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

TL;DR

This paper proves that for non-split, alternating links, any two same-genus Seifert surfaces are smoothly isotopic in the 4-ball, leading to the conclusion that certain surfaces in 4-sphere are unknotted.

Contribution

It establishes the uniqueness of Seifert surfaces of the same genus for non-split, alternating links in the 4-ball, a new result in knot theory and 4-manifold topology.

Findings

Any two same-genus Seifert surfaces for such links are smoothly isotopic fixing boundary.

Surfaces obtained by gluing Seifert surfaces are always smoothly unknotted in $S^4$.

The result applies to non-split, alternating links specifically.

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.