Quantum invariants of hyperbolic knots and extreme values of trigonometric products

TL;DR

This paper explores the connection between a quantum invariant of the figure-eight knot and Sudler's trigonometric product, revealing unique asymptotic behaviors and establishing bounds relevant to number theory and quantum topology.

Contribution

It identifies the relation between the quantum invariant and Sudler's product along continued fraction convergents, showing deviations from universal asymptotics and providing bounds for Sudler's products.

Findings

$J_{4_1,0}$ matches Sudler's product up to a constant along quadratic irrational convergents.

Asymptotics of $J_{4_1,0}$ deviate from universal limits for large partial quotients.

Established asymptotic bounds for Sudler's trigonometric products.

Abstract

In this paper we study the relation between the function $J_{4_1,0}$, which arises from a quantum invariant of the figure-eight knot, and Sudler's trigonometric product. We find $J_{4_1,0}$ 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 $J_{4_1,0}$ 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.