Cubic graphs induced by bridge trisections

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 $1$-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. Tools used to prove the main theorem include two new operations on bridge trisections, crosscap summation and tubing, which may be of independent interest.