Knots, $q$-series, and modular formsMPIM

by Matthias Storzer (UC Cork)

Wednesday May 13, 2026, 2:30 PM → 3:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

o study knots, invariants like the colored Jones polynomials (CJP) are used. For alternating knots, it is known that the coefficients of the CJP stabilize and thus, they converge to a well-defined $q$-series, the tail of the CJP. For several but not all knots with up to 10 crossings, the tail of the CJP can be written as a product of (partial) theta functions and thus has modular properties. In this talk, we present a general formula for a class of knots, and argue that the tail of the CJP for other knots does not have any modular properties. We will also discuss a potential connection to exceptional hyperbolic surgery on knots.

This talk is based on joint work with Robert Osburn (Cork).