Stably slice disks of links

Abstract

We define the stabilizing number $\mathrm{sn}(K)$ 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 $\mathrm{sn}(K)$ is bounded below by signatures and Casson-Gordon invariants and bounded above by the topological $4$-genus $g_4^{\mathrm{top}}(K)$. We provide an infinite family of examples with $\mathrm{sn}(K)<g_4^{\mathrm{top}}(K)$.

arXiv comment40 pages, 11 figures, implements suggestions from a referee report and corrects a mistake in the appendix

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Cited by in corpus (9)

• Relative genus bounds in indefinite four-manifolds
• Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space
• Deep and shallow slice knots in 4-manifolds
• Knot concordance invariants from Seiberg-Witten theory and slice genus bounds in 4-manifolds
• On the slice genus of quasipositive knots in indefinite 4-manifolds
• Involutions, knots, and Floer K-theory
• Stabilization and satellite construction of doubly slice links
• Unknotting via null-homologous twists and multi-twists
• On Arf invariants of colored links