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

Abstract

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