2022/03/30 by Prarit Agarwal, Agarwal, Prarit, Dongmin Gang +5
Mathematics · Medicine · #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.2203.15985
openalex publication_date 2022/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement S3\backslash 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.