Bridge trisections of knotted surfaces in 4--manifolds

TL;DR

This paper introduces a new way to decompose and represent knotted surfaces in 4-manifolds using generalized bridge trisections and shadow diagrams, enabling classification and analysis of complex surface embeddings.

Contribution

The authors extend bridge trisection theory to all smoothly embedded surfaces in 4-manifolds, providing diagrammatic tools and classification results, including examples like complex curves in projective space.

Findings

Every surface can be isotoped to a bridge position with respect to a trisection.

Shadow diagrams offer a new diagrammatic representation of knotted surfaces.

Existence of exotic 4-manifolds with specific trisection properties.

Abstract

We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4--manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside $\mathbb{CP}^2$. Using these examples, we prove that there exist exotic 4--manifolds with $(g,0)$--trisections for certain values of $g$. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams.