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

TL;DR

This paper computes the Kauffman bracket skein module of the complement of certain torus knots using braid techniques, establishing a basis and exploring implications for 3-manifolds obtained by surgery.

Contribution

It introduces a braid-based method to compute the skein module of torus knot complements and extends the approach to 3-manifolds from surgery, providing explicit bases and relations.

Findings

Established a basis for the skein module of torus knot complements.

Linked skein modules of knot complements to those of handlebodies.

Analyzed braid moves corresponding to 3-manifold surgeries.

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 $S^3$, the closure of which are mixed links in $S^3$ 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 $B_{2, n}$ 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, $H_2$, and we establish a relation between $KBSM(T_{(2, 2p+1)}^c)$ and $KBSM(H_2)$. In particular, we show that in order to compute $KBSM(T_{(2, 2p+1)}^c)$ it suffices to consider a basis of $KBSM(H_2)$ and study the effect of combing on elements in this basis. We consider the standard basis of $KBSM(H_2)$ 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 $\mathcal{B}_{T_{(2, 2p+1)}^c}$, 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 $S^3$ 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, $KBSM(Tr^c)$, and we study the effect of braid band moves on elements in the basis of $KBSM(Tr^c)$. These moves reflect isotopy in $M$ and are similar to the second Kirby moves.