Complexified tetrahedrons, fundamental groups, and volume conjecture for double twist knots

TL;DR

This paper proves the volume conjecture for double twist knots using complexified tetrahedra and their associated SL(2,C) representations, linking geometric and quantum invariants.

Contribution

It introduces the use of complexified tetrahedra to establish the volume conjecture for a class of knots, connecting geometric structures with quantum invariants.

Findings

Proved the volume conjecture for double twist knots.

Established a link between complexified tetrahedra and quantum invariants.

Expressed colored Jones polynomial via quantum 6j symbols.

Abstract

In this paper, the volume conjecture for double twist knots are proved. The main tool is the complexified tetrahedron and the associated $\mathrm{SL}(2, \mathbb{C})$ representation of the fundamental group. A complexified tetrahedron is a version of a truncated or a doubly truncated tetrahedron whose edge lengths and the dihedral angles are complexified. The colored Jones polynomial is expressed in terms of the quantum $6j$ symbol, which corresponds to the complexified tetrahedron.