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

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}.