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Title: Bridge trisections of knotted surfaces in 4-manifolds
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 generalized bridge trisection, extends the authors’ definition of bridge trisections for surfaces in S 4 . Using this construction, we give diagrammatic representations called 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 ℂ ℙ 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.  more » « less
Award ID(s):
1664578
PAR ID:
10099438
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Proceedings of the National Academy of Sciences
Volume:
115
Issue:
43
ISSN:
0027-8424
Page Range / eLocation ID:
10880 to 10886
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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