# Modularity and value distribution of quantum invariants of hyperbolic knots

## TL;DR

This paper establishes a modularity relation for quantum invariants of hyperbolic knots, linking Zagier's conjecture to arithmeticity, and proves it for certain knots, also deriving a law of large numbers for colored Jones polynomial values.

## Contribution

It introduces an exact modularity relation for the $q$-Pochhammer symbol and connects Zagier's modularity conjecture to the arithmeticity conjecture for hyperbolic knots, proving it for specific cases.

## Key findings

- Zagier's conjecture holds for hyperbolic knots with up to seven crossings, except 7_2.
- A reciprocity formula for the figure-eight knot (4_1) is proved.
- A law of large numbers for colored Jones polynomials at roots of unity is established.

## Abstract

We obtain an exact modularity relation for the $q$-Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture for a knot $K$ essentially reduces to the arithmeticity conjecture for $K$. In particular, we show that Zagier's conjecture holds for hyperbolic knots $K\neq 7_2$ with at most seven crossings. For $K=4_1$, we also prove a complementary reciprocity formula which allows us to prove a law of large numbers for the values of the colored Jones polynomials at roots of unity. We conjecture a similar formula holds for all knots and we show that this is the case if one assumes a suitable version of Zagier's conjecture.