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

2024/06/17 by Kim, Seungwon, Miller, Maggie, Yoo, Jaehoon
#57K10 #57K45 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2406.11718

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 S⁴ obtained by gluing two Seifert surfaces for a non-split alternating link is always smoothly unknotted.

Related

• Seifert surfaces in the 4-ball
• Non-isotopic Seifert surfaces in the 4-ball
• Bridge number and twist equivalence of Seifert surfaces
• Unknotting Nonorientable Surfaces of Genus 4 and 5
• Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds