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

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.