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Modularity and value distribution of quantum invariants of hyperbolic knots

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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.

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Notes

  1. Notice that the series is in fact a finite sum at roots of unities.

  2. In what follows, a matrix in \(\text {SL}(2, {\mathbb {R}})\) acts on \({\mathbb {C}}\) by homography.

  3. This continuity property at rationals follows from the modularity conjecture but only when approaching a rational \(h/k=[0;b_1,\dots ,b_r]\) with fractions essentially of the form \([0;b_1,\dots ,b_r,N]\) with \(N\rightarrow \infty \); and not, for example, with \([0;b_1,\dots ,b_r,N_1,N_2]\) with both \(N_1,N_2\in {\mathbb {N}}\) going to infinity.

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Acknowledgements

This paper was partially written during a visit of of S. Bettin at the Aix-Marseille University, a visit of S. Drappeau at the University of Genova, and a visit of both authors at ICTP Trieste. The authors thank these Institution for the hospitality and Aix-Marseille University, INdAM and ICTP for the financial support for these visits. The authors wish to thank Don Zagier for useful discussions, Brian Conrey for putting us in contact with him, and Hitoshi Murakami for many helpful comments on this work.

S. Bettin is member of the INdAM group GNAMPA and his work is partially supported by PRIN 2017 “Geometric, algebraic and analytic methods in arithmetic”.

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Authors and Affiliations

  1. DIMA-Dipartimento di Matematica, Via Dodecaneso, 35, 16146, Genoa, Italy

    S. Bettin

  2. Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France

    S. Drappeau

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  1. S. Bettin
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Correspondence to S. Bettin.

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Communicated by Kannan Soundararajan.

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Bettin, S., Drappeau, S. Modularity and value distribution of quantum invariants of hyperbolic knots. Math. Ann. 382, 1631–1679 (2022). https://doi.org/10.1007/s00208-021-02288-2

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