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Frobenius Algebras Derived from the Kauffman Bracket Skein Algebra

Nel Abdiel, Charles Frohman

Abstract

In this paper we study the skein modules of the surfaces, $Σ_{i,j}$ $(i,j)\in \{(0,2),(0,3),(1,0),(1,1)\}$ at $2N$-th roots of unity where $N\geq 3$ is an odd counting number and construct Frobenius algebras from them.

Frobenius Algebras Derived from the Kauffman Bracket Skein Algebra

Abstract

In this paper we study the skein modules of the surfaces, at -th roots of unity where is an odd counting number and construct Frobenius algebras from them.

Paper Structure

This paper contains 6 sections, 33 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The Annulus
  4. The Pair of Pants
  5. The Torus
  6. The once punctured torus

Key Result

Lemma 2.1

If $p(z)$ is a polynomial so that for all $q\in \mathbb{C}-\{0\}$, $p(q+q^{-1})=q^k+q^{-k}$ then $p(z)=T_k(z)$.

Theorems & Definitions (40)

  • Lemma 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 30 more