On the HOMFLYPT skein module of $$S^1 \times S^2$$ via braids

In this paper we study the HOMFLYPT skein module of $$S^1 \times S^2\, \cong \, L(0, 1)$$ , denoted $$\mathcal {S}(S^1 \times S^2)$$ , using braid-theoretic techniques. We extend the Lambropoulou invariant, X, for links in the solid torus ST to links in $$S^1 \times S^2$$ , by solving an infinite system of equations of the form $$X_{\widehat{a}} = X_{\widehat{bbm(a)}}$$ , where bbm(a) denotes all possible band moves applied to a, for all a in a basis of $$\mathcal {S}(ST)$$ . We show that the free part of $$\mathcal {S}(S^1 \times S^2)$$ is generated by the empty link, while all other elements lies in the torsion submodule.