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On Geometric triangulations of double twist knots

Dionne Ibarra, Daniel V. Mathews, Jessica S. Purcell

TL;DR

This work provides two explicit, geometrically realized triangulations for the complements of double twist knots $K(p,q)$ arising from Dehn fillings of the Borromean rings: a canonical (Sakuma–Weeks) triangulation and a conjecturally minimal, geometric triangulation. It establishes the geometricity of the minimal construction via angle-structure methods and leverages the Neumann–Zagier/Ptolemy framework to derive explicit PSL–A-polynomial equations, enabling direct computation of deformation varieties and A-polynomials for these knots. The paper also demonstrates a concrete Dehn-filling pipeline that starts from well-understood Borromean rings triangulations and yields tractable, edge-friendly representations of the knot complements, with explicit results for small cases (e.g., $K(2,2)$ and $K(3,3)$) and a general eight-equation system for $p,q rac{ o}{ o}3$. Overall, the approach links explicit triangulations, canonical/cusped Dehn fillings, and A-polynomial computation, contributing practical tools for studying deformation spaces and the AJ conjecture in this knot family.

Abstract

In this paper we construct two different explicit triangulations of the family of double twist knots $K(p,q)$ using methods of triangulating Dehn fillings, with layered solid tori and their double covers. One construction yields the canonical triangulation, and one yields a triangulation that we conjecture is minimal. We prove that both are geometric, meaning they are built of positively oriented convex hyperbolic tetrahedra. We use the conjecturally minimal triangulation to present eight equations cutting out the A-polynomial of these knots.

On Geometric triangulations of double twist knots

TL;DR

This work provides two explicit, geometrically realized triangulations for the complements of double twist knots arising from Dehn fillings of the Borromean rings: a canonical (Sakuma–Weeks) triangulation and a conjecturally minimal, geometric triangulation. It establishes the geometricity of the minimal construction via angle-structure methods and leverages the Neumann–Zagier/Ptolemy framework to derive explicit PSL–A-polynomial equations, enabling direct computation of deformation varieties and A-polynomials for these knots. The paper also demonstrates a concrete Dehn-filling pipeline that starts from well-understood Borromean rings triangulations and yields tractable, edge-friendly representations of the knot complements, with explicit results for small cases (e.g., and ) and a general eight-equation system for . Overall, the approach links explicit triangulations, canonical/cusped Dehn fillings, and A-polynomial computation, contributing practical tools for studying deformation spaces and the AJ conjecture in this knot family.

Abstract

In this paper we construct two different explicit triangulations of the family of double twist knots using methods of triangulating Dehn fillings, with layered solid tori and their double covers. One construction yields the canonical triangulation, and one yields a triangulation that we conjecture is minimal. We prove that both are geometric, meaning they are built of positively oriented convex hyperbolic tetrahedra. We use the conjecturally minimal triangulation to present eight equations cutting out the A-polynomial of these knots.

Paper Structure

This paper contains 15 sections, 12 theorems, 42 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Relation to canonical triangulation
  3. Application: deformation variety equations
  4. Acknowledgements
  5. Triangulation of the Borromean rings complement
  6. Borromean rings: 8 tetrahedra
  7. Borromean rings: 10 tetrahedra
  8. Dehn filling
  9. Layered solid tori.
  10. Construction of (conjecturally) minimal triangulation
  11. Construction of canonical triangulation
  12. Geometric triangulations
  13. Triangulations and Neumann--Zagier equations
  14. Background on Ptolemy equations
  15. A-polynomials of double twist knots

Key Result

Lemma 3.1

Let $m = -1/p = -\mu+p \lambda$ be a slope on a torus with generators $\mu, \lambda$ and $p \notin \{0, \pm 1\}$. Construct a layered solid torus $X$ by beginning in the Farey triangle with vertices $(1/0, 0/1, -1/1)$ and step to the triangle with slope $-1/2p$. Then, the double cover of $X$, denote

Figures (12)

  • Figure 1.1: Left: $K(2,2)$. Right: The double twist knot $K(p,q)$ is obtained by Dehn filling the Borromean rings.
  • Figure 2.1: Constructing the cusp neighbourhood.
  • Figure 2.2: Cusp neighbourhoods of the Borromean rings, 8-tetrahedron triangulation.
  • Figure 2.3: Gluing the crossing circle cusps of the Borromean rings.
  • Figure 2.4: Replacing four tetrahedra per crossing circle. Here, the boldface numbers index the ideal tetrahedra.
  • ...and 7 more figures

Theorems & Definitions (26)

  • Conjecture 1.1
  • Lemma 3.1: LST double cover
  • proof
  • Theorem 4.1
  • Definition 4.2
  • Lemma 4.19
  • proof
  • Lemma 4.20
  • proof
  • Lemma 4.21
  • ...and 16 more