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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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Proceedings of the National Academy of Sciences
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Proceedings of the National Academy of Sciences
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Article No. e2025436118
Medium: X
Sponsoring Org:
National Science Foundation
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