  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}.
