2021/06/09 by Ioannis Diamantis, Diamantis, Ioannis · 1 citation
Computer Science · Mathematics · #20C08 #20F36 #20F38 #57K10 #57K12 #57K14 #57K35 #57K45 #57K99 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2106.04965
openalex publication_date 2021/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, 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 B_T(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). 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.