Date(s) : 07/07/2020
11 h 00 min - 12 h 00 min
WEBINAIRE (lien: https://webconf.lal.cloud.math.cnrs.fr/b/pie-hmu-en9)
Résumé : Quantum invariants of hyperbolic knots have rich arithmetic structure. In this talk we consider an extension to the rationals of the so-called Kashaev invariant of the figure eight knot, a quantum modular form in the sense of Zagier. We find its asymptotics along the sequence of best rational approximations to a given quadratic irrational, and show that this asymptotics deviates from that of well approximable irrationals recently established by Bettin and Drappeau. Our proof exploits the close connection between the Kashaev invariant and Sudler’s trigonometric product; in particular, we use a recent result of Grepstad, Neumuller and Zafeiropoulos on the convergence of certain normalized and shifted versions of Sudler’s products at quadratic irrationals.
Joint work with Christoph Aistleitner.
Bence Borda (TU Graz)