Seifert surfaces in the 4-ball

TL;DR

This paper constructs and distinguishes Seifert surfaces in the 4-ball for knots in $S^3$, using Khovanov homology cobordism maps to identify non-isotopic surfaces in various topological and smooth categories.

Contribution

It provides explicit examples of Seifert surfaces that are not isotopic in $B^4$, employing Khovanov homology to differentiate them, and explores their stability under satellite operations.

Findings

Examples of non-isotopic Seifert surfaces in $B^4$

Distinction of surfaces using Khovanov homology cobordism maps

Demonstration of stability of these maps under satellite operations

Abstract

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in $S^3$ that do not become isotopic when their interiors are pushed into $B^4$. In particular, we identify examples where the surfaces are not even topologically isotopic in $B^4$, 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.