Cubic graphs induced by bridge trisections

TL;DR

This paper establishes a correspondence between Tait-colored cubic graphs and the 1-skeletons of bridge trisections of unknotted surfaces in 4-spheres, providing a new way to understand surface embeddings via graph theory.

Contribution

It proves that every Tait-colored cubic graph can be realized as the 1-skeleton of a bridge trisection of an unknotted surface, linking graph colorings to surface topology.

Findings

Every Tait-colored cubic graph corresponds to an unknotted surface's bridge trisection.

Such embeddings exist for nonorientable surfaces with any normal Euler number.

Tri-plane diagrams of knotted surfaces can be simplified to unknotted ones through crossing changes and Reidemeister moves.

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 $\Gamma$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $\Gamma$ 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.