is not clear however whether thishomomorphism is injective. One point at issue is that, as well as the extra elements introduced, AHAS AND SKEIN THEORY 5 the additional relations between them might have the effect of collapsing the algebra considerably.Nonetheless, we make the following:
Conjecture 1.5.
The map BSk n ( T , ∗ ) → Sk n ( T , ∗ ) from the braid skein algebra to the tangleskein algebra is an isomorphism. Remark 1.6.
This may seem surprising, since tangles can contain closed curves, which meansthat a-priori, the tangle algebra is “much bigger” and the map in question should not be surjective.Indeed, for other surfaces the analogous map is not surjective. However, on the torus, an embeddedclosed curve “on one side of the puncture” is isotopic to the same curve “on the other side of thepuncture,” and we use this fact in Theorem 4.1 to show that the map in the conjecture is surjective.There is an algebra map from the (classical) Homflypt skein module Sk( T − D ) of closed linksin the thickened punctured torus to our skein algebra Sk n ( T , ∗ ) of tangles with a base string, givenby “filling in the puncture with the identity braid and the base string.” If we assume Conjecture1.5, this gives us an algebra map Sk( T − D ) → ¨ H n for any n . We can compose this map withmultiplication by the symmetrizer e in the finite Hecke algebra to obtain a map Sk( T − D ) → e ¨ H n e to the so-called spherical subalgebra .Schiffmann and Vasserot showed that the elliptic Hall algebra is the n → ∞ limit of the sphericalsubalgebras (see Theorem 2.3 for a precise statement). Let Sk + ( T − D ) be the subalgebra gener-ated by “curves lifted from the closed torus which only cross the y -axis positively” (see Definition5.6 for a precise statement). We then show the following, which we view as evidence for Conjecture1.3. Theorem (see Thm. 5.7) . Assuming Conjecture 1.5 holds, there is a surjective algebra map Sk + ( T − D ) ։ E + σ, ¯ σ . One corollary of this theorem (again assuming Conjecture 1.5) is that the generator u x of theelliptic Hall algebra has a simple interpretation as a sum W x of certain closed curves on the torus,with homology class x ∈ Z = H ( T − D ). In fact, these elements are lifts of the exact sameelements in Sk( T ) that were used in [MS17]. The subtle point here is that not all the relationsbetween W x that were proved in [MS17] hold in the punctured torus, since the proofs of some ofthese relations used global isotopies on T that don’t lift to the punctured torus. Roughly, theproblem is that some curves “get caught on the puncture.” When pushed into our skein algebraof tangles with a base string, these curves can once again be pushed through the puncture, but atthe cost of some “lower order terms” involving braids, and these lower order terms contribute tothe generating series relations in the elliptic Hall algebra. See also Remark 4.3.This suggests one purely algebraic question of possible interest. The Schiffmann-Vasserot ele-ments u x are in the spherical DAHA e n ¨ H q,t e n , where e n is the symmetrizer in the finite Heckealgebra. However, the images of our elements W x most naturally lie in the centralizer Z ¨ H n ( H n ) ofthe finite Hecke algebra H n inside the double affine Hecke algebra. This suggests there may be aninteresting limit of these centralizers which would include the elliptic Hall algebra a subalgebra.Let us briefly comment on related or future work. In [JV17], Jordan and Vazarani used factoriza-tion homology to construct representations of the braid-skein algebra BSk n ( T , ∗ ), and more skein-theoretic techniques to construct representations are being used in work in progress of Vazarani andWalker. We hope that some combination of these approaches could be used to prove Conjecture1.5, but we don’t discuss this in the present paper.We also note that the so-called A q,t algebra introduced by Carlesson and Mellit in [CM18] has arelation that looks like a 3-term version of the skein relation involving the base string. Discussionswith Jordan and Mellit indicate that more precise versions of this statement are available, but thiswill be left to future work.A summary of the contents of the paper is as follows. In Section 2 we recall algebraic backgroundinvolving DAHAs and the elliptic Hall algebra. In Section 3 we define the braid skein algebra and H. R. MORTON AND P. SAMUELSON show it is isomorphic to the DAHA, and in Section 4 we discuss the tangle skein algebra. In Section5 we compare this and the classical skein algebra of the punctured torus to the elliptic Hall algebra.
Acknowledgements:
This work was initiated during the authors participation in the Research inPairs program at Oberwolfach in the spring of 2015, and we gratefully acknowledge their supportfor our stay there, and for their excellent working conditions. More work was done at conferencesat the Isaac Newton Institute and at BIRS in Banff, and we gratefully acknowledge their support.Parts of the travel of the second author were supported by a Simons Travel Grant. We thank E.Gorsky, A. Negut, A. Oblomkov, O. Schiffmann, E. Vasserot, M. Vazirani, and K. Walker for theirinterest and discussions of this and/or their work over the years. We especially thank D. Jordanand A. Mellit for many discussions closely related to this paper.2.
Algebraic background
In this section we recall the algebraic definitions and results that we need in the rest of the paper.In particular, we define the elliptic Hall algebra and double affine Hecke algebras (DAHAs), andwe recall results of Schiffmann and Vasserot relating the two. In later sections we use these resultsto relate the skein algebra of the punctured torus to the elliptic Hall algebra.2.1.
The Elliptic Hall algebra.
Let us recall the definition of the elliptic Hall algebra E = E σ, ¯ σ of Burban and Schiffmann [BS12], using the conventions of [SV11]. It is an algebra over the ring Q ( σ, ¯ σ ), and it is generated by elements u x for x ∈ Z , subject to the following relations:(1) If x and x ′ belong to the same line in Z , then [ u x , u x ′ ] = 0.(2) Assume that x is primitive and that the triangle with vertices 0, x , and x + y has no interiorlattice points. Then(2.1) [ u y , u x ] = ǫ x , y θ x + y α where the elements θ z with z ∈ Z are obtained by the generating series identity X i θ i x z i = exp X i ≥ α i u i x z i for x ∈ Z primitive.In the above relations we used the constants ǫ x , y = sign(det( x y )) and α i = (1 − σ i )(1 − ¯ σ i )(1 − ( σ ¯ σ ) − i ) /i We also define the following subsets of Z := Z :(2.2) Z > := { ( x, y ) | x > } , Z + := Z > ⊔ { (0 , y ) | y ≥ } We also use this notation to define subalgebras of E , for example, E + := h u x | x ∈ Z + } We will use similar notation for other algebras generated by elements indexed by Z . Finally, let d ( x ) be the greatest common denominator of the entries of x ∈ Z .2.2. Limits of DAHAs.
We now recall the definition of the double affine Hecke algebra ¨ H n ,following the conventions given in [SV11]. This is an algebra over Z [ t ± / , q ± ] with generators { T i } , ≤ i ≤ n − , { X j } , { Y j } , ≤ j ≤ n AHAS AND SKEIN THEORY 7 and relations ( T i + t / )( T i − t − / ) = 0(2.3) T i T i +1 T i = T i +1 T i T i +1 (2.4) [ T i , T j ] = 0 , | i − j | > T i , X j ] = [ T i , Y j ] = 0 , j = i, i + 1(2.6) [ X i , X j ] = [ Y i , Y j ] = 0(2.7) X i +1 = T i X i T i , (2.8) Y i +1 = T − i Y i T − i (2.9) X − Y = Y X − T − (2.10) Y X · · · X n = qX · · · X n Y (2.11)Let e n be the symmetrizing idempotent in the finite Hecke algebra (which is generated by the T i ’s), which is characterized by T j e n = e n T j = t / e n for all j . The spherical DAHA is thesubalgebra S ¨ H nq,t := e n ¨ H nq,t e n of ¨ H nq,t , and it is also Z -graded. There is an SL ( Z ) action on thesubalgebra S ¨ H nq,t (see the paragraph above Lemma 2.1 in [SV11]).Following [SV11, Sec. 2.2] (except for the notational change P → Q ), for k > Q n ,k = e n X i Y ki e n Elements Q n x for x ∈ Z are defined using the SL ( Z ) action. We define S ¨ H n,>q,t to be the subalgebraof S ¨ H nq,t generated by Q na,b with a > σ = q − and ¯ σ = t − . Then Schiffmann and Vasserot proved thefollowing theorem relating the elliptic Hall algebra and spherical DAHAs. Theorem 2.1 ([SV11, Thm. 3.1]) . The assignment u x q d ( x ) − Q n x extends uniquely to a Z -graded SL ( Z ) -equivariant surjective algebra homomorphism φ n : E q,t ։ S ¨ H nq,t Given the previous theorem, a natural question is whether there is some type of limit one cantake as n → ∞ . It turns out that there is, but to describe it Schiffmann and Vasserot first had toprove the following theorem. Theorem 2.2 ([SV13, Prop. 4.1]) . The assignment Q n x Q n − x for each x ∈ Z + extends to aunique surjective algebra map Φ n : S ¨ H n, + q,t → S ¨ H n − , + q,t . This theorem allows us to construct a projective limit lim ←− S ¨ H nq,t . Also, the generators Q n x provideelements in this projective limit, and we let S ¨ H ∞ , + q,t be the subalgebra generated by these elementsfor x ∈ Z + . Theorem 2.1 shows that there is a map from the elliptic Hall algebra to S ¨ H ∞ , + q,t . Theorem 2.3 ([SV13, Thm. 4.6]) . The induced map φ ∞ : E + q,t → S ¨ H ∞ , + q,t is an isomorphism. Summarizing this work of Schiffmann and Vasserot, we obtain the following corollary which weuse below.
Corollary 2.4.
Suppose A is an algebra generated by elements a x for x ∈ S ⊂ Z + . Suppose thereare algebra maps A → S ¨ H n, + q,t for each n such that a x Q x . Then there is an algebra map A → E + q,t sending a x ( q d ( x ) − u x . H. R. MORTON AND P. SAMUELSON Skeins with a base string
We will describe some skeins which use the framed Homfly relations on oriented framed curvesand braids in the thickened torus T × I , together with a single fixed base string {∗} × I ⊂ T × I .In this section we define the braid skein algebra BSk n ( T , ∗ ) in terms of Z [ s ± , c ± ]-linear com-binations of braids, and their composites, and prove that it is isomorphic to the double affine Heckealgebra ¨ H n (following the conventions in [SV11]). (See Theorem 3.7.)3.1. Isotopies of braids in the punctured torus.
We start by considering the group of n -braidsin the punctured torus T − {∗} . We will work with the thickened torus T × I with a single fixedbase string {∗} × I to determine by the puncture ∗ ∈ T . Braids are made up of n strings orientedmonotonically from T × { } to T × { } which do not intersect each other or the base string.Braids are considered equivalent when the strings are isotopic avoiding the base string.Composition of braids is defined by placing one on top of the other, using the convention that AB means braid A lying below braid B .As in [MS17] we shall regard T as given by identifying opposite pairs of sides in the unit square[0 , × [0 , ∗ to be the centre (1 / , /
2) of the square. Fix n > ,
0) and ∗ as the end points for n -string braids in T × I − {∗} × I .We can draw the thickened torus in plan view as a square with opposite pairs of edges identified.We show the braid points and the base string position in the figure below, including a line alongthe diagonal through them as a visual help to keep track of them. * We can indicate some simple braids where only one or two of the points move by drawing thepath of the moving points on the plan view, rather as in the diagrams in [AM98]. In this view thebraid product is given by concatenation of the paths.For example, write x i for the braid in which point i moves uniformly around the (1 ,
0) curve inthe torus, and y i where point i moves around the (0 ,
1) curve, with all other points remaining fixed.These are shown in plan view as x i = * , y i = * and in a side view in figures 2 and 3, where the colouring of the edges being identified is consistentwith that used in the plan view. Similarly the braid σ i appears in plan view as in figure 1,concentrating only on the region around the braid points.A side elevation for x i viewed in the (0 ,
1) direction is shown in figure 2, and y i viewed in the( − ,
0) direction is seen in elevation in figure 3.
AHAS AND SKEIN THEORY 9 σ i = i n Figure 1.
Plan view of σ i x i = i ∗ Figure 2.
Side view of x i y i = i ∗ Figure 3.
Side view of y i Using either of these two elevation views the braids σ i appear in their usual form above, and itis immediate from these views that σ − i x i σ − i = x i +1 (3.1) σ i y i σ i = y i +1 . (3.2)In a plan view we assume that paths are projections of braid strings which rise monotonicallyfrom their initial braid point to their final braid point. The product of two braids corresponds tothe concatenation of their paths.We can see that the braids { x i } commute among themselves, since their paths in the plan vieware disjoint. The same applies to the braids { y i } , and equally the braids σ i commute with x j and y j when j = i, i + 1.The relations x x = x x , y y = y y become x σ − x σ − = σ − x σ − x , (3.3) y σ y σ = σ y σ y (3.4)in terms of the generators x , y . We can use the plan view for a braid where two paths cross, taking the usual convention of knotcrossings to show which strand lies at a higher level. For example in the plan view of x y the pathof point 1 lies below that of point 2, giving views of x y and y x in figure 4. x y = * y x = * Figure 4.
Plan views of x y and y x When two braids are composed there may be a path on the plan view that passes through abraid point at an intermediate stage. The plan can be altered to avoid such intermediate calls, bydiverting the path slightly away from the braid point. For example the braid x y starts with aplan view in figure 5. When the intermediate visit to braid point 1 is diverted a plan view for x y is shown in figure 6 along with a view for y x . * Figure 5.
Plan view of x y x y = * , y x = * Figure 6.
Smoothed plan view of x y and y x With further smoothing we get the plan view of the commutator x y x − y − as shown in figure7. From its elevation view in figure 8 we can write it as x y x − y − = σ σ · · · σ n − P σ n − · · · σ σ . AHAS AND SKEIN THEORY 11 x y x − y − = * Figure 7.
Plan view of x y x − y − ∗ Figure 8.
Elevation of x y x − y − Here P =is the braid taking string n once round the base string, with plan view * This gives an expression P = σ − n − · · · σ − x y x − y − σ − · · · σ − n − . as a braid in the punctured torus, in terms of the generators x , y , σ i .As a further help in using the plan view for paths we can alter the view near the projection ofone of the braid points, where a path starts out at the lowest level from the braid point and finishesat the highest level. Then another path crossing nearby (with either orientation) can be movedacross the braid point as shown locally in figure 9. = Figure 9.
Moving an arc past a braidpointApply this to the view of y x by moving the path from braid point 1 across braid point 2. Thisgives y x = * = * = x α where α = * = σ y , and thus x y − = y − x σ . (3.5)We can rewrite this equation in terms of the generators x and y as σ − x σ − y − = y − σ − x σ and further σ − x y − = y − x . A similar argument, moving one path across braid points 2 . . . n , shows that y x x · · · x n = * = * = x x · · · x n α n AHAS AND SKEIN THEORY 13 in the punctured braid group, where α n = * = β n y , with β n = * as in figure 10, giving y x · · · x n = x · · · x n β n y . ∗ Figure 10.
Side view of the braid β n = σ σ · · · σ n − σ n − · · · σ σ Bellingeri [Bel04, theorem 1.1] gives a presentation for the group of n -braids in the puncturedtorus with generators σ , · · · , σ n − , a, b, and relations σ i σ j = σ j σ i , | i − j | > σ i σ i +1 σ i = σ i +1 σ i σ i +1 (3.7) σ i a = aσ i , i > σ i b = bσ i , i > aσ − aσ − = σ − aσ − a (3.10) bσ − bσ − = σ − bσ − b (3.11) bσ − aσ = σ − aσ − b (3.12)In our notation this corresponds to a presentation with generators x , y , σ i taking a = y and b = x − and σ − i in place of σ i .Bellingeri’s relations involving a and b correspond to the equations x x = x x y y = y y x y − = y − x σ when written in terms of the generators x , y , σ . A presentation for the algebra
BSk n ( T , ∗ ) .Definition 3.1. The braid skein algebra BSk n ( T , ∗ ) is defined to be Z [ s ± , c ± ]-linear combina-tions of n -braids in the punctured torus, up to equivalence, subject to the local relations(3.13) − = ( s − s − )and(3.14) ∗ = c ∗ between braids.By the term local relation in this definition we mean that the braids in the relations only differas shown inside a 3-ball. We would like to find a “small” generating set for the ideal defined bythese relations, which we do in the following three theorems. (To simplify exposition, Theorems3.2 and 3.3 are proved in Subsection 3.3.) Theorem 3.2.
Suppose that α, β, γ are three n -braids in the punctured torus whose diagrams canbe isotoped in ( T − {∗} ) × I , fixing the boundary, so that they differ only inside a ball as α = , β = , γ = . Then there exist braids A and B such that α = Aσ B, β = Aσ − B, γ = AB.
Theorem 3.3.
Suppose that δ, ǫ are two n -braids in the punctured torus whose diagrams can beisotoped in ( T − D ) × I , fixing the boundary, so that they differ only in a ball as δ = ∗ , ǫ = ∗ . Then there exist braids A and B such that δ = AP B, ǫ = AB, where P is the braid taking string n once round the base string, shown here in plan and elevation. P = * = ∗ Theorem 3.4.
The ideal generated by (3.13) and (3.14) is the same as the ideal defined by (3.15) σ − σ − = ( s − s − )(3.16) P = c . AHAS AND SKEIN THEORY 15
Proof.
Clearly the equations σ − σ − = s − s − and P = c are special cases of (3.13) and (3.14) .Conversely given any braids α, β, γ whose diagrams differ in some ball as α = , β = , γ = . By theorem 3.2 we can write α − β = A ( σ − σ − ) B Then equation (3.15) shows that α − β = ( s − s − ) AB = ( s − s − ) γ, showing that α, β and γ satisfy equation (3.13).To deduce equation (3.14) for braids δ and ǫ as in theorem 3.3 write δ = AP B and apply equation (3.16) to get δ = AP B = c AB = c ǫ. (cid:3) We can now adjoin these relations to Bellingeri’s presentation for the braid group of the puncturedtorus to give a presentation of the algebra BSk n ( T , ∗ ). Theorem 3.5.
The algebra
BSk n ( T , ∗ ) can be presented by the braids σ , · · · , σ n − , x , y , with relations σ i σ j = σ j σ i , | i − j | > σ i σ i +1 σ i = σ i +1 σ i σ i +1 (3.18) σ i x = x σ i , i > σ i y = y σ i , i > x σ − x σ − = σ − x σ − x (3.21) y σ y σ = σ y σ y (3.22) x − σ y σ − = σ y σ x − (3.23) ( σ − s )( σ + s − ) = 0(3.24) x y x − y − = c σ σ · · · σ n − σ n − · · · σ σ (3.25) Proof.
In our notation Bellingeri’s generators are a = y , b = x − , and our σ i is Bellingeri’s σ − i .Relations (3.17) to (3.23) then present the algebra of n -braids in the punctured torus, by [Bel04].Relation (3.24) is equivalent to relation (3.15). Relation (3.25) is equivalent to the relation (3.16), P = c , since x y x − y − = σ σ · · · σ n − P σ n − · · · σ σ . (cid:3) Remark 3.6.
As confirmation that our conventions are consistent with these relations note thatwith x = σ − x σ − and y = σ y σ the relations between the generators x and y become x x = x x , y y = y y and y x − = x − y σ − . These relations have already been demonstratedin our illustrations above. Theorem 3.7.
The skein algebra
BSk n ( T , ∗ ) is isomorphic to the double affine Hecke algebra ¨ H n .Proof. We construct inverse homomorphisms between the two algebras. • Define a homomorphism from BSk n ( T , ∗ ) to ¨ H n by sending x , y , σ i to X , Y , T − i and s , c to t, q − .To show that this gives a homomorphism it is enough to check that the relations in thepresentation of BSk n ( T , ∗ ) hold after the assignment of generators in ¨ H n .The only relation for which this is not immediately clear is relation (3.25) in BSk n ( T , ∗ ).Relation (3.25) can be written x y x − = c β n y . We also know that y x · · · x n = x · · · x n β n y . The relation can then be rewritten as c − x y x − = ( x · · · x n ) − y x · · · x n . In our assignment to ¨ H n we can see that each x i is sent to X i . It is then enough to checkthat qX Y X − = ( X · · · X n ) − Y X · · · X n in ¨ H n . This follows immediately from the last relation for ¨ H n and the fact that the elements X i all commute. • We can define an inverse homomorphism from ¨ H n to BSk n ( T , ∗ ) by sending X i , Y i , T i to x i , y i , σ − i and t, q to s , c − . Our illustrations above confirm that the relations from ¨ H n hold in BSk n ( T , ∗ ) after this assignment. (cid:3) Isotopies of relations.
In this section we prove Theorems 3.2 and 3.3.
Proof of theorem 3.2.
For convenience of drawing we include an extra crossing inside the ball sothat α = , β = , γ =inside the ball and all three agree outside it. The two strings crossing in the ball in the diagram for α must belong to different braid strings, otherwise the diagram of γ would have a closed component,and so could not be a braid. Premultiply all three diagrams by the same braid C so that these twostrings start at braid points 1 and 2. Then postmultiply all three diagrams by the braid β − C − to get three diagrams of braids α ′ , β ′ , γ ′ which again agree except inside the ball D , as indicated infigure 11.In this figure we show a diagram for β ′ , indicating schematically that the strings in D areconnected to points 1 and 2 at the top and bottom of the braids. However the strings in thediagram are not assumed to run monotonically from bottom to top in T × I in the complement of D .What we do know is that the braid β ′ is the identity braid, so all n strings could be straightenedout simultaneously.In the braid α ′ we can then straighten out all the strings simultaneously, except for the firststring. Classical results of Birman [Bir74], following Fadell and Neuwirth [FN62], on pure braidsin a surface, state that the group of braids in which all but one string is fixed is isomorphic to thefundamental group of the complement of the fixed strings. The isomorphism carries a braid to thehomotopy class of the non-fixed string in the complement of the others. Bellingeri [Bel04] gives agood account of this. We can then determine the braid α ′ by finding the homotopy class of the AHAS AND SKEIN THEORY 17 D Figure 11.
A diagram of the braid β ′ and the ball D where the diagrams differfirst string in the complement of the remaining strings. The braid α ′ is shown in figure 12, withthe first string highlighted in red. Figure 12.
A diagram of the braid α ′ , with the first string highlightedNow the first string of α ′ can be drawn as a loop by taking the string back from the top point1 down to the bottom point along the first string of the identity braid β ′ , so that it follows a path l up to D , then a meridian loop m around string 2 inside D , and returning along the inverse path l − , as shown in figure 13.Break up this loop into the composite of three loops, up to homotopy, by making an intermediatediversion to and from the base point 1 along a path p from D following close to string 2 to thebottom, and then going directly along the bottom back to 1, shown in figure 14. The loop is isthen homotopic to the product of the loops lp , p − mp and p − l − .Write L for the pure braid represented by lp , and M for the braid represented by p − mp . Wethen can write α ′ = LM L − . Now the meridian loop m can be pulled down string 2 by the path p to the bottom, to see that p − mp represents σ , and so α ′ = Lσ L − . Figure 13.
A diagram of the loop represented by the first string of α ′ l mp Figure 14.
The loop as a compositeThe classical homotopy results can also be applied in a two-stage process to the pure braid L − γ ′ Lσ − , by firstly removing string 1 and showing that string 2 is trivial, and then showing thatstring 1 is trivial, so that the pure braid L − γ ′ Lσ − is the identity. This gives the result γ ′ = Lσ L − . The outcome is that α = Cα ′ β − C − = ( CLσ ) σ ( L − β − C − )while β = Cβ ′ β − C − = Cβ − C − = ( CLσ ) σ − ( L − β − C − )and γ = Cγ ′ β − C − = ( CLσ )( L − β − C − ) . The result now follows, taking A = CLσ and B = L − β − C − . (cid:3) Proof of theorem 3.3.
Premultiply both braids by a braid C so that the braid string in the ball D where the diagrams differ is string n , and then postmultiply both braids by ǫ − C − to get δ ′ and ǫ ′ .Then ǫ ′ is the identity braid, while δ ′ is a braid in which all strings, including the base string, arestraight, except for string n . We can then use the same classical technique to find δ ′ by consideringthe homotopy class represented by string n in the complement of the other strings.As in the previous case we represent this homotopy class as a product of three loops lp , p − mp and p − l − , where the path l runs up string n to the ball D , and p runs alongside the base string AHAS AND SKEIN THEORY 19 to the bottom and then back to the point n , while m runs round the meridian of the base stringinside the ball D . We can then write α ′ = LM L − , where L is the pure braid represented by lp , and M is the braid represented by p − mp .Now M is the braid P , which takes string n once round the base string, so we have δ ′ = LP L − = Cδǫ − C − , Then δ = ( C − L ) P ( L − Cǫ ) , giving the result δ = AP B, ǫ = AB , where A = C − L and B = L − Cǫ . (cid:3) Remark 3.8.
It is important in our argument here to know that all the diagrams are braids, evenalthough they may have been isotoped out of position relative to the direction in the interval I . Itis only under these circumstances that we can analyse the diagrams by considering the homotopyclass (rather than the isotopy class) of one string in the complement of the others.4. The tangle skein algebra
In this section we generalize the definition of the braid skein algebra using framed tangles, andwe conjecture that this produces the same algebra as the braid skein algebra. In the next sectionwe use this conjecture to relate the classical skein algebra of the punctured torus to the elliptic Hallalgebra.In Homflypt skein theory we consider oriented banded curves in a 3-manifold M , possibly withmarked input and output points on its boundary.Here are some such piecesWe can think of these as made of flat tape rather than rope. The only difference from rope isthat the tapes can have extra twists in them such asorTwists may be dealt with by drawing little kinks in the diagram, replacingbyand byWhen there are boundary points the curves will include oriented arcs joining input to outputpoints. In addition we can have some closed oriented curves.The general Homflypt skein Sk( M ) is defined to be Z [ s ± , v ± ] -linear combinations of bandedlinks, up to isotopy, with the basic linear relations(1) − = ( s − s − )(2) = v −
10 H. R. MORTON AND P. SAMUELSON between banded links whose diagrams differ only locally as shown.Special cases of interest to us are where M = F × I for a surface F , with or without boundary.In such cases we write Sk( F ) for the skein Sk( M ), which has the structure of an algebra, withproduct induced by stacking curves in the direction of the interval I .In [MS17] we have looked at the case where F = T , and given a presentation for Sk( T ).The case C = Sk( A ), where A is the annulus, is a commutative algebra. It has been widelystudied, originally by Turaev [Tur97], and subsequently by Morton and others.In our present work we will incorporate the skein of the torus with one hole, Sk( T − D ),including elements which map to the generators of Sk( T ) under the homomorphism induced bythe inclusion T − D → T .Again in the case M = F × I we will consider the case where we fix n input points in F × { } ,and take the corresponding n output points in F × { } . Stacking in the I direction will give thisskein the structure of an algebra over Z [ s ± , v ± ] which we denote by Sk n ( F ).The simplest case of this, when F = D , gives the algebra Sk n ( D ). This algebra is a version ofthe Hecke algebra H n ( z ) of type A , based on the quadratic relation σ i = zσ i + 1, where z = s − s − .In anticipation of the next section we are led to consider the skein Sk n ( T −{∗} ) of the puncturedtorus. In order to incorporate our algebra BSk n ( T , ∗ ) into this framework we will adjoin therelation ∗ = c ∗ to allow a string to cross through the fixed string {∗} × I in T × I which defines the puncture.With this extra relation in place we use the notation Sk n ( T , ∗ ) for the resulting algebra over Z [ s ± , v ± , c ± ]. Theorem 4.1.
There is an algebra homomorphism F n : BSk n ( T , ∗ ) ∼ = ¨ H n → Sk n ( T , ∗ ) . Proof.
The homomorphism F n is defined on a braid by making a consistent choice of framing for it.Braids in T can be framed by fixing a direction in T , say the (1 ,
0) direction, and taking the bandon each braid string in this direction, which is transverse to the string at all points. This appearsto give the framing used in [BWPV14]. Under any braid isotopy the bands will be preserved, andthe relations between braids will satisfy the skein relations between banded curves. (cid:3)
In this section and the next we consider the algebra Sk n ( T , ∗ ), which incorporates general framedtangles along with closed curves, with the elliptic Hall algebra and its relation to the algebras ¨ H n in mind. One outcome of this is the following result, which will be established in the next section. Theorem 4.2.
The homomorphism F n : BSk n ( T , ∗ ) → Sk n ( T , ∗ ) is surjective.Proof. We must look at elements of the skein Sk n ( T , ∗ ) and reduce them to combinations of braids.In particular we have to deal with closed curves as well as braids.Choose a disc D in T which contains the n braid points and the puncture ∗ . A suitable choiceis a neighbourhood of the line through the braid points 1 , · · · , n and the base point ∗ , shown infigure 15. The inclusion of T − D in T , combined with the identity braid on the n strings,induces an algebra homomorphism ϕ n : Sk( T − D ) → Sk n ( T , ∗ ) . AHAS AND SKEIN THEORY 21 * Figure 15.
A choice for the disc D in T The closed curves we particularly want to use can be described in the skein Sk( T − D ). Thisskein is an algebra with a homomorphismSk( T − D ) → Sk( T )induced by filling in the disc. A presentation of the algebra Sk( T ) is given in [MS17], withgenerators W x , x ∈ Z , and relations[ W x , W y ] = ( s m − s − m ) W x + y where m = det( xy ).Where z ∈ Z is primitive there is an embedded curve W z ⊂ T which is taken to represent theelement W z ∈ Sk( T ). The same curve will give a well-defined element of Sk( T − D ), which wewill also write as W z . Write A z ⊂ T − D for the annulus with W z as its core. Further elements W x , where x = m z , are defined in section 5.1 as in [MS17] by suitably chosen combinations ofcurves in A z , and determine the generators of Sk( T ) above.The elements W x do not generate the whole of the algebra Sk( T − D ), nor do all the commu-tation relations from [MS17] hold in Sk( T − D ). The main problem is that the disc D gets inthe way of isotopies that can be made in T . Remark 4.3.
It is worth noting however that if y and z are primitive, and the curves W y and W z intersect in just one point, then the commutation relation[ W x , W y ] = ( s m − s − m ) W x + y holds for x = m z . This is because the argument from [MS17] only involves curves in the union ofthe annuli A y and A z . Hence in particular we have (4.1) [ W ( m, , W (0 , ] = − ( s m − s − m ) W ( m, in the image of ¨ H n in Sk n ( T , ∗ ). Note that under the map Sk + ( T − D ) → E q,t described inTheorem 5.7, we have the assignments W m,
7→ { m } s u m, , W ,
7→ { } s u , , W m,
7→ { } s u m, where we have used the notation { d } s := s d − s − d . Expanding the relation (2.1) in the case y = ( m,
0) and x = (0 , u m, , u , ] = − u m, Thus, we see that that relation (4.1) in Sk + ( T − D ) gets mapped to the relation (4.2) in E q,t under the algebra map in Theorem 5.7 (whose existence depends on Conjecture 1.5). The sign difference between the right hand side of (4.1) and the corresponding relation in [MS17] comes from thefact that our convention for the product in the skein algebra here (left element goes below the right element) is theopposite from [MS17]. The existence of this map depends on the assumption that Conjecture 1.5 is true.
To prove theorem 4.2 we first show that for each n the image of Sk( T − D ) is represented bybraids. In other words we establish Lemma 4.4. ϕ n (Sk( T − D )) ⊂ F n (BSk n ( T , ∗ )) . This depends on two results. • We can write any curve C ⊂ T − D as a polynomial in the elements { W x } plus a linearcombination of braids. • Each W x is a linear combination of braids.The proof of theorem 4.2 is completed by showing that a general tangle can be written as aproduct of braids and closed curves which avoid D . Lemma 4.5.
The algebra
Sk( T − D ) is generated by totally ascending single curves C ⊂ T − D .Proof. This is a standard skein theory exercise. We proceed by induction on the number of crossingsin a diagram in T − D . Order the components of the diagram and choose a starting point oneach component. A totally ascending diagram is one in which each crossing appears first as anundercrossing, when working along the component from the chosen starting point. Go through thecomponents in turn, switching crossings, and using the skein relation, to end up with a diagram L in which the components are totally ascending, along with a linear combination of diagrams withfewer crossings.The components of L are then stacked one above the other, and so represent the product ofsingle totally ascending curves. (cid:3) The ability to alter the starting point of a totally ascending diagram of a curve, using inductionon the number of crossings in the diagram, will prove useful in the arguments which follow.
Lemma 4.6.
Suppose that C and D are two diagrams differing only in the signs of their crossings,then D = C + ( s − s − ) P ± D α where each D α is a -component diagram with fewer crossings than C .Proof. This follows immediately, using the skein relation at each crossing of C to be switched. (cid:3) Lemma 4.7.
Let C be a curve in T − D which is totally ascending from a starting point imme-diately beside the boundary of D . The curve C ′ given by diverting C around the other side of D ,which is also totally ascending and is isotopic to C in T but not in T − D , satisfies the relation ϕ n ( q ± C ′ − C ) = ± ( s − s − ) n X i =1 β i ( C ) where β i ( C ) are braids.Proof. Divert the curve C successively past the braid points to give curves C = C , C i , C n , andthen C ′ , with C i crossing D between points i and i + 1, interpreting ∗ as point n + 1. The skeinrelation gives C i = C i − ± ( s − s − ) β i ( C ) where β i ( C ) is a braid in which only string i moves,because the curve C i − is totally ascending at the point of crossing string i . The result follows,given that C ′ = q ± C n . (cid:3) Remark 4.8.
The sign depends on the direction of the curves C i across the disc. It will be+ if C crosses the line of braid points in the x direction. The curve C represents an element w C ( x, y ) ∈ π ( T − D ) based at the starting point. This fundamental group is the free group on 2generators, and the braid β ( C ) is then β ( C ) = w ( x , y ) while the successive braids β i ( C ) satisfy β i ( C ) = σ i − · · · σ β ( C ) σ · · · σ i − . AHAS AND SKEIN THEORY 23
We are now in a position to prove that a curve C ⊂ T − D can be written in Sk n ( T , ∗ ) as apolynomial in { W x } modulo im( F n ). Work by induction on the number of crossings in the diagramof C . We can assume by lemma 4.5 that C is totally ascending.Write c ∈ Z for the homology class of C in H ( T ), and write c = m z with z primitive. If c = (0 ,
0) choose any primitive as z .Fix a simple closed curve Z in T in the direction of z . Now perform a sequence of moves atthe expense of braids and other elements W x , at each stage replacing C by C ′ homologous to C .Firstly move strands of C across D until there are no strands of C lying between Z and D toone side of Z . At each move we need to change the starting point to lie on the strand to be movedacross D , using lemma 4.7.Now suppose that Z is crossed l times by C in the direction away from D . If l = 0 then C liesentirely in the annulus A z on the side away from D . It then represents an element in the skein ofthis annulus, Sk( A z ), which can be written as a polynomial in the commuting elements { W k z } .Otherwise Z is also crossed l times by C in the opposite sense, because C is homologous to m z .Choose a crossing in the direction away from D where the next crossing is towards D , and takethis as the starting point for C , again using the induction on the number of crossings. The arc of C between the starting crossing and the next one can now be isotoped across Z , without crossing D , to reduce l .In equation (5.2), it is shown that W ( m, can be written as a linear combination of braids, andcombining this with the SL ( Z ) action shows that any curve in T − D can be represented inSk n ( T , ∗ ) by a linear combination of braids. This completes the proof of lemma 4.4.The last step in the proof of theorem 4.2 is to deal with a general framed n -tangle in ( T , ∗ ).This will consist of n arcs along with some closed curves. We use the plan view, modified slightlyto separate the top and bottom points, and work by induction on the number of crossing points ina diagram to show that it represents a polynomial in braids and closed curves avoiding D .Make the tangle diagram totally ascending, choosing starting points first at the bottom point ofeach arc and then on the closed curves. This can be done by the induction. The resulting diagramrepresents a product of a braid with closed curves, since the arcs are each totally ascending, and thecrossings with closed curves all lie above them. It remains to show that a single totally ascendingcurve C in the torus which avoids the n + 1 points { , · · · , n, ∗} can be altered at the expense ofbraids to avoid the line through the n + 1 points which determines D .Suppose that C crosses the connecting line immediately to the right of point i . Make C totallyascending at this crossing position, and then move C across i to lie immediately to its left, at theexpense of a braid β i ( C ) as in the proof of lemma 4.7. Continue moving intersections to the left,and eventually past the point 1 at the end of the line, to finish by avoiding the connecting linealtogether, and hence lying in T − D . Lemma 4.4 then completes the proof of theorem 4.2 (cid:3) Remark 4.9.
It is not clear whether the homomorphism F n is injective. There can be the questionof possible further relations between elements in the image of BSk n ( T , ∗ ) ∼ = ¨ H n coming from theadditional closed curves that can be used in Sk n ( T , ∗ ).Despite the previous remark, we conjecture (see Conjecture 1.5 in the introduction) that thealgebra map F n : ¨ H n → Sk n ( T , ∗ ) in Theorem 4.1 is an isomorphism.5. Relations with the elliptic Hall algebra
In [SV11] the authors relate the double affine Hecke algebras ¨ H n to the elliptic Hall algebra. Aspart of their construction they make use of the sums of powers X i X li , X i Y li ∈ ¨ H n which have a very useful skein theoretic description, and which led us to try including closed curvesin our skein BSk n ( T , ∗ ). We will show that the images of these elements in Sk n ( T , ∗ ) agree withthe images of certain natural elements in Sk( T − D ). In Theorem 5.7, we combine this with resultsof Schiffmann and Vasserot to show that Conjecture 1.5 implies a weakened version of Conjecture1.3.5.1. Certain closed curves.
For the moment consider the Homflypt skein Sk n ( A ) where A isan annulus, using oriented diagrams in the thickened annulus A × I with n output points on thetop A × { } , and n matching input points on A × { } . We also allow closed components in thediagrams.When restricted to braid diagrams the skein BSk n ( A ) is used by Graham and Lehrer as a modelfor the affine Hecke algebra ˙ H n , where composition is again induced by composition of braids.Write Z i and Z i for the elements represented in Sk n ( A ) by the diagrams shown here. Take theframing of the closed component as given by the plane of the diagram. Z i = i , Z i = i It is readily established that − = ( s − s − ) X Z i = ( s − s − ) X Z i There is also a well-established element P m for each m in the skein Sk( A ) = C of the annuluswith no boundary points which satisfies the relation P m − P m = ( s m − s − m ) X Z mi = ( s m − s − m ) X Z mi . A detailed account of P m can be found, for example, in [Mor02]. AHAS AND SKEIN THEORY 25
When we embed A into T around the (1 ,
0) curve, matching the braid points suitably, theinduced homomorphism from Sk n ( A ) to Sk n ( T , ∗ ) gives the equation( s m − s − m ) X i x mi = * P m − * P m = (1 − c m ) * P m Similarly, taking A around the (0 ,
1) curve on T we get( s m − s − m ) X i y mi = * P m − * P m = ( c − m − * P m In Sk n ( A ), taking account of the crossing signs, we also have P m − P m = − ( s m − s − m ) X Z − mi = − ( s m − s − m ) X Z − mi . Placing A along the (1 ,
0) curve then gives − ( s m − s − m ) X x − mi = * P m − * P m = (1 − c − m ) * P m , while placing A along the (0 ,
1) curve gives − ( s m − s − m ) X i y − mi = * P m − * P m = ( c m − * P m In [SV11] there is a description of the elliptic Hall algebra which involves generators u x for everynon-zero x ∈ Z . These elements satisfy certain commutation relations, and the comparison withthe algebras ¨ H n requires the prescription of an image for each u x , and a check on their commutationproperties.We can give a version of this comparison by using the skein Sk n ( T , ∗ ), and the skein Sk( T − D ).Fix a disc D in T which includes the braid points and the base point. A suitable choice for ourpurposes is a neighbourhood of the diagonal in the square. In the previous section we introducedthe homomorphism(5.1) ϕ n : Sk( T − D ) → Sk n ( T , ∗ ) AHAS AND SKEIN THEORY 27 defined by taking the banded curves in T − D along with the identity n -braid in Sk n ( T , ∗ ),consisting of n vertical strings in D × I and the base string.Now any oriented embedded curve in T − D is determined up to isotopy by a primitive element z ∈ Z , representing the homology class of the curve. This curve, framed by its neighbourhood in T defines an element W z ∈ Sk( T − D ). For any other non-zero x ∈ Z write x = m z with m > z primitive, and define W x to be W z with the closed curve decorated by the element P m .We will write W x also for its image in the skein Sk n ( T , ∗ ). We then have plan views of W ( ± m, and W (0 , ± m ) as W ( m, = * P m , W ( − m, = * P m , W (0 ,m ) = * P m , W (0 , − m ) = * P m ,Our equations above show that(1 − c m ) W ( m, = ( s m − s − m ) X x mi , (5.2) ( c − m − W ( − m, = ( s m − s − m ) X x − mi ( c − m − W (0 ,m ) = ( s m − s − m ) X y mi (1 − c m ) W (0 , − m ) = ( s m − s − m ) X y − mi . Comparison with the algebraic approach.
For non-zero x ∈ Z Schiffman and Vasserotin [SV11] define elements Q x in the spherical algebra S ¨ H n , where S ¨ H n is defined as e n ¨ H n e n , with e n ∈ H n being the symmetrizer. They use the elements Q x in setting up their comparisons withthe elliptic Hall algebra.Using the identification of BSk n ( T , ∗ ) with ¨ H n , where q = c − , s = t , we show now that ourelements W x are closely related to Q x ∈ S ¨ H n , when mapped into the full skein algebra Sk n ( T , ∗ ).Before doing this we note the construction of the symmetrizer e n ∈ H n ⊂ ¨ H n in the braid skeinsetting, as used by Aiston and Morton in [AM98].We use the model of the Hecke algebra H n described in [MT90], and further in [AM98]. Thesymmetrizer is given there as a multiple of the quasi-idempotent a n = P s l ( π ) ω π , where ω π is thepositive permutation braid associated to the permutation π with length l ( π ) in the symmetric group.The symmetrizer is then e n = α n a n where α n is given by the equation a n a n = α n a n [Luk05, AM98].Using the quasi-idempotent b n = P ( − s ) − l ( π ) ω π in a similar way gives the antisymmetrizer.We prefer to avoid the notation S for the symmetrizer, because of conflict with the notation forthe symmetric group. In [SV11] the element a n is denoted by ˜ S , and the symmetrizer by S . Theorem 5.1.
For x ∈ Z we have the following equality in Sk n ( T , ∗ ) : ( q m − e n W x e n = ( s m − s − m ) Q x , where x = m y with y primitive and m > .Proof. We start from the definition in [SV11] which sets Q (0 ,m ) := e n P Y mi e n for m > x = (0 , m ), since( q m − e n W (0 ,m ) e n = ( s m − s − m ) e n X Y mi e n = ( s m − s − m ) Q (0 ,m ) . When x = ( ± m, , (0 , − m ) the values of Q x are shown in [SV11, Eq. 2.16-2.18] to be Q ( − m, = e n X X − mi e n Q (0 , − m ) = q m e n X Y − mi e n Q ( m, = q m e n X X mi e n . The theorem follows immediately from (5.2) in these cases too, since( q m − W ( − m, = ( s m − s − m ) X x − mi , giving the case x = ( − m, q m − W ( m, = ( s m − s − m ) q m X x mi and ( q m − W (0 , − m ) = ( s m − s − m ) q m X y − mi , giving the other two cases.We use automorphisms of ¨ H n , and their counterpart in the skein models BSk n ( T , ∗ ) andSk n ( T , ∗ ) to establish the proof for general x .Firstly, in our skein model, a right-hand Dehn twist about the (unoriented) (1 ,
0) curve in T − D induces an automorphism τ of Sk( T − D ), which carries W x to W y with y = (cid:18) (cid:19) x . A left-hand Dehn twist about the unoriented (0 ,
1) curve in T − D induces an automorphism τ of Sk( T − D ), which carries W x to W y with y = (cid:18) (cid:19) x . These two automorphisms generate all homeomorphisms of T which fix D , up to isotopy fixing ∂D . This group of automorphisms is isomorphic to the braid group B with τ and τ − playingthe roles of the usual Artin generators σ , σ . The kernel of the map to SL (2 , Z ) is infinite cyclic,generated by ( τ τ − τ ) , which is the right-hand Dehn twist about ∂D .For any x with d ( x ) = m > γ so that x = γ ((0 , m )).Now the effect of τ on the generators σ i , x i , y i of BSk n ( T , ∗ ) is τ ( σ i ) = σ i τ ( x i ) = x i τ ( y i ) = η i where η i = y i x i δ i and δ i = σ i − . . . σ σ . . . σ i − . AHAS AND SKEIN THEORY 29
The effect of τ is τ ( σ i ) = σ i τ ( x i ) = ξ i τ ( y i ) = y i where ξ i = x i y i δ − i . The automorphisms ρ and ρ used in [SV11] agree with τ and τ , given the correspondence of x i , y i , σ i with X i , Y i , T − i respectively.Since Q x is given from Q (0 ,m ) by applying a suitable product of ρ and ρ , the same automorphismwill carry W (0 ,m ) to W x and the theorem will follow. (cid:3) Without the symmetrizer.
Theorem 5.1, which refers to elements of Sk n ( T , ∗ ), suggeststhat Q x could be defined unambiguously from an element ˜ Q x in ¨ H n ∼ = BSk( T , ∗ ) before passingto S ¨ H n . The kernel of the map from B to SL (2 , Z ) is generated by ( τ τ − τ ) . In the skein modelthis is a Dehn twist about the boundary of the disc D , and so in this model we expect the followingtheorem, which we can prove algebraically. Proposition 5.2.
For any Z ∈ ¨ H n ∼ = BSk n ( T , ∗ ) we have ( τ τ − τ ) Z = ∆ − Z ∆ , where ∆ is the full twist braid in the finite Hecke algebra H n .Proof. It is enough to prove this when Z = x and Z = y , since these elements, along with σ i ,generate BSk n ( T , ∗ ). In the case Z = σ i we have τ ( σ i ) = τ ( σ i ) = σ i , while the full twist ∆ commutes with each σ i .We also know that τ ( x ) = x τ ( y ) = y x τ − ( x ) = x y − τ − ( y ) = y Writing τ τ − τ = θ we get θ ( x ) = y − , θ ( y ) = y x y − so θ ( x ) = ( θ ( y )) − = y x − y − = ( y x ) x − ( y x ) − θ ( y ) = θ ( y ) θ ( x )( θ ( y )) − = ( y x ) y − ( y x ) − Finally θ ( x ) = θ ( y x ) θ ( x − )( θ ( y x )) − = ( y x )( y − x − ) x ( x y )( y x ) − = [ x , y ] − x [ x , y ] θ ( y ) = [ x , y ] − x [ x , y ]Now [ x , y ] = c β n so θ ( x ) = β − n x β n θ ( y ) = β − n y β n The result now follows since∆ = w ( σ , · · · , σ n − ) σ σ · · · σ n − σ n − · · · σ σ = wβ n and the braid w commutes with x and y . (cid:3) Remark 5.3.
Simmental, [Sim, Lemma 2.4.20], in notes which are part of a seminar series at MITand Northeastern in 2017, makes a similar observation when applied to the spherical algebra S ¨ H n ,to demonstrate the construction of the elements Q x .We can go further and define ˜ Q ,m for m >
0, by˜ Q ,m = y m + y m + · · · + y mn . Then Q ,m = e n ˜ Q ,m e n , in [SV11]. We follow the same procedure as in [SV11] to define ˜ Q x from ˜ Q ,m by applying anautomorphism from SL (2 , Z ) which takes (0 , m ) to x .This gives a well-defined element ˜ Q x , provided we can show that ( τ τ − τ ) = θ acts triviallyon ˜ Q ,m = P y mi . So we prove Lemma 5.4. ∆ − ( y m + · · · + y mn )∆ = y m + · · · + y mn . Proof.
It is enough to show that y m + · · · + y mn commutes with σ i for all i . Now σ i commutes with y j for j = i, i + 1. So we just need to show that σ i commutes with y mi + y mi +1 .This in turn follows once we prove that σ i ( y i + y i +1 ) = ( y i + y i +1 ) σ i σ i ( y i y i +1 ) = ( y i y i +1 ) σ i Now σ i ( y i + y i +1 ) = σ i y i + σ i y i σ i = σ i y i + y i σ i + ( s − s − ) σ i y i σ i = y i σ i + σ i y i σ i = ( y i + y i +1 ) σ i ,σ i ( y i y i +1 ) = σ i y i σ i y i σ i = y i +1 y i σ i = ( y i y i +1 ) σ i . This completes the proof. (cid:3)
So we have constructed elements ˜ Q x ∈ ¨ H n with Q x = e n ˜ Q x e n , which are related even moredirectly to the elements W x in Sk( T , ∗ ), in the following enhancement of theorem 5.1. Theorem 5.5.
For every non-zero x ∈ Z we have ( q m − W x = ( s m − s − m ) ˜ Q x , where x = m y with y primitive and m > . The punctured torus and elliptic Hall algebra.
In this subsection, we use the previousresults in this section to show that Conjecture 1.5 implies a weakened version of Conjecture 1.3.Recall that Z + ⊂ Z = Z is defined by Z + := { ( a, b ) ∈ Z | a > } ⊔ { (0 , b ) | b ≥ } Definition 5.6.
Let Sk + ( T − D ) be the subalgebra of Sk( T − D ) generated by W x for x ∈ Z + . Theorem 5.7.
If Conjecture 1.3 is true, then there is a surjective algebra map Sk + ( T − D ) ։ E + σ, ¯ σ sending W x to ( s d ( x ) − s − d ( x ) ) u x . AHAS AND SKEIN THEORY 31
Proof.
By Conjecture 1.5, the map F n : ¨ H n → Sk n ( T , ∗ ) is an isomorphism, and we can composeits inverse with the natural map ϕ n : Sk( T − D ) → Sk n ( T , ∗ )to obtain a map Sk( T − D ) → ¨ H n . By Theorem 5.1, this map satisfies the following equation: W x s d ( x ) − s − d ( x ) q d ( x ) − Q x By Corollary 2.4, this proves the existence of the algebra map stated in the theorem, and surjectivityfollows immediately from the definition of the subalgebra E + σ, ¯ σ . (cid:3) The simplest relations between the W x are easy to check in the elliptic Hall algebra independentlyof Conjecture 1.5 (see Remark 4.3). However, describing all relations between the W x in thepunctured torus is an open problem. Remark 5.8.
It would be desirable to extend the map in Theorem 5.7 to a much larger subalgebraof Sk( T − D ), instead of the subalgebra Sk + ( T − D ) generated by W x for x ∈ Z + . It seems thatthe main difficulty is showing compatibility between the Schiffmann-Vasserot projections betweenspherical DAHAs and the maps from Sk( T − D ) to the spherical DAHAs. Ideally this wouldfollow from a topological interpretation of the Schiffmann-Vasserot projections as some kind ofpartial trace, but it isn’t clear if such an interpretation exists. We do note that we have definedalgebra maps from the entire skein algebra Sk( T − D ) to the DAHAs (and not just the positivesubalgebra). The technical difficulty here is that the Schiffmann-Vasserot projections betweenspherical DAHAs of different ranks are only defined on the “positive subalgebras.” References [AM98] A. K. Aiston and H. R. Morton,
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