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An upper bound conjecture for the Yokota invariant

Giulio Belletti

TL;DR

This work addresses the asymptotic growth of the Yokota invariant for polyhedral graphs and its connection to hyperbolic volumes. It introduces an upper bound conjecture for $Y_r(\Gamma,col)$ and proves it in a broad class of planar graphs by leveraging Barrett's Fourier transform to relate dual graphs' invariants. The results yield new instances of the Turaev-Viro Volume Conjecture for infinite families of hyperbolic 3-manifolds, derived from octahedral decompositions and rectified graphs. Overall, the paper strengthens the bridge between quantum invariants and hyperbolic geometry, providing practical tools to confirm volume conjectures across a wider set of manifolds.

Abstract

We conjecture an upper bound on the growth of the Yokota invariant of polyhedral graphs, extending a previous result on the growth of the $6j$-symbol. Using Barrett's Fourier transform we are able to prove this conjecture in a large family of examples. As a consequence of this result, we prove the Turaev-Viro Volume Conjecture for a new infinite family of hyperbolic manifolds.

An upper bound conjecture for the Yokota invariant

TL;DR

This work addresses the asymptotic growth of the Yokota invariant for polyhedral graphs and its connection to hyperbolic volumes. It introduces an upper bound conjecture for and proves it in a broad class of planar graphs by leveraging Barrett's Fourier transform to relate dual graphs' invariants. The results yield new instances of the Turaev-Viro Volume Conjecture for infinite families of hyperbolic 3-manifolds, derived from octahedral decompositions and rectified graphs. Overall, the paper strengthens the bridge between quantum invariants and hyperbolic geometry, providing practical tools to confirm volume conjectures across a wider set of manifolds.

Abstract

We conjecture an upper bound on the growth of the Yokota invariant of polyhedral graphs, extending a previous result on the growth of the -symbol. Using Barrett's Fourier transform we are able to prove this conjecture in a large family of examples. As a consequence of this result, we prove the Turaev-Viro Volume Conjecture for a new infinite family of hyperbolic manifolds.

Paper Structure

This paper contains 10 sections, 15 theorems, 48 equations, 26 figures.

Table of Contents

  1. Introduction
  2. The Kauffman bracket and the Yokota invariant
  3. The Kauffman bracket
  4. The definition of the Yokota invariant from the Kauffman bracket
  5. Volume Conjecture for polyhedra
  6. The Volume Conjecture for polyhedra
  7. The upper bound conjecture
  8. The Fourier Transform
  9. The Turaev-Viro Volume Conjecture
  10. Appendix: Numerical evidence for Conjecture \ref{['conjmk']}

Key Result

Theorem 1.1

For any polyhedral graph $\Gamma$, where $P$ varies among all proper generalized hyperbolic polyhedra with $1$-skeleton $\Gamma$ and $\overline{\Gamma}$ is the rectification of $\Gamma$.

Figures (26)

  • Figure 1: Truncating a vertex
  • Figure 2: Triangulating a face
  • Figure 3: An $r$-admissible coloring for a tetrahedron
  • Figure 4: A Theta graph
  • Figure 5: Desingularization of a vertex of valence $6$
  • ...and 21 more figures

Theorems & Definitions (54)

  • Conjecture 1: The Turaev-Viro Volume Conjecture
  • Conjecture 2: The Upper Bound Conjecture
  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 44 more