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On the asymptotics of the meromorphic 3D-index

Craig D. Hodgson, Andrew J. Kricker, Rafał M. Siejakowski

Abstract

In their recent work, Garoufalidis and Kashaev extended the 3D-index of an ideally triangulated 3-manifold with toroidal boundary to a well-defined topological invariant which takes the form of a meromorphic function of 2 complex variables per boundary component and which depends in addition on a quantisation parameter q. In this paper, we study asymptotics of this invariant as q approaches 1 and develop a conjectural asymptotic approximation in the form of a sum of contributions associated to conjugacy classes of certain boundary parabolic PSL(2,C) representations of the fundamental group. Furthermore, we study the coefficients appearing in these contributions, which include the hyperbolic volume, the '1-loop invariant' of Dimofte and Garoufalidis, as well as a new topological invariant of 3-manifolds with torus boundary, which we call the 'beta invariant'. The technical heart of our analysis is the expression of the state-integral of the Garoufalidis--Kashaev invariant as an integral over one connected component of the space of circle-valued angle structures introduced by Luo. Our stationary phase analysis of the asymptotics of this integral reveals many connections to the theory of angle structures and volume optimization. This investigation was motivated by extensive numerical experiments. In addition, we prove a variety of theorems about the quantities appearing in the analysis which support the overall conjectural picture.

On the asymptotics of the meromorphic 3D-index

Abstract

In their recent work, Garoufalidis and Kashaev extended the 3D-index of an ideally triangulated 3-manifold with toroidal boundary to a well-defined topological invariant which takes the form of a meromorphic function of 2 complex variables per boundary component and which depends in addition on a quantisation parameter q. In this paper, we study asymptotics of this invariant as q approaches 1 and develop a conjectural asymptotic approximation in the form of a sum of contributions associated to conjugacy classes of certain boundary parabolic PSL(2,C) representations of the fundamental group. Furthermore, we study the coefficients appearing in these contributions, which include the hyperbolic volume, the '1-loop invariant' of Dimofte and Garoufalidis, as well as a new topological invariant of 3-manifolds with torus boundary, which we call the 'beta invariant'. The technical heart of our analysis is the expression of the state-integral of the Garoufalidis--Kashaev invariant as an integral over one connected component of the space of circle-valued angle structures introduced by Luo. Our stationary phase analysis of the asymptotics of this integral reveals many connections to the theory of angle structures and volume optimization. This investigation was motivated by extensive numerical experiments. In addition, we prove a variety of theorems about the quantities appearing in the analysis which support the overall conjectural picture.

Paper Structure

This paper contains 55 sections, 28 theorems, 329 equations, 16 figures.

Table of Contents

  1. Introduction
  2. A sketch of the meromorphic 3D-index
  3. Circle-valued angle structures
  4. Asymptotic conjecture for the meromorphic 3D-index
  5. Contributions to the asymptotics from non-real representations
  6. The tau-invariant
  7. Contributions to the asymptotics from real representations and beta invariant
  8. Example: the SnapPea census manifold 'm011'
  9. Preliminaries on ideal triangulations and angle structures
  10. Notations and conventions for ideal triangulations
  11. Thurston's gluing equations
  12. Angle structures on ideal triangulations
  13. Tangential angle structures
  14. Differentiating the peripheral angle-holonomies along leading-trailing deformations.
  15. Variational approach to the gluing equations
  16. ...and 40 more sections

Key Result

Theorem 1.1

There exist maps satisfying the following conditions:

Figures (16)

  • Figure 1.1: The edge variables which appear in the factor in the integrand of the state-integral corresponding to a quad type. See Equation \ref{['eq:stateIntegralFactor']}.
  • Figure 1.2: Comparison between the values $\mathcal{I}_{M,(0,0)}(-1/\kappa)$ of the meromorphic 3D-index for the SnapPy census manifold $M=\mathtt{m011}$ with the approximation $\mathcal{A}(\kappa)$ of \ref{['predicted-approx-m011']}.
  • Figure 2.1: The cyclic ordering of normal quadrilaterals $\square\to\square'\to\square"\to\square$.
  • Figure 3.1: A labelled tetrahedron $\Delta$ equipped with dihedral angles.
  • Figure 6.1: The shaped Pachner $2\mathord{\rightarrow}3$ move on a labelled bipyramid. The triangulations $\mathcal{T}_2$ and $\mathcal{T}_3$ are identical outside of the bipyramid.
  • ...and 11 more figures

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 70 more