Cubic graphs induced by bridge trisections

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.