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

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 $\mathcal{S}(\mathrm{ST})$ 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 $\mathcal{S}(\mathrm{ST})$, $\Lambda$, 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}.

28 pages, 21 Figures. arXiv admin note: substantial text overlap with arXiv:1604.06163, arXiv:1412.3642