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The twisted 1-loop invariant and the Jacobian of Ptolemy coordinates

Seokbeom Yoon

Abstract

We present an alternative definition of the twisted 1-loop invariant in terms of the Jacobian of Ptolemy coordinates. As an application, we prove that the twisted 1-loop invariant is equal to the adjoint twisted Alexander polynomial for all hyperbolic once-punctured torus bundles. This implies that the 1-loop conjecture proposed by Dimofte and Garoufalidis holds for all hyperbolic once-punctured torus bundles.

The twisted 1-loop invariant and the Jacobian of Ptolemy coordinates

Abstract

We present an alternative definition of the twisted 1-loop invariant in terms of the Jacobian of Ptolemy coordinates. As an application, we prove that the twisted 1-loop invariant is equal to the adjoint twisted Alexander polynomial for all hyperbolic once-punctured torus bundles. This implies that the 1-loop conjecture proposed by Dimofte and Garoufalidis holds for all hyperbolic once-punctured torus bundles.

Paper Structure

This paper contains 15 sections, 14 theorems, 89 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The twisted 1-loop invariant
  3. The Jacobian of $\sigma$-Ptolemy coordinates
  4. A single tetrahedron
  5. The Jacobian of the $\sigma$-Ptolemy equation
  6. $\sigma$-Ptolemy assignments for a 3-manifold
  7. An alternative definition of the twisted 1-loop invariant
  8. Once-punctured torus bundles
  9. The canonical triangulation of $M_\varphi$
  10. The twisted 1-loop invariant of $\mathcal{T}_\varphi$
  11. $\sigma$-Ptolemy assignments of $\mathcal{T}_\varphi$
  12. Adjoint twisted Alexander polynomials
  13. Proofs
  14. Proof of Theorem \ref{['thm.main2']}
  15. Proof of Corollary \ref{['thm.main3']}

Key Result

Theorem 1.3

The twisted 1-loop invariant $\tau^\mathrm{CS}(\mathcal{T}_\varphi,t)$ equals to the adjoint twisted Alexander polynomial $\tau(M_\varphi,t)$ of $M_\varphi$ for all hyperbolic once-punctured torus bundles $M_\varphi$.

Figures (6)

  • Figure 1: An ideal tetrahedron with shape parameters $z^\square$.
  • Figure 2: An edge notation for a truncated tetrahedron.
  • Figure 3: Four copies of $F$ and the way of attaching $\Delta_i$.
  • Figure 4: The induced triangulation of the peripheral torus for $\varphi = R^2L^3$. The triangles with index $1 \leq i \leq 5$ are from $\Delta_i$, and the dots indicate where the parameter $z_i$ is assigned.
  • Figure 5: The compact surface $\overline{F}$.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Conjecture 1.1: DG1
  • Conjecture 1.2: GY21
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1: GY21
  • Remark 2.2
  • Definition 3.1: Yoon19
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 27 more