# Stably slice disks of links

## TL;DR

This paper introduces the stabilizing number of a knot, a new invariant measuring how many $S^2 imes S^2$ summands are needed for the knot to bound a nullhomotopic disc in a stabilized 4-manifold, and explores its properties and bounds.

## Contribution

It defines the stabilizing number for knots, establishes bounds using signatures and Casson-Gordon invariants, and provides examples where this number is less than the topological 4-genus.

## Key findings

- $ ext{sn}(K)$ is bounded below by signatures and Casson-Gordon invariants.
- $ ext{sn}(K)$ is bounded above by the topological 4-genus $g_4^{ ext{top}}(K)$.
- An infinite family of knots with $ ext{sn}(K) < g_4^{ ext{top}}(K)$ is constructed.

## Abstract

We define the stabilizing number $\operatorname{sn}(K)$ of a knot $K \subset S^3$ as the minimal number $n$ of $S^2 \times S^2$ connected summands required for $K$ to bound a nullhomotopic locally flat disc in $D^4 \# n S^2 \times S^2$. This quantity is defined when the Arf invariant of $K$ is zero. We show that $\operatorname{sn}(K)$ is bounded below by signatures and Casson-Gordon invariants and bounded above by the topological $4$-genus $g_4^{\operatorname{top}}(K)$. We provide an infinite family of examples with $\operatorname{sn}(K)<g_4^{\operatorname{top}}(K)$.