Bridge trisections of knotted surfaces in $S^4$

TL;DR

This paper introduces bridge trisections for knotted surfaces in four-spheres, providing a new way to decompose, classify, and analyze their complexity and fundamental groups, extending classical three-dimensional concepts to four dimensions.

Contribution

It establishes the existence and uniqueness (up to stabilization) of bridge trisections for knotted surfaces in $S^4$, introduces a diagrammatic calculus, and defines a new complexity measure called bridge number.

Findings

Every knotted surface admits a bridge trisection.

Two bridge trisections of the same surface are related by stabilizations/destabilizations.

Existence of knotted surfaces with arbitrarily large bridge number.

Abstract

We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the three-sphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and describe a set of moves that suffice to pass between two diagrams for the same surface. Using these decompositions, we define a new complexity measure: the bridge number of a knotted surface. In addition, we classify bridge trisections with low complexity, we relate bridge trisections to the fundamental groups of knotted surface complements, and we prove that there exist knotted surfaces with arbitrarily large bridge number.