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

In this paper we give a new basis, Λ, for the Homflypt skein module of the solid torus, S(ST)S(ST), which topologically is compatible with the handle sliding moves and which was predicted by J.H. Przytycki. The basis Λ is different from the basis Λ′Λ′, discovered independently by Hoste and Kidwell [1] and Turaev [2] with the use of diagrammatic methods, and also different from the basis of Morton and Aiston [3]. For finding the basis Λ we use the generalized Hecke algebra of type B, H1,nH1,n, which is generated by looping elements and braiding elements and which is isomorphic to the affine Hecke algebra of type A [4]. More precisely, we start with the well-known basis Λ′Λ′ of S(ST)S(ST) and an appropriate linear basis ΣnΣn of the algebra H1,nH1,n. We then convert elements in Λ′Λ′ to sums of elements in ΣnΣn. Then, using conjugation and the stabilization moves, we convert these elements to sums of elements in Λ by managing gaps in the indices, by ordering the exponents of the looping elements and by eliminating braiding tails in the words. Further, we define total orderings on the sets Λ′Λ′ and Λ and, using these orderings, we relate the two sets via a block diagonal matrix, where each block is an infinite lower triangular matrix with invertible elements in the diagonal. Using this matrix we prove linear independence of the set Λ, thus Λ is a basis for S(ST)S(ST).S(ST)S(ST) plays an important role in the study of Homflypt skein modules of arbitrary c.c.o. 3-manifolds, since every c.c.o. 3-manifold can be obtained by integral surgery along a framed link in S3S3 with unknotted components. In particular, the new basis of S(ST)S(ST) is appropriate for computing the Homflypt skein module of the lens spaces. In this paper we provide some basic algebraic tools for computing skein modules of c.c.o. 3-manifolds via algebraic means.