Modularity and value distribution of quantum invariants of hyperbolic knots

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 $$K\ne 7_2$$ 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.