Modularity and value distribution of quantum invariants of hyperbolic\n knots

Abstract

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