Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Knot complements decomposing into prisms

Jason DeBlois, Arshia Gharagozlou, Neil R Hoffman

Abstract

We describe four hyperbolic knot complements in $\mathbb{S}^3$, each of which covers a prism orbifold: the quotient of $\mathbb{H}^3$ by the action of a discrete group generated by reflections in the faces of a polyhedron that has the combinatorial type of a triangular prism. The prism orbifolds are rigid-cusped and contain compact, totally geodesic hyperbolic triangle sub-orbifolds; as a result, the knot complements covering them have hidden symmetries and contain closed, embedded, totally geodesic surfaces.

Knot complements decomposing into prisms

Abstract

We describe four hyperbolic knot complements in , each of which covers a prism orbifold: the quotient of by the action of a discrete group generated by reflections in the faces of a polyhedron that has the combinatorial type of a triangular prism. The prism orbifolds are rigid-cusped and contain compact, totally geodesic hyperbolic triangle sub-orbifolds; as a result, the knot complements covering them have hidden symmetries and contain closed, embedded, totally geodesic surfaces.

Paper Structure

This paper contains 25 sections, 54 theorems, 59 equations, 19 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Hidden symmetries of knot complements
  3. The Menasco-Reid conjecture
  4. Computational Tools
  5. Background: the classification and geometry of prism orbifolds
  6. The classification of prism orbifolds
  7. Explicit embeddings and maximal cusp neighborhoods
  8. Totally geodesic surfaces
  9. Geometric particulars of O2 and O3
  10. Initial observations
  11. Volume computation
  12. Maximal horoballs and commensurability
  13. Proof of the main theorem
  14. Degree-24 covers and their properties.
  15. Spines, and a first crack at first homology
  16. ...and 10 more sections

Key Result

Theorem 1.1

There exist hyperbolic knot complements in $\mathbb{S}^3$ that cover prism orbifolds.

Figures (19)

  • Figure 1: A hyperbolic prism (left), and its double across a face, with group generators for $\widetilde{\Pi}(\mathbf{e})$ from the presentation (\ref{['able']}) shown.
  • Figure 2: The compact triangle from \ref{['corandreev']}.
  • Figure 3: The compact prisms $P^+$ of \ref{['corandreev']}'s proof, for $O^{333}_2$ and $O^{333}_3$.
  • Figure 4: $P^{333}_1$ and $P^{333}_2$ (left), and $P^{333}_3$ (right), viewed from above, with the circles bounding the hemispheres that contain their non-vertical faces.
  • Figure 5: Some translates of $\Delta$ and points of tangency.
  • ...and 14 more figures

Theorems & Definitions (123)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 2.1
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 113 more