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

Jun Murakami

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arXiv · 2501.00225

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

Jun Murakami

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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.

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Jun Murakami. 2024-12-31. Complexified tetrahedrons, fundamental groups, and volume conjecture for double twist knots. https://arxiv.org/abs/2501.00225

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