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Seifert surfaces in the 4-ball

  • Kyle Hayden

    Rutgers University, Newark, USA
    ORCID

  • Seungwon Kim

    Sungkyunkwan University, Suwon, South Korea
    ORCID

  • Maggie Miller

    The University of Texas at Austin, USA
    ORCID

  • JungHwan Park

    Korea Advanced Institute of Science and Technology, Daejeon, South Korea
    ORCID

  • Isaac Sundberg

    Max Planck Institute for Mathematics, Bonn, Germany
    ORCID
Seifert surfaces in the 4-ball cover

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Abstract

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in that do not become isotopic when their interiors are pushed into . In particular, we identify examples where the surfaces are not even topologically isotopic in , examples that are topologically but not smoothly isotopic, and examples of infinite families of surfaces that are distinct only up to isotopy rel. boundary. Our main proofs distinguish surfaces using the cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.

Cite this article

Kyle Hayden, Seungwon Kim, Maggie Miller, JungHwan Park, Isaac Sundberg, Seifert surfaces in the 4-ball. J. Eur. Math. Soc. (2025), published online first

DOI 10.4171/JEMS/1703