HOMFLY-PT skein module of singular links in the three-sphere

TL;DR

This paper computes the HOMFLY-PT skein module of singular links in the three-sphere, providing explicit results and relating it to the Conway skein module, thus advancing the understanding of link invariants in topology.

Contribution

It explicitly calculates the HOMFLY-PT skein module for singular links in S^3 under certain invertibility conditions, connecting it to the Conway skein module.

Findings

Computed the HOMFLY-PT skein module for singular links in S^3.

Established conditions for invertibility of key elements in the ring.

Derived the Conway skein module as a special case.

Abstract

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local relations, called skein relations, that involve $t$ and $x$. We compute the HOMFLY-PT skein module of singular links for any $R$ such that $(t^{-1}-t+x)$ and $(t^{-1}-t-x)$ are invertible. In particular, we deduce the Conway skein module of singular links.