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Lambda lengths in The figure eight knot complement

Joshua A. Howie, Dionne Ibarra, Daniel V. Mathews, Lecheng Su

Abstract

In the complete hyperbolic structure on the complement of the figure eight knot, we determine the set of lambda lengths from the maximal cusp to itself. Using the correspondence between spinors and spin-decorated horospheres, we show that these lambda lengths are precisely the Eisenstein integers, up to multiplication by a unit. We also show that the inter-cusp distances from the maximal cusp to itself are precisely the norms of Eisenstein integers.

Lambda lengths in The figure eight knot complement

Abstract

In the complete hyperbolic structure on the complement of the figure eight knot, we determine the set of lambda lengths from the maximal cusp to itself. Using the correspondence between spinors and spin-decorated horospheres, we show that these lambda lengths are precisely the Eisenstein integers, up to multiplication by a unit. We also show that the inter-cusp distances from the maximal cusp to itself are precisely the norms of Eisenstein integers.

Paper Structure

This paper contains 5 sections, 5 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Eisenstein Integers
  3. Figure Eight Knot Complement
  4. Spinors
  5. Actions of ${\Gamma}$ on $\hat{{\mathbb{C}}}$ and ${\mathbb{C}}^2$

Key Result

Theorem 1.1

If $\smqty(\xi\\ \eta)\in{\mathbb{C}}^2$ is a spinor arising in the canonical hyperbolic structure on $S^3\setminus K$, then $\xi$ and $\eta$ are relatively prime Eisenstein integers. Moreover, given a pair of relatively prime Eisenstein integers $(\xi,\eta)$, there exists a unit Eisenstein integer

Figures (2)

  • Figure 1: The fundamental domain of the figure eight knot complement consists of two regular ideal tetrahedra.
  • Figure 2:

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 5.1
  • proof
  • Example 5.2