Basis for KBSM of fibered torus with multiplicity two exceptional fiber

We construct a family of bases for the Kauffman bracket skein module (KBSM) of the product of an annulus and a circle. Using these bases, we find a new basis for the KBSM of $(\beta ,2)$-fibered torus as a first step toward developing techniques for computing KBSM of a family of small Seifert fibered $3$-manifolds.