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Knots, Perturbative Series and Quantum Modularity

Stavros Garoufalidis, Don Zagier

Abstract

We introduce an invariant of a hyperbolic knot which is a map $α\mapsto \boldsymbolΦ_α(h)$ 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 $(σ_0,σ_1)$ entry, where $σ_0$ is the trivial and $σ_1$ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity ${\rm e}^{2π{\rm i} α}$ 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 $\boldsymbolΦ$ are fundamental solutions of a linear $q$-difference equation; (d) the matrix defines an ${\rm SL}_2(\mathbb{Z})$-cocycle $W_γ$ 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 $\boldsymbolΦ(h)$ to actual functions. The two invariants $\boldsymbolΦ$ and $W_γ$ 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.

Knots, Perturbative Series and Quantum Modularity

Abstract

We introduce an invariant of a hyperbolic knot which is a map from to matrices with entries in and with rows and columns indexed by the boundary parabolic representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their entry, where is the trivial and the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity 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 are fundamental solutions of a linear -difference equation; (d) the matrix defines an -cocycle in matrix-valued functions on that conjecturally extends to a smooth function on and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series to actual functions. The two invariants and are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the , and pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent -series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.

Paper Structure

This paper contains 45 sections, 10 theorems, 220 equations, 5 figures.

Table of Contents

  1. The original quantum modularity conjecture
  2. A collection of formal power series
  3. The indexing set $\boldsymbol{\mathcal{P}_K}$
  4. Four constructions of the power series $\boldsymbol{\Phi^{(K,\sigma)}_\alpha(h)}$
  5. Interrelations among the power series $\boldsymbol{\Phi_\alpha^{(\sigma)}(h)}$
  6. The generalized quantum modularity conjecture
  7. Lifting the QMC from constant terms to power series
  8. Quadratic relations
  9. Asymptotics of the coefficients
  10. Refining the quantum modularity conjecture
  11. Improving the quantum modularity conjecture: optimal truncation
  12. New elements of the Habiro ring
  13. Smoothed optimal truncation
  14. Strengthening the generalized quantum modularity conjecture
  15. The refined quantum modularity conjecture
  16. ...and 30 more sections

Key Result

Lemma 3.1

For all $\gamma, \gamma'\in \mathrm{PSL}_2(\mathbb Z)$ and $x\in\mathbb Q\smallsetminus\{{\gamma'}^{-1}(\infty),{(\gamma\gamma')}^{-1}(\infty)\}$, we have

Figures (5)

  • Figure 1: The functions $\log(J(x))$ and $\log(J(x)/J(-1/x))$ for the $4_1$ knot.
  • Figure 2: Plots of the six nontrivial entries (rescaled) of the matrix $W^{(4_1)}_S(x)$.
  • Figure 3: The Witten cylinder.
  • Figure 4: Plots of the six nontrivial entries of the matrix $W_S(x)$ for the $4_1$ knot.
  • Figure 5: A tetrahedron with shape parameters.

Theorems & Definitions (12)

  • Lemma 3.1
  • Conjecture 5.1
  • Proposition 5.2: assuming RQMC
  • Corollary 5.3
  • Conjecture 5.4
  • Proposition 8.1
  • Proposition 8.2
  • Theorem 9.1
  • Lemma 9.2
  • Theorem 9.3
  • ...and 2 more