Seifert surfaces in the 4-ball

Kyle Hayden; Seungwon Kim; Maggie Miller; JungHwan Park; Isaac Sundberg

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arXiv · 2205.15283

Seifert surfaces in the 4-ball

Kyle Hayden·Seungwon Kim·Maggie Miller·JungHwan Park·Isaac Sundberg

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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 $S^3$ that do not become isotopic when their interiors are pushed into $B^4$. In particular, we identify examples where the surfaces are not even topologically isotopic in $B^4$, 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.

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BibTeX RIS

Kyle Hayden, Seungwon Kim, Maggie Miller, JungHwan Park, Isaac Sundberg. 2022-05-30. Seifert surfaces in the 4-ball. https://arxiv.org/abs/2205.15283

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