HOMFLYPT skein sub-modules of the lens spaces $L(p, 1)$

In this paper we work toward the HOMFLYPT skein module of $L(p, 1)$, $\mathcal{S}(L(p,1))$, via braids. Our starting point is the linear Turaev-basis, $\Lambda^{\prime}$, of the HOMFLYPT skein module of the solid torus ST, $\mathcal{S}({\rm ST})$, which can be decomposed as the tensor product of the "positive" ${\Lambda^{\prime}}^+$ and the "negative" ${\Lambda^{\prime}}^-$ sub-modules, and the Lambropoulou invariant, $X$, for knots and links in ST, that captures $S({\rm ST})$. It is a well-known result by now that $\mathcal{S}(L(p, 1))=\frac{\mathcal{S}(ST)}{<bbm's>}$, where bbm's (braid band moves) denotes the isotopy moves that correspond to the surgery description of $L(p, 1)$. Namely, a HOMFLYPT-type invariant for knots and links in ST can be extended to an invariant for knots and links in $L(p, 1)$ by imposing relations coming from the performance of bbm's and solving the infinite system of equations obtained that way. \smallbreak In this paper we work with a new basis of $\mathcal{S}({\rm ST})$, $\Lambda$, and we relate the infinite system of equations obtained by performing bbm's on elements in $\Lambda^+$ to the infinite system of equations obtained by performing bbm's on elements in $\Lambda^-$ via a map $I$. More precisely we prove that the solutions of one system can be derived from the solutions of the other. Our aim is to reduce the complexity of the infinite system one needs to solve in order to compute $\mathcal{S}(L(p,1))$ using the braid technique. Finally, we present a generating set and a potential basis for $\frac{\Lambda^+}{<bbm's>}$ and thus, we obtain a generating set and a potential basis for $\frac{\Lambda^-}{<bbm's>}$. We also discuss further steps needed in order to compute $\mathcal{S}(L(p,1))$ via braids.