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

2019/05/06 by Sandro Bettin, Bettin, Sandro, Sary Drappeau +1 · 1 citation
Mathematics · #11B65 (primary) #11F03 #57M27 #60F05 (secondary) #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Number Theory (math.NT) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.1905.02045

openalex publication_date 2019/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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\≠ 7₂ with at\nmost seven crossings.\n For K=4₁, 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

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