Geometric triangulations and the Teichm\"uller TQFT volume conjecture for twist knots

TL;DR

This paper constructs new ideal triangulations for twist knot complements, proves their geometric nature, and uses them to verify the Teichmüller TQFT volume conjecture through explicit partition function calculations.

Contribution

It introduces a new family of triangulations for twist knots and confirms the volume conjecture for all twist knots using these triangulations.

Findings

Triangulations provide new upper bounds for Matveev complexity.

Triangulations are proven to be geometric.

Volume conjecture verified for all twist knots.

Abstract

We construct a new infinite family of ideal triangulations and H-triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichm\"uller TQFT and to prove the associated volume conjecture for all twist knots, using the saddle point method.