Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 055, 87 pages      arXiv:2111.06645      https://doi.org/10.3842/SIGMA.2024.055

Knots, Perturbative Series and Quantum Modularity

Stavros Garoufalidis a and Don Zagier bc
a) International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen, P.R. China
b) Max Planck Institute for Mathematics, Bonn, Germany
c) International Centre for Theoretical Physics, Trieste, Italy

Received April 25, 2023, in final form May 26, 2024; Published online June 24, 2024

Abstract
We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto \boldsymbol{\Phi}_\alpha(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the boundary parabolic ${\rm SL}_2(\mathbb{C})$ representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their $(\sigma_0,\sigma_1)$ entry, where $\sigma_0$ is the trivial and $\sigma_1$ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity ${\rm e}^{2\pi{\rm i} \alpha}$ as an element of the Habiro ring, and the remaining entries belong to generalized Habiro rings of number fields; (b) the first column is given by the perturbative power series of Dimofte-Garoufalidis; (c) the columns of $\boldsymbol{\Phi}$ are fundamental solutions of a linear $q$-difference equation; (d) the matrix defines an ${\rm SL}_2(\mathbb{Z})$-cocycle $W_\gamma$ in matrix-valued functions on $\mathbb{Q}$ that conjecturally extends to a smooth function on $\mathbb{R}$ and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series $\boldsymbol{\Phi}(h)$ to actual functions. The two invariants $\boldsymbol{\Phi}$ and $W_\gamma$ are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the $4_1$, $5_2$ and $(-2,3,7)$ pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent $q$-series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.

Key words: quantum topology; knots; 3-manifolds; Jones polynomial; Kashaev invariant; volume conjecture; Chern-Simons theory; asymptotics; quantum modularity conjecture; quantum modular forms; hyperbolic 3-manifolds; dilogarithm; cocycles; $\mathrm{SL}_2(\mathbb{Z})$; denominators; Habiro-like functions; functions near $\mathbb{Q}$; Neumann-Zagier matrices; Nahm sums; $q$-holonomic modules.

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