The Kauffman bracket skein module of the complement of (2,2p + 1)-torus knots via braids

Abstract

In this paper we compute the Kauffman bracket skein module of the complement of (2,2p+1)-torus knots, KBSM(T(2,2p+1)c), via braids. We start by considering geometric mixed braids in S3, the closure of which are mixed links in S3 that represent links in the complement of (2,2p+1)-torus knots, T(2,2p+1)c. Using the technique of parting and combing geometric mixed braids, we obtain algebraic mixed braids, that is, mixed braids that belong to the mixed braid group B2,n and that are followed by their “coset” part, that represents T(2,2p+1)c. In that way we show that links in T(2,2p+1)c may be pushed to the genus 2 handlebody, H2, and we establish a relation between KBSM(T(2,2p+1)c) and KBSM(H2). In particular, we show that in order to compute KBSM(T(2,2p+1)c) it suffices to consider a basis of KBSM(H2) and study the effect of combing on elements in this basis. We consider the standard basis of KBSM(H2) and we show how to treat its elements in KBSM(T(2,2p+1)c), passing through many different spanning sets for KBSM(T(2,2p+1)c). These spanning sets form the intermediate steps in order to reach at the set BT(2,2p+1)c, which, using an ordering relation and the notion of total winding, we prove that it forms a basis for KBSM(T(2,2p+1)c). Note that elements in BT(2,2p+1)c have no crossings on the level of braids, and in that sense, BT(2,2p+1)c forms a more natural basis of KBSM(T(2,2p+1)c) in our setting. We finally consider c.c.o. 3-manifolds M obtained from S3 by surgery along the trefoil knot and we discuss steps needed in order to compute the Kauffman bracket skein module of M. We first demonstrate the process described before for computing the Kauffman bracket skein module of the complement of the trefoil, KBSM(Trc), and we study the effect of braid band moves on elements in the basis of KBSM(Trc). These moves reflect isotopy in M and are similar to the second Kirby moves.The “braid” method that we propose for computing Kauffman bracket skein modules seem promising in computing KBSM of arbitrary c.c.o. 3-manifolds M. The only difficulty lies in finding the sufficient relations that reduce elements in the basis of our underlying genus g-handlebody, Hg. These relations come from combing in the case of knot complements and from combing and braid band moves for 3-manifolds obtained by surgery along a knot in S3. Our aim is to set the necessary background of this “braid” approach in order to compute Kauffman bracket skein modules of arbitrary 3-manifolds.