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We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement S^3∖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,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 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.

No abstract uploaded!

No abstract uploaded!

3D shape classification plays a fundamental role in geometric big data analysis. This work proposes a novel method for shape classification based on optimal mass transportation theory. The Riemann surfaces are mapped onto canonical domains conformally based on uniformization theorem. The conformal factor function is treated as probability distribution on the canonical domain. For each pair of probability distributions, the optimal mass transportation map is computed by solving Monge-Amper´e equation. The transportation cost is theWasserstein distance between two distributions. By using this distance, geometric classification based on clustering can be performed. The method is applied to 3D human facial expression recognition, which demonstrates the efficiency and efficacy of the method.