The (2, ∞)-Skein Module of Lens Spaces: A Generalization of the Jones Polynomial

Abstract

We extend the Jones polynomial for links in S3 to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S1×S2 the skein module is infinitely generated.

Estimating a skein module with 𝑆𝐿₂(ℂ) characters

“…Furthermore, every weight relation except +2 ∅ is trivial. This is because they are equivalent to the framing relations used in earlier definitions of K(M ) [8,5], which are easily seen to be trivial if A = −1. The fact that framing ceases to be relevant seems to make A = −1 a more natural specialization than A = 1.…”

“…The motivation behind this work is a pattern of behavior demonstrated by all the known examples [2,3,5,6,8] of Kauffman bracket skein modules of compact orientable 3-manifolds. Namely, the module is finitely generated if and only if the underlying manifold does not contain an incompressible surface.…”

On the relation between the A-polynomial and the Jones polynomial

“…If we add a 2-handle to the cylinder over the torus along the meridian, the skein module becomes C[t, t −1 , α], where α is the curve which is the image of the longitude, which is an infinite dimensional free module. But Hoste and Przytycki [7] have shown that adding a 2-handle induces a set of algebraic relations at the level of the skein module, in particular the same relations are introduced in K t (T 2 × I) and in its image through φ. Since adding these relations produces different modules, φ is not an isomorphism, so it has a kernel.…”

The Kauffman bracket skein module of a twist knot exterior

“…The only other successful method [1,7,8] has been to consider the effect of adding a single 2-handle to a handlebody. This creates a presentation with fairly simple generators (any basis for the module of the handlebody), but having an unwieldy set of relations.…”

“…This has been managed for all genus one manifolds [7,8], and for (2, q)-torus knot exteriors [1]. In principle, the combinatorics ought to be accessible for genus two manifolds with toral boundary (one added handle), but the computations are quite daunting in practice.…”