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The Kauffman Bracket Skein Module at an irreducible representation

Mohammad Farajzadeh-Tehrani, Charles Frohman, Joanna Kania-Bartoszynska

Abstract

In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to an irreducible representation of the fundamental group of the manifold, forms a one-dimensional free module over the localized unreduced coordinate ring of the character variety. We apply this by proving that the dimension of the skein module of a homology sphere with finite character variety, when the order of the root of unity is not divisible by $4$, is greater than or equal to the dimension of the unreduced coordinate ring of the character variety. This leads to a computation of the dimension of the skein module with coefficients in rational functions for homology spheres with tame universal skein module.

The Kauffman Bracket Skein Module at an irreducible representation

Abstract

In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to an irreducible representation of the fundamental group of the manifold, forms a one-dimensional free module over the localized unreduced coordinate ring of the character variety. We apply this by proving that the dimension of the skein module of a homology sphere with finite character variety, when the order of the root of unity is not divisible by , is greater than or equal to the dimension of the unreduced coordinate ring of the character variety. This leads to a computation of the dimension of the skein module with coefficients in rational functions for homology spheres with tame universal skein module.
Paper Structure (13 sections, 17 theorems, 112 equations, 1 figure)

This paper contains 13 sections, 17 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the Kauffman bracket skein modules and algebras
  3. Review of the algebraic foundations
  4. Localization, factorization and Artinian rings
  5. Left and right ideals of a matrix algebra
  6. Locally free modules
  7. The Azumaya Locus
  8. Towards the Kauffman bracket skein module of a closed three-manifold
  9. Localized skein module modules at an irreducible representation
  10. Proof of Theorem \ref{['main_thm']}
  11. Applications and further directions
  12. Estimating the dimension of $K_\zeta(X)$
  13. The dimension of the $\mathbb{Q}(q)$-skein module

Key Result

Theorem 2.2

Bu, PS1. The algebra $K_\epsilon(X)$ is isomorphic to the unreduced coordinate ring of the $\textnormal{SL}_2\mathbb{C}$-character variety of $\pi_1(X)$

Figures (1)

  • Figure 1: Handlebody of genus $2$

Theorems & Definitions (25)

  • Remark 1.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 3.1
  • Theorem 3.2: Azumaya
  • Proposition 3.3
  • Remark 3.4
  • Theorem 4.1
  • Theorem 4.2: FKL
  • ...and 15 more