TY - JOUR
T1 - Quantum invariants of hyperbolic knots and extreme values of trigonometric products
AU - Aistleitner, Christoph
AU - Borda, Bence
N1 - Funding Information:
CA is supported by the Austrian Science Fund (FWF), projects F-5512, I-3466, I-4945, I-5554, P-34763, P-35322 and Y-901. BB is supported by FWF project Y-901. We want to thank Agamemnon Zafeiropoulos for drawing our attention to the papers of Bettin and Drappeau, and for keeping us informed about his joint work with Grepstad and Neumüller.
Publisher Copyright:
© 2022, The Author(s).
PY - 2022/10
Y1 - 2022/10
N2 - In this paper, we study the relation between the function J41,0, which arises from a quantum invariant of the figure-eight knot, and Sudler’s trigonometric product. We find J41,0 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 J41,0 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.
AB - In this paper, we study the relation between the function J41,0, which arises from a quantum invariant of the figure-eight knot, and Sudler’s trigonometric product. We find J41,0 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 J41,0 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.
KW - Colored Jones polynomial
KW - Continued fraction
KW - Kashaev invariant
KW - Quadratic irrational
KW - Quantum modular form
KW - Sudler product
UR - http://www.scopus.com/inward/record.url?scp=85134353346&partnerID=8YFLogxK
U2 - 10.1007/s00209-022-03086-5
DO - 10.1007/s00209-022-03086-5
M3 - Article
AN - SCOPUS:85134353346
SN - 0025-5874
VL - 302
SP - 759
EP - 782
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 2
ER -