Bridge trisections of knotted surfaces in 4--manifolds

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.

17 pages, 5 figures. Comments welcome

References in corpus (3)

• Bridge trisections of knotted surfaces in $S^4$
• Classification of trisections and the Generalized Property R Conjecture
• Genus two trisections are standard

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• Doubly pointed trisection diagrams and surgery on 2-knots
• Kirby-Thompson distance for trisections of knotted surfaces
• Symplectic 4-manifolds admit Weinstein trisections
• Diagrams of *-Trisections
• Thin Position through the lens of trisections of 4-manifolds
• 3-manifolds lying in trisected 4-manifolds