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

TL;DR

The paper proves the complex volume conjecture for hyperbolic double twist knots by recasting colored Jones invariants in terms of complexified tetrahedra tied to $SL(2,\mathbb{C})$ representations and quantum $6j$ symbols. It develops the framework of a complexified tetrahedron as a deformation of ideal polyhedra, connects the Neumann–Zagier potential function to the asymptotics via Poisson summation and saddle-point analysis, and shows that $J_{N-1}(K)$ grows like $e^{N\zeta(K)}$ with $\zeta(K)=\sqrt{-1}(Vol(K)+i\,CS(K))$. The main results establish that for Borromean rings, twisted Whitehead links, and hyperbolic double twist knots, the asymptotics satisfy $2\pi\lim_{N\to\infty}\frac{\log J_{N-1}(K)}{N}=Vol(K)+CS(K)\,\sqrt{-1}$ (mod $\pi^2\sqrt{-1}\mathbb{Z}$). These methods unify the hyperbolic-geometric interpretation with quantum-algebraic invariants, and extend the volume conjecture to a broad family of links and knots via complex polyhedral decompositions.

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.

Figures (29)

• Figure 1: Knots and links handled in this paper.
• Figure 2: A usual tetrahedron, a truncated tetrahedron and a doubly truncated tetrahedron. Any face which truncate a vertex is perpendicular to the original three faces of the tetrahedron which are adjacent to the vertex.
• Figure 3: Complexify the angle and the length at an edge. The parameter $i\theta$ is modified to $\ell_\theta + i\theta$ and the parameter $\ell$ is modified to $\ell + i \theta_\ell$. The shaded faces correspond to the truncated faces.
• Figure 4: Elements of $\pi_1(S^3\setminus B)$. The base point is located above the plane.
• Figure 5: Regular ideal octahedra $O_1$, $O_2$ in the upper half space whose union is the fundamental domain of the action of $\rho(\pi_1(S^3\setminus B))$.

Theorems & Definitions (38)

• Conjecture 1: Volume conjecture MM
• Conjecture 2: Complexified volume conjecture MMOTY
• Theorem 1
• Remark 1
• Remark 2
• Theorem 2
• Theorem 3
• proof
• Proposition 3.1
• proof