Seifert surfaces in the 4-ball
Abstract
We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in that do not become isotopic when their interiors are pushed into . In particular, we identify examples where the surfaces are not even topologically isotopic in , examples that are topologically but not smoothly isotopic, and examples of infinite families of surfaces that are distinct only up to isotopy rel. boundary. Our main proofs distinguish surfaces using the cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.
Cite
@article{arxiv.2205.15283,
title = {Seifert surfaces in the 4-ball},
author = {Kyle Hayden and Seungwon Kim and Maggie Miller and JungHwan Park and Isaac Sundberg},
journal= {arXiv preprint arXiv:2205.15283},
year = {2023}
}
Comments
31 pages + bibliography, 28 figures. Some computational details available in ancillary file. Compared to v1, we added Theorems 1.4 and 1.5 producing infinite families of Seifert surfaces that are pairwise not isotopic rel. boundary in B^4. (In v3, just corrected floats in Fig. 27.)