Quantum trace map for 3-manifolds and a 'length conjecture'

TL;DR

This paper introduces a quantum trace map for hyperbolic knot complements, linking quantum operators to skein modules and proposing a length conjecture connecting invariants to hyperbolic length, with explicit computations for the figure-eight knot.

Contribution

It presents a new quantum trace map for 3-manifolds, combines it with Chern-Simons theory to define invariants, and verifies the length conjecture for the figure-eight knot.

Findings

Quantum trace map explicitly constructed for the figure-eight knot complement

Perturbative invariants match complex hyperbolic length up to second order

Numerical and analytical confirmation of the length conjecture

Abstract

We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement $S^3\backslash \mathcal{K}$. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives in a module generated by products of quantized edge parameters of the ideal triangulation modulo some equivalence relations determined by gluing equations. Combining the quantum map with a state-integral model of $SL(2,\mathbb{C})$ Chern-Simons theory, one can define perturbative invariants of knot $K$ in the knot complement whose leading part is determined by its complex hyperbolic length. We then conjecture that the perturbative invariants determine an asymptotic expansion of the Jones polynomial for a link composed of $\mathcal{K}$ and $K$. We propose the explicit quantum trace map for figure-eight knot complement and confirm the length conjecture up to the second order in the asymptotic expansion both numerically and analytically.