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Octahedral coordinates from the Wirtinger presentation

Calvin McPhail-Snyder

Abstract

Let $ρ$ be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into $\operatorname{SL}_2(\mathbb{C})$ expressed in terms of the Wirtinger generators of a diagram $D$. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. $ρ$ induces a hyperbolic structure on the complement of $D$, and in this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition induced by this structure. Our formula gives a new, explicit criterion for whether $ρ$ occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function.

Octahedral coordinates from the Wirtinger presentation

Abstract

Let be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into expressed in terms of the Wirtinger generators of a diagram . This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. induces a hyperbolic structure on the complement of , and in this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition induced by this structure. Our formula gives a new, explicit criterion for whether occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function.
Paper Structure (5 sections, 12 theorems, 67 equations, 5 figures)

This paper contains 5 sections, 12 theorems, 67 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Octahedral colorings
  3. The holonomy of an octahedral coloring
  4. Gauge transformations
  5. Shape parameters and the potential function

Key Result

proposition 2

Let $D$ be a diagram of a tangle $T$. The choice of $D$ gives a presentation for $\pi(T)$ with one generator $w_i$ for each arc and one relation at each crossing, where the indices are as in fig:crossing-types. (At a positive crossing $1$ and $1'$ are the same arc.) We call this the Wirtinger presentation of $\pi(T)$ and denote it by $\pi(D)$. If $i$ is a segment of $D$ we write $w_{i}$ for the a

Figures (5)

  • Figure 1: Positive (left) and negative (right) crossings with our standard labeling.
  • Figure 2: Rules for decorated $\operatorname{SL}_2(\mathbb{C})$ colorings.
  • Figure 3: This tangle has regions $0, 1, 2, 3$ and segments $1, 2, 3$. The over path for region $2 = 3^{\uparrow}$ is $s^+ = x_1^+ x_2^+$, so the Wirtinger generator $w_3 \in \pi(D)$ of segment $3$ is mapped to $\mathcal{F}(w_3) = s^+ x_3^+ (x_3^-)^{-1} (s^+)^{-1} = x_1^+ x_2^+ x_3^+ \left(x_3^- \right)^{-1} \left(x_2^+\right)^{-1} \left(x_1^+\right)^{-1} \in \Pi\mathopen{}\left(D\right).$
  • Figure 4: Graphical description of gauge transformations. The thick strands represent bundles of incoming and outgoing segments (possibly empty) with arbitrary colorings.
  • Figure 5: One can divide the octahedron at a crossing into four tetrahedra, one for each region touching the crossing. The shape parameters are then given as ratios of the $b$ and $m$-coordinates, as in \ref{['eq:b-shapes']}.

Theorems & Definitions (42)

  • definition 1
  • proposition 2
  • definition 3
  • definition 4
  • definition 5
  • proposition 6
  • definition 7
  • definition 8
  • lemma 9
  • proof
  • ...and 32 more