Some Results About the Kauffman Bracket Skein Module of the Twist Knot Exterior

Abstract

Abstract. In this paper, we list in explicit form the factoring relations of the Kauffman bracket skein module (KBSM for short) of a twist knot exterior. This is done using curves decorated by characters of irreducible SL(2, C)-representations. In the process, we exhibit a relation which holds in the KBSM of the knot exterior, called the minimal relation. In the final section we prove that, when specializing the variable of the Kauffman bracket at t = −1, the minimal relation becomes the defining polynomial of the SL(2, C)-character variety of the twist knot.1. Motivation and background 1.1. Motivation. The Kauffman bracket skein module of a 3-manifold was introduced by J. Przyticki as a natural generalization of the Kauffman bracket to general 3-manifolds. It was later linked to the character variety of SL(2, C)-representations of the fundamental group of the manifold [B1], [PS]. In this perspective, Kauffman bracket skein modules were used to give an alternate description of the A-polynomial of Cooper, Culler, Gillet, Long, and Shalen, and to generalize it to a noncommutative setting [FGL]. The computation of the noncommutative generalization of the A-polynomial of a knot relies heavily on the good understanding of the skein module of the knot complement.The noncommutative generalization of the A-polynomial was computed for the unknot in [FGL], trefoil knot in [G1], partially for (2, 2p + 1)-torus knots in [GS1], and for the figure-eight knot in [GS2]. In those papers, and also in [G2] this knot invariant was linked to the Jones polynomial. This relation led to new developments in the study of colored Jones polynomials [GL]. Let us also point out that such computations yield an alternative way of finding the classical A-polynomial, as it was discussed in [N].The computation itself is done in three stages. The first stage consists of the understanding of the Kauffman bracket skein module of the knot complement. As knot complements can be obtained by attaching 2-handles to a handlebody, and such topological operations yield algebraic factorizations at the level of the skein module, it is necessary to write the factoring relations in explicit form. The second stage of the computation is concerned with determining the action of the Kauffman bracket skein algebra of the torus on the skein module of the knot complement, while the third stage is about finding the annihilator of the empty skein.The present paper describes the first step in the computation of the noncommutative version of the A-polynomial for twist knots. Our convention is to count twists Key words and phrases. A-polynomial, colored Kauffman brackets, Kauffman bracket skein module, tunnel number.The second author has been supported by JSPS Research Fellowships for Young Scientists.

On minimal elements for a partial order of prime knots

“…and let R m (x, y) be the second factor of R m (x, y). It follows from [5] using the KBSM that X(K m ) is described as Expressions of X(G) are sometimes quite important when we discuss geometric properties of X(G). For example, the presentation in (1.3) is more convenient than that of (1.1) or (1.2) to research the number of irreducible components of X(K m ) at least in the sense that the presentation does not depend on m even or odd.…”