A new basis for the Homflypt skein module of the solid torus

Abstract

Abstract. In this paper we give a new basis, Λ, for the Homflypt skein module of the solid torus, S(ST), which was predicted by Jozef Przytycki using topological interpretation. The basis Λ is different from the basis Λ ′ , discovered independently by Hoste-Kidwell [HK] and Turaev [Tu] with the use of diagrammatic methods, and also different from the basis of Morton-Aiston [MA]. For finding the basis Λ we use the generalized Hecke algebra of type B, H1,n, defined by the second author in [La2], which is generated by looping elements and braiding elements and which is isomorphic to the affine Hecke algebra of type A. Namely, we start with the wellknown basis of S(ST), Λ ′ , and an appropriate linear basis Σn of the algebra H1,n. We then convert elements in Λ ′ to linear combinations of elements in the new basic set Λ. This is done in two steps: First we convert elements in Λ ′ to elements in Σn. Then, using conjugation and the stabilization moves, we convert these elements to linear combinations of elements in Λ by managing gaps in the indices of the looping elements and by eliminating braiding tails in the words. Further, we define an ordering relation in Λ ′ and Λ and prove that the sets are totally ordered. Finally, using this ordering, we relate the sets Λ ′ and Λ via a block diagonal matrix, where each block is an infinite lower triangular matrix with invertible elements in the diagonal and we prove linear independence of the set Λ. The infinite matrix is then "invertible" and thus, the set Λ is a basis for S(ST). IntroductionLet M be an oriented 3-manifold, R = Z[u ±1 , z ±1 ], L the set of all oriented links in M up to ambient isotopy in M and let S the submodule of RL generated by the skein expressions u −1 L + − uL − − zL 0 , where L + , L − and L 0 are oriented links that have identical diagrams, except in one crossing, where they are as depicted in Figure 1. For convenience we allow the empty knot, ∅, and add the relation u −1 ∅ − u∅ = zT 1 , where T 1 denotes the trivial knot. Then the Homflypt skein module of M is defined to be:

Topological steps toward the Homflypt skein module of the lens spaces L(p,1) via braids

“…The new basis, Λ, of S(ST). In [DL1] we give a different basis Λ for S(ST), which was predicted by the third author. The new basic set is described in Eq.…”

“…In [DL1] a new basis, Λ, of S(ST) is presented (see Theorem 5 in this paper). For an illustration see Figure 11.…”

“…• Then, we reduce the equations obtained from elements in the H 1,n (q)-module L by performing braid band moves on any strand, to equations obtained from elements in the H 1,n (q)-module Λ ( [DL1]) by performing braid band moves on any strand.…”

“…The paper is organized as follows: In Section 1 we recall the algebraic setting and results needed from [La2,LR2,DL1]. We present the generalized Iwahori-Hecke algebra of type B, which is related to the knot theory of the solid torus and which plays a crucial role for this paper.…”

“…In particular, we establish the connection between S(ST), the Homflypt skein module of the solid torus ST, and S(L(p, 1)) and arrive at an infinite system, whose solution corresponds to the computation of S(L(p, 1)). We start from the Lambropoulou invariant X for knots and links in ST, the universal analogue of the Homflypt polynomial in ST, and a new basis, Λ, of S(ST) presented in [DL1]. We show that S(L(p, 1)) is obtained from S(ST) by considering relations coming from the performance of braid band moves (bbm) on elements in the basis Λ, where the braid band moves are performed on any moving strand of each element in Λ.…”