Non-isotopic splitting spheres for a split link in S⁴

2023/07/22 by Mark Hughes, Hughes, Mark, Seungwon Kim +3
Mathematics · Engineering · Biochemistry, Genetics and Molecular Biology · #Geometric and Algebraic Topology #Advanced Materials and Mechanics #Connective tissue disorders research

paper · pdf · doi:10.48550/arxiv.2307.12140

Abstract

We show that there exist split, orientable, 2-component surface-links in S⁴ with non-isotopic splitting spheres in their complements. In particular, for non-negative integers m,n with m≥ 4, the unlink L_m,n consisting of one component of genus m and one component of genus n contains in its complement two smooth splitting spheres that are not topologically isotopic in S⁴∖ L_m,n. This contrasts with link theory in the classical dimension, as any two splitting spheres in the complement of a 2-component split link L⊂ S³ are isotopic in S³∖ L.

Related

• Splitting spheres for unlinked S²'s in S⁴
• Splitting links by integer homology spheres
• Non-isotopic Seifert surfaces in the 4-ball
• Non-split, alternating links bound unique Seifert surfaces in the 4-ball
• Isotoping 2-spheres in 4-manifolds