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From a handle-theoretic perspective, the simplest contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant ν is an invariant of the trace of a knot, i.e. the smooth 4-manifold obtained by attaching a 2-handle to the 4-ball along K. This provides a computable, integer-valued diffeomorphism invariant that is effective at distinguishing exotic smooth structures on knot traces and other simple 4-manifolds, including when other adjunction-type obstructions are ineffective. We also show that the concordance invariants τ and ϵ are not knot trace invariants. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct surgeries to $$S^1 \times S^2$$, resolving a question from Problem 1.16 in Kirby's list.
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Comparing Bennequin-type inequalities
The slice-Bennequin inequality gives an upper bound for the self-linking number of a knot in terms of its four-ball genus. The s-Bennequin and tau-Bennequin inequalities provide upper bounds on the self-linking number of a knot in terms of the Rasmussen s invariant and the Ozsváth-Szabó tau invariant. We exhibit examples in which the difference between self-linking number and four-ball genus grows arbitrarily large, whereas the s-Bennequin inequality and the tau-Bennequin inequality are both sharp.
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- Award ID(s):
- 2104309
- PAR ID:
- 10232797
- Date Published:
- Journal Name:
- New York journal of mathematics
- Volume:
- 27
- ISSN:
- 1076-9803
- Page Range / eLocation ID:
- 124-140
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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