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Seifert surfaces in the 4-ball

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arxiv 2205.15283 v3 pith:IIQHRB2W submitted 2022-05-30 math.GT

Kyle Hayden , Seungwon Kim , Maggie Miller , JungHwan Park , Isaac Sundberg This is my paper · ORCID
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keywords surfacesexamplesisotopicmapsseiferttopologicallyanswerball
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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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  1. Lecture notes on link homologies and knotted surfaces

    math.GT 2025-07 conditional novelty 2.0 of 10

    Lecture notes presenting the cobordism maps on Khovanov and link Floer homology as invariants of knotted surfaces, with worked examples and exercises.

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