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Cubic graphs induced by bridge trisections

Jeffrey Meier, Abigail Thompson, Alexander Zupan

Abstract

Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the $\textit{1-skeleton}$ of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph $Γ$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $Γ$ such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves.

Cubic graphs induced by bridge trisections

Abstract

Every embedded surface in the 4-sphere admits a bridge trisection, a decomposition of into three simple pieces. In this case, the surface is determined by an embedded 1-complex, called the of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph that inherits a natural Tait coloring, a 3-coloring of the edge set of such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves.

Paper Structure

This paper contains 13 sections, 9 theorems, 1 equation, 17 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bridge trisections
  4. Unknotted surfaces
  5. Cubic graphs and surfaces
  6. Operations on bridge trisections and cubic graphs
  7. Connected summation
  8. Elementary perturbation
  9. Crosscap summation
  10. Tubing
  11. Compression of cubic graphs
  12. Proof of the main theorem
  13. The examples $\Sigma_O$ and $\Sigma_N$

Key Result

Theorem 1.1

If $\Gamma$ is a cubic graph with a Tait coloring $\mathcal{C}$, then there exists a bridge trisection $\mathcal{T}$ of an unknotted surface $\mathcal{U} \subset S^4$ such that the 1-skeleton $\mathcal{T}$ is graph isomorphic to $\Gamma$, with the coloring $\mathcal{C}$ induced by $\mathcal{T}$. Mor

Figures (17)

  • Figure 1: Two examples of Tait-colored cubic graphs
  • Figure 2: Examples of genus zero shadow diagrams.
  • Figure 3: An example of an induced surface.
  • Figure 4: Distinct Tait colorings of $\Gamma$ inducing different surfaces
  • Figure 5: Embedding of $K_{3,3}$ in $\mathbb{RP}^2$, where antipodal points of the disk are identified. No Tait coloring of $K_{3,3}$ induces $\mathbb{RP}^2$.
  • ...and 12 more figures

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.4
  • proof
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • ...and 12 more