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 $operatorname{sn}(K)<g_4^{operatorname{top}}(K)$.
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