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Title: Four-dimensional aspects of tight contact 3-manifolds

We conjecture a four-dimensional characterization of tightness: A contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces inY×[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: If a fibered link L induces a tight contact structure on Y, then its fiber surface maximizes the Euler characteristic among all surfaces inY×[0,1]with boundary L. We provide evidence for both conjectures by proving them for contact structures with nonvanishing Ozsváth–Szabó contact invariant.

 
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Award ID(s):
1709016
NSF-PAR ID:
10231071
Author(s) / Creator(s):
;
Publisher / Repository:
Proceedings of the National Academy of Sciences
Date Published:
Journal Name:
Proceedings of the National Academy of Sciences
Volume:
118
Issue:
22
ISSN:
0027-8424
Page Range / eLocation ID:
Article No. e2025436118
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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