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An edge-type state integral over local field and A-polynomials

Honghuai Fang

Abstract

To each local field, Garoufalidis and Kashaev recently associate a quantum dilogarithm that satisfies a pentagon identity and some symmetries. By employing an angled version of these quantum dilogarithms, they developed three generalized TQFTs, one given by a face state-integral and others by edge state integrals. These TQFTs produce distributional invariants for one-cusped three-manifolds, which are believed to be related to counting points on the A-polynomial curve. In this paper, we will calculate partition functions of an edge-type generalized TQFT over a local field for several examples and prove the appearance of A-polynomial in these new invariants.

An edge-type state integral over local field and A-polynomials

Abstract

To each local field, Garoufalidis and Kashaev recently associate a quantum dilogarithm that satisfies a pentagon identity and some symmetries. By employing an angled version of these quantum dilogarithms, they developed three generalized TQFTs, one given by a face state-integral and others by edge state integrals. These TQFTs produce distributional invariants for one-cusped three-manifolds, which are believed to be related to counting points on the A-polynomial curve. In this paper, we will calculate partition functions of an edge-type generalized TQFT over a local field for several examples and prove the appearance of A-polynomial in these new invariants.
Paper Structure (7 sections, 4 theorems, 71 equations, 1 figure)

This paper contains 7 sections, 4 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Teichmüller TQFT over local field
  3. Quantum dilogarithm over local field
  4. Face-type Teichmüller TQFT over local field
  5. Two edge-type Teichmüller TQFT over local field
  6. Examples of $\mathsf{B}$-type
  7. A theorem connecting A-polynomial and state-integral

Key Result

Theorem 1.1

The function $P_K(x,y)$ coincides with the $\operatorname{PSL}(2,\mathbb{C})$ A-polynomial of knot $K$.

Figures (1)

  • Figure 1: 3-2 and 2-3 angled Pachner moves.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Remark 4.1