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Knot Floer homology and relative adjunction inequalities

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Abstract

We establish inequalities that constrain the genera of smooth cobordisms between knots in 4-dimensional cobordisms. These “relative adjunction inequalities” improve the adjunction inequalities for closed surfaces which have been instrumental in many topological applications of gauge theory. The relative inequalities refine the latter by incorporating numerical invariants of knots in the boundary associated to Heegaard Floer homology classes determined by the 4-manifold. As a corollary, we produce a host of concordance invariants for knots in a general 3-manifold, one such invariant for every non-zero Floer class. We apply our results to produce analogues of the Ozsváth–Szabó–Rasmussen concordance invariant for links, allowing us to reprove the link version of the Milnor conjecture, and, furthermore, to show that knot Floer homology detects strongly quasipositive fibered links.

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Notes

  1. Here, we are implicitly assuming the relative homology class of the surface in W equals that of the Seifert surface used to define \(\tau \), under the inclusion induced map \(H_2(Y,K)\rightarrow H_2(W,K)\).

  2. In fact \(n=\tau _{\xi _{std}}(K_n)\), since \(\tau \) invariants are bounded by the Seifert genus, and there is a genus n Seifert surface for \(K_n\) by construction.

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Acknowledgements

This paper evolved over many years, and enjoyed the benefit of interest and input from a number of people, whom we warmly thank: John Baldwin, Inanc Baykur, Alberto Cavallo, Georgi Gospodinov, Eli Grigsby, Miriam Kuzbary, Adam Levine, Chuck Livingston, Tom Mark, Peter Ozsváth, Olga Plamenevskaya, Danny Ruberman, Sucharit Sarkar, Linh Truong, Zhongtao Wu, and Ian Zemke. KR also thanks Bryn Mawr College for hosting her as a research associate and the Max Planck Institute for Mathematics. MH thanks MSRI and AIM for the supportive environments provided for portions of this work.

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  1. Department of Mathematics, Michigan State University, East Lansing, USA

    Matthew Hedden & Katherine Raoux

  2. Department of Mathematical Sciences, University of Arkansas, Arkansas, Fayetteville, USA

    Katherine Raoux

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  1. Matthew Hedden
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Correspondence to Matthew Hedden.

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MH gratefully acknowledges support from NSF grants DMS-0706979, DMS-0906258, CAREER DMS-1150872, DMS-1709016, DMS-2104664 and an Alfred P. Sloan Research Fellowship. KR was partially supported by an AWM Mentoring Travel Grant.

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Hedden, M., Raoux, K. Knot Floer homology and relative adjunction inequalities. Sel. Math. New Ser. 29, 7 (2023). https://doi.org/10.1007/s00029-022-00810-1

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