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

Abstract

In this paper we give a new basis, $\Lambda$, for the Homflypt skein module of the solid torus, $\mathcal{S}({\rm ST})$, which was predicted by Jozef Przytycki, using topological interpretation. The basis $\Lambda$ is different from the basis $\Lambda^{\prime}$, discovered independently by Hoste--Kidwell \cite{HK} and Turaev \cite{Tu} with the use of diagrammatic methods. For finding the basis $\Lambda$ we use the generalized Hecke algebra of type B, $\textrm{H}_{1,n}$, defined by the second author in \cite{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 well-known basis of $\mathcal{S}({\rm ST})$, $\Lambda^{\prime}$, and an appropriate linear basis $\Sigma_n$ of the algebra $\textrm{H}_{1,n}$. We then convert elements in $\Lambda^{\prime}$ to linear combinations of elements in the new basic set $\Lambda$. This is done in two steps: First we convert elements in $\Lambda^{\prime}$ to elements in $\Sigma_n$. Then, using conjugation and the stabilization moves, we convert these elements to linear combinations of elements in $\Lambda$ 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 $\Lambda^{\prime}$ and $\Lambda$ and prove that the sets are totally ordered. Finally, using this ordering, we relate the sets $\Lambda^{\prime}$ and $\Lambda$ 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 $\Lambda$. The infinite matrix is then "invertible" and thus, the set $\Lambda$ is a basis for $\mathcal{S}({\rm ST})$. The aim of this paper is to provide the basic algebraic tools for computing skein modules of c.c.o. $3$-manifolds via algebraic means.