A basis for the Kauffman skein module of the product of a surface and a circle

The Kauffman bracket skein module $S(M)$ of a 3-manifold $M$ is a $\mathbb{Q}(A)$-vector space spanned by links in $M$ modulo the so-called Kauffman relations. In this article, for any closed oriented surface $\mathrm{\Sigma }$ we provide an explicit spanning family for the skein modules $S(\Sigma\times S^1)$. Combined with earlier work of Gilmer and Masbaum, we answer their question about the dimension of $S(\mathrm{\Sigma }\times S^1)$ being $2^{2g+1}+2g-1$.