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 $L(p,1)$, $$\mathcal{S}$\left(L(p,1) \right)$, via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, $X$, for knots and links in ST, the universal analogue of the HOMFLYPT polynomial in ST. The relation between $\mathcal{S}(L(p,1))$ and $\mathcal{S}(\mathrm{S}\mathrm{T})$ is established in \cite{DLP} and it is shown that in order to compute $$\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 $\mathcal{S}(\mathrm{S}\mathrm{T})$, $\mathrm{\Lambda }$, 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}.