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