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Non-split, alternating links bound unique Seifert surfaces in the 4-ball

Published online by Cambridge University Press:  15 December 2025

Seungwon Kim Open the ORCID record for Seungwon Kim [Opens in a new window] ,
Maggie Miller  and
Jaehoon Yoo
Seungwon Kim
Affiliation:
Department of Mathematics, Sungkyunkwan University, Jangan-gu, Suwon, Gyeonggi-do, Republic of Korea seungwon.kim@skku.edu
Maggie Miller
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, TX, USA maggie.miller.math@gmail.com
Jaehoon Yoo
Affiliation:
Department of Mathematics, Indiana University Bloomington, Bloomington, IN, USA jaehyoo@iu.edu
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Abstract

We show that any two same-genus, oriented, boundary parallel surfaces bounded by a non-split, alternating link into the 4-ball are smoothly isotopic relative to the boundary. In other words, any same-genus Seifert surfaces for a non-split, alternating link become smoothly isotopic relative to the boundary once their interiors are pushed into the 4-ball. We conclude that a smooth surface in $S^4$ obtained by gluing two Seifert surfaces for a non-split alternating link is always smoothly unknotted.

Keywords

Seifert surfacesalternating linksKakimizu complexfour-dimensionalknot theory

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 10 , October 2025 , pp. 2712 - 2723
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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