The Kauffman bracket skein module of the handlebody of genus 2 via braids

In this paper we present two new bases, B-H2' and B-H2, for the Kauffman bracket skein module of the handlebody of genus 2 H-2, KBSM(H-2). We start from the well-known Przytycki-basis of KBSM(H-2), B-H2, and using the technique of parting we present elements in B-H2 in open braid form. We define an ordering relation on an augmented set L consisting of monomials of all different "loopings" in H-2, that contains the sets B-H2, B-H2' and B-H2 as proper subsets. Using the Kauffman bracket skein relation we relate B-H2 to the sets B-H2' and B-H2 via a lower triangular infinite matrix with invertible elements in the diagonal. The basis B-H2' is an intermediate step in order to reach at elements in B-H2 that have no crossings on the level of braids, and in that sense, B-H2 is a more natural basis of KBSM(H-2). Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed-connected-oriented (c.c.o.) 3-manifolds M that are obtained from H-2 by surgery, since isotopy moves in M are naturally described by elements in B-H2.