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In this paper we give a new basis, $\Lambda$, for the Homflypt skein moduleof the solid torus, $\mathcal{S}({\rm ST})$, which was predicted by JozefPrzytycki, using topological interpretation. The basis $\Lambda$ is differentfrom the basis $\Lambda^{\prime}$, discovered independently by Hoste--Kidwell\cite{HK} and Turaev \cite{Tu} with the use of diagrammatic methods. Forfinding the basis $\Lambda$ we use the generalized Hecke algebra of type B,$\textrm{H}_{1,n}$, defined by the second author in \cite{La2}, which isgenerated by looping elements and braiding elements and which is isomorphic tothe affine Hecke algebra of type A. Namely, we start with the well-known basisof $\mathcal{S}({\rm ST})$, $\Lambda^{\prime}$, and an appropriate linear basis$\Sigma_n$ of the algebra $\textrm{H}_{1,n}$. We then convert elements in$\Lambda^{\prime}$ to linear combinations of elements in the new basic set$\Lambda$. This is done in two steps: First we convert elements in$\Lambda^{\prime}$ to elements in $\Sigma_n$. Then, using conjugation and thestabilization moves, we convert these elements to linear combinations ofelements in $\Lambda$ by managing gaps in the indices of the looping elementsand by eliminating braiding tails in the words. Further, we define an orderingrelation in $\Lambda^{\prime}$ and $\Lambda$ and prove that the sets aretotally ordered. Finally, using this ordering, we relate the sets$\Lambda^{\prime}$ and $\Lambda$ via a block diagonal matrix, where each blockis an infinite lower triangular matrix with invertible elements in the diagonaland we prove linear independence of the set $\Lambda$. The infinite matrix isthen "invertible" and thus, the set $\Lambda$ is a basis for $\mathcal{S}({\rmST})$. The aim of this paper is to provide the basic algebraic tools forcomputing skein modules of c.c.o. $3$-manifolds via algebraic means.

A new basis for the Homflypt skein module of the solid torus