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Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms I: the figure eight knot complement

Tushar Pandey, Ka Ho Wong

Abstract

We propose a generalization of the Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms, by relating the asymptotics of the invariants with certain hyperbolic cone structure on the mapping torus determined by the choice of the invariant puncture weights. We prove the conjecture for the once-punctured torus bundle with the diffeomorphism given by the word $LR$.

Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms I: the figure eight knot complement

Abstract

We propose a generalization of the Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms, by relating the asymptotics of the invariants with certain hyperbolic cone structure on the mapping torus determined by the choice of the invariant puncture weights. We prove the conjecture for the once-punctured torus bundle with the diffeomorphism given by the word .
Paper Structure (17 sections, 30 theorems, 138 equations, 1 figure)

This paper contains 17 sections, 30 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Discrete quantum dilogarithm function
  4. Continuous quantum dilogarithm function
  5. Classical dilogarithm function and Lobachesky function
  6. Explicit Formula of the quantum invariants of surface diffeomorphisms
  7. Computation of $(\tilde{l}, \tilde{m}, \tilde{n})$
  8. Explicit formula of $\mathrm{Trace}\text{ } \Lambda_{\varphi, r, p_v}^q$
  9. Geometry of the potential functions
  10. Geometric triangulation for the hyperbolic cone metric on $\Sigma_\eta$
  11. Critical point equations of the potential functions
  12. Critical values of the potential functions
  13. Hessians of the potential functions
  14. Neumann-Zagier potential function
  15. Asymptotics of the leading Fourier coefficients
  16. ...and 2 more sections

Key Result

Theorem 1.3

The asymptotic expansion formula of $|\mathrm{Trace} \Lambda_{\varphi, r, p_v}^q |$ is given by where $C(n)$ is a sequence of real numbers satisfying $A< C(n) < B$ for some constants $A,B>0$.

Figures (1)

  • Figure 1: Triangulation of the boundary torus. The part drawn by solid lines and the one drawn by dotted lines are copies of the fundamental domain. The red curve $\mu_1$ and the blue curve $\mu_2$ are homotopic curves representing the meridian $\mu_v$. The orange curve represents a curve $l$ that intersects the meridian once. The purple curve represents $2l$.

Theorems & Definitions (50)

  • Conjecture 1.1: Conjecture 6, BWYI
  • Conjecture 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1: Lemma 22, BWYI
  • Proposition 2.2: Lemma 7, BWYII
  • Proposition 2.3: Proposition 18, BWYII
  • Proposition 2.4: Proposition 4, BWYII
  • Proposition 2.5: Corollary 5, BWYII
  • Proposition 2.6: Proposition 3, BWYII
  • ...and 40 more