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

Seungwon Kim; Maggie Miller; Jaehoon Yoo

doi:10.1112/s0010437x25102765

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arXiv · 2406.11718

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

Seungwon Kim·Maggie Miller·Jaehoon Yoo

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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 fixing boundary. In other words, any same-genus Seifert surfaces for a non-split, alternating link become smoothly isotopic fixing 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.

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Seungwon Kim, Maggie Miller, Jaehoon Yoo. 2024-06-17. Non-split, alternating links bound unique Seifert surfaces in the 4-ball. https://doi.org/10.1112/s0010437x25102765

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