Quantum invariants of hyperbolic knots and extreme values of trigonometric products
Christoph Aistleitner, Bence Borda
Abstract
Abstract In this paper, we study the relation between the function $$J_{4_1,0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mrow> <mml:msub> <mml:mn>4</mml:mn> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> , which arises from a quantum invariant of the figure-eight knot, and Sudler’s trigonometric product. We find $$J_{4_1,0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mrow> <mml:msub> <mml:mn>4</mml:mn> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> 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 $$J_{4_1,0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mrow> <mml:msub> <mml:mn>4</mml:mn> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> 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.