An important step for the computation of the HOMFLYPT skein module of the lens spaces $L(p,1)$ via braids

TL;DR

This paper demonstrates a method to compute the HOMFLYPT skein module of lens spaces L(p,1) by reducing the problem to solving an infinite system of equations involving braid band moves on a specific basis of the solid torus skein module.

Contribution

It shows that computing the skein module of L(p,1) can be achieved by focusing on braid band moves on the first strand within an augmented basis, simplifying previous approaches.

Findings

Reduction of the problem to an infinite system of equations

Use of braid band moves on the first strand

Augmentation of the basis for the solid torus skein module

Abstract

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, $\mathcal{S}({\rm ST})$, 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 $\Lambda^{aug}$, augmenting the basis $\Lambda$ of $\mathcal{S}({\rm ST})$.