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

TL;DR

This paper introduces a new algebraic basis for the Homflypt skein module of the solid torus, using generalized Hecke algebras, and proves its linear independence and basis properties.

Contribution

The paper constructs and proves the validity of a new basis for the Homflypt skein module of the solid torus, differing from previous bases, via algebraic methods involving generalized Hecke algebras.

Findings

Established a new basis $\Lambda$ for the skein module.

Proved the linear independence of the basis $\Lambda$.

Provided algebraic tools for computing skein modules of 3-manifolds.

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.