Congruences for Taylor expansions of quantum modular forms

TL;DR

This paper explores the congruence properties of quantum modular forms, extending known results on Fishburn numbers to a broader class of combinatorial sequences and their connections to knot invariants.

Contribution

It develops a general theory of congruences for quantum modular forms and applies it to new combinatorial sequences related to knot invariants and half-derivatives of Andrews-Gordon functions.

Findings

Established conditions for linear congruences modulo at least 50% of primes.

Connected combinatorial sequences to quantum modular forms and knot invariants.

Extended congruence results to sequences associated with half-derivatives of Andrews-Gordon functions.

Abstract

Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of these congruences to arbitrary powers of the primes involved. Here, we take a different perspective and explain the general theory of such congruences in the context of an important class of quantum modular forms. As one example, we obtain an infinite series of combinatorial sequences connected to the "half-derivatives" of the Andrews-Gordon functions and with Kashaev's invariant on $(2m+1,2)$ torus knots, and we prove conditions under which the sequences satisfy linear congruences modulo at least $50\%$ of primes of primes.