Topological steps toward the Homflypt skein module of the lens spaces $L(p,1)$ via braids

TL;DR

This paper develops a braid-based method to compute the Homflypt skein module of lens spaces $L(p,1)$, connecting it to the skein module of the solid torus and establishing an infinite system of equations for the calculation.

Contribution

It introduces a new basis for the skein module of the solid torus and relates skein modules of lens spaces to braid band moves, advancing the understanding of 3-manifold skein modules.

Findings

Established the connection between skein modules of the solid torus and lens spaces.

Derived an infinite system of equations for computing the skein module of $L(p,1)$.

Proved the equivalence of systems considering all braid band moves and a basic subset.

Abstract

In this paper we work toward the Homflypt skein module of the lens spaces $L(p,1)$, $\mathcal{S}(L(p,1))$, using braids. In particular, we establish the connection between $\mathcal{S}({\rm ST})$, the Homflypt skein module of the solid torus ST, and $\mathcal{S}(L(p,1))$ and arrive at an infinite system, whose solution corresponds to the computation of $\mathcal{S}(L(p,1))$. We start from the Lambropoulou invariant $X$ for knots and links in ST, the universal analogue of the Homflypt polynomial in ST, and a new basis, $\Lambda$, of $\mathcal{S}({\rm ST})$ presented in \cite{DL1}. We show that $\mathcal{S}(L(p,1))$ is obtained from $\mathcal{S}({\rm ST})$ by considering relations coming from the performance of braid band moves (bbm) on elements in the basis $\Lambda$, where the braid band moves are performed on any moving strand of each element in $\Lambda$. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set $\Lambda$. The importance of our approach is that it can shed light to the problem of computing skein modules of arbitrary c.c.o. $3$-manifolds, since any $3$-manifold can be obtained by surgery on $S^3$ along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.