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:

Abstract. In this paper we work toward the Homflypt skein module of the lens spaces L(p, 1), S(L(p, 1)), using braids. 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. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set Λ.The importance of our approach is that it can shed light to the problem of computing skein modules of arbitrary c.c.o. 3-manifolds, since any 3-manifold can be obtained by surgery on S 3 along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.

In this paper we describe braid equivalence for knots and links in a 3-manifold M obtained by rational surgery along a framed link in S 3 . We first prove a sharpened version of the Reidemeister theorem for links in M . We then give geometric formulations of the braid equivalence via mixed braids in S 3 using the L-moves and the braid band moves. We finally give algebraic formulations in terms of the mixed braid groups Bm,n using cabling and the techniques of parting and combing for mixed braids. We also provide concrete formuli of the braid equivalence in the case where M is a lens space, a Seifert manifold or a homology sphere obtained from the trefoil. The algebraic classification of knots and links in a 3-manifold via mixed braids is a useful tool for studying skein modules of 3-manifolds.2010 Mathematics Subject Classification. 57M27, 57M25, 57N10.

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces L(p, 1) from the HOMFLYPT skein module of the solid torus, S(ST), it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant X for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set Λ aug , augmenting the basis Λ of S(ST).

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