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

In this paper, the volume conjecture for double twist knots are proved. The main tool is the complexified tetrahedron and the associated $\mathrm{S}\mathrm{L}(2,\mathbb{C})$ 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.