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We introduce a quantum trace map for an ideally triangulated hyperbolic knotcomplement $S^3\backslash \mathcal{K}$. The map assigns a quantum operator toeach element of Kauffmann Skein module of the 3-manifold. The quantum operatorlives in a module generated by products of quantized edge parameters of theideal triangulation modulo some equivalence relations determined by gluingequations. Combining the quantum map with a state-integral model of$SL(2,\mathbb{C})$ Chern-Simons theory, one can define perturbative invariantsof knot $K$ in the knot complement whose leading part is determined by itscomplex hyperbolic length. We then conjecture that the perturbative invariantsdetermine an asymptotic expansion of the Jones polynomial for a link composedof $\mathcal{K}$ and $K$. We propose the explicit quantum trace map forfigure-eight knot complement and confirm the length conjecture up to the secondorder in the asymptotic expansion both numerically and analytically.

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