Quantum invariants of hyperbolic knots and extreme values of trigonometric products

2020/06/15 by Christoph Aistleitner, Aistleitner, Christoph, Bence Borda +1 · 1 citation
Mathematics · #11L03 #2020: 11J70 #26D05 #57K16 #FOS: Mathematics #FOS: Physical sciences #General Topology (math.GN) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2006.08578

openalex publication_date 2020/06/15 · openalex created_date 2022/07/17 · openalex updated_date 2026/07/28

Abstract

In this paper we study the relation between the function J_41,0, which arises from a quantum invariant of the figure-eight knot, and Sudler's trigonometric product. We find J_41,0 up to a constant factor along continued fraction convergents to a quadratic irrational, and we show that its asymptotics deviates from the universal limiting behavior that has been found by Bettin and Drappeau in the case of large partial quotients. We relate the value of J_41,0 to that of Sudler's trigonometric product, and establish asymptotic upper and lower bounds for such Sudler products in response to a question of Lubinsky.

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• On the behavior of the colored Jones polynomial of the figure-eight knot under modular transformations

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