Knots, $q$-series, and modular forms

To 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).