International Seminar on Automorphic Forms

Matthias Storzer (University College Dublin): Knots, q-series, and modular forms

To study knots, we use knot invariants like the colored Jones polynomials (CJP). For alternating knots, it is known that the CJP 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.Moreover, we argue that the tail of the CJP for some knots does not have any modular properties. We also briefly discuss potential topological interpretations of the (non-)modularity.This is joint work with Robert Osburn.

https://tu-darmstadt.zoom.us/j/68048280736

The password is the first Fourier coefficient of the modular j-function (as digits)