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

Abstract

In this paper we present two new bases, \(B^{\prime}_{H_2}\) and \(\mathcal{B}_{H_2}\), 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(\(H_2\)), \(B_{H_2}\), and using the technique of parting we present elements in \(B_{H_2}\) 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 \(B_{H_2}\), \(B^{\prime}_{H_2}\) and \(\mathcal{B}_{H_2}\) as proper subsets. Using the Kauffman bracket skein relation we relate \(B_{H_2}\) to the sets \(B^{\prime}_{H_2}\) and \(\mathcal{B}_{H_2}\) via a lower triangular infinite matrix with invertible elements in the diagonal. The basis \(B^{\prime}_{H_2}\) is an intermediate step in order to reach at elements in \(\mathcal{B}_{H_2}\) that have no crossings on the level of braids, and in that sense, \(\mathcal{B}_{H_2}\) is a more natural basis of KBSM(\(H_2\)). Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of 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 \(\mathcal{B}_{H_2}\).