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The (twisted/$L^2$)-Alexander polynomial of ideally triangulated 3-manifolds

Stavros Garoufalidis, Seokbeom Yoon

Abstract

We establish a connection between the Alexander polynomial of a knot and its twisted and $L^2$-versions with the triangulations that appear in 3-dimensional hyperbolic geometry. Specifically, we introduce twisted Neumann--Zagier matrices of ordered ideal triangulations and use them to provide formulas for the Alexander polynomial and its variants, the twisted Alexander polynomial and the $L^2$-Alexander torsion.

The (twisted/$L^2$)-Alexander polynomial of ideally triangulated 3-manifolds

Abstract

We establish a connection between the Alexander polynomial of a knot and its twisted and -versions with the triangulations that appear in 3-dimensional hyperbolic geometry. Specifically, we introduce twisted Neumann--Zagier matrices of ordered ideal triangulations and use them to provide formulas for the Alexander polynomial and its variants, the twisted Alexander polynomial and the -Alexander torsion.
Paper Structure (12 sections, 6 theorems, 60 equations, 5 figures)

This paper contains 12 sections, 6 theorems, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Twisted Neumann--Zagier matrices
  3. Alexander invariants from twisted NZ matrices
  4. Alexander polynomial
  5. Twisted Alexander polynomial
  6. $L^2$-Alexander torsion
  7. Fox calculus and twisted NZ matrices
  8. Fox calculus
  9. Curves in triangulations
  10. Determinants of NZ matrices and the (twisted) Alexander polynomial
  11. FK determinants of NZ matrices and the $L^2$-Alexander torsion
  12. Example

Key Result

Theorem 3.1

Fix $M$,$\mathcal{T}$ and $\alpha$ as in $(\dagger)$. Then either $\det \mathbf{B}_\alpha(t)=0$ or for some $n \geq 0$ and $m \geq 1$.

Figures (5)

  • Figure 1: A tetrahedron with shape parameters.
  • Figure 2: Three ways of smoothing $\mathcal{D}^{(1)}$.
  • Figure 3: Homotope $Z$-curves to peripheral curves.
  • Figure 4: Two generators joined by B-smoothing.
  • Figure 5: An ordered ideal triangulation of $4_1$.

Theorems & Definitions (15)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Remark 4.4
  • ...and 5 more