A proof of the Teichm\"{u}ller TQFT volume conjecture for $7_3$ knot

TL;DR

This paper proves the Teichmüller TQFT volume conjecture for the hyperbolic knot 7_3 by demonstrating that the hyperbolic volume can be derived from quantum invariants associated with a specific tetrahedral decomposition.

Contribution

It provides the first rigorous proof of the volume conjecture for the 7_3 knot within the framework of Teichmüller TQFT, confirming the conjecture's validity for this case.

Findings

Confirmed the volume conjecture for 7_3 knot

Established a method to extract hyperbolic volume from quantum invariants

Validated the tetrahedral decomposition approach for this knot

Abstract

In the generalized topological quantum field theory constructed by Andersen and Kashaev, invariants of 3-manifolds are defined given the combinatorial structure of a tetrahedral decomposition. Furthermore, a variant of the volume conjecture has been proposed in which the hyperbolic volume can be extracted from this invariant of the complementary space of the hyperbolic knot in an oriented $3$-dimensional closed manifold. We prove that the reformulated volume conjecture holds for the complementary space of the hyperbolic knot $7_3$ in $S^3$, given a specific tetrahedral decomposition.