# The braid approach to the HOMFLYPT skein module of the lens spaces $L(p,1)$

## TL;DR

This paper discusses a braid-based method for computing the HOMFLYPT skein module of lens spaces $L(p,1)$, building on knot theory in the solid torus and an invariant called $X$, with the full solution to be detailed in a sequel.

## Contribution

It introduces a braid approach to relate the skein module of lens spaces to that of the solid torus, reducing the problem to solving an infinite system of equations.

## Key findings

- Established the relation between $ ext{S}(L(p,1))$ and $ ext{S}( ext{ST})$
- Reduced the computation to solving braid band move equations
- Presented a basis $ ext{Lambda}$ for $ ext{S}( ext{ST})$

## Abstract

In this paper we present recent results toward the computation of the HOMFLYPT skein module of the lens spaces $L(p,1)$, $\mathcal{S}\left(L(p,1) \right)$, via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, $X$, for knots and links in ST, the universal analogue of the HOMFLYPT polynomial in ST. The relation between $\mathcal{S}\left(L(p,1) \right)$ and $\mathcal{S}({\rm ST})$ is established in \cite{DLP} and it is shown that in order to compute $\mathcal{S}\left(L(p,1) \right)$, it suffices to solve an infinite system of equations obtained by performing all possible braid band moves on elements in the basis of $\mathcal{S}({\rm ST})$, $\Lambda$, presented in \cite{DL2}. The solution of this infinite system of equations is very technical and is the subject of a sequel paper \cite{DL3}.