Slice obstructions from genus bounds in definite 4-manifolds

2023/03/19 by Paolo Aceto, Nickolas A. Castro, Aceto, Paolo +7 · 1 citation
Mathematics · Medicine · #Geometric and Algebraic Topology #Botulinum Toxin and Related Neurological Disorders

paper · pdf · doi:10.48550/arxiv.2303.10587

Abstract

We discuss an obstruction to a knot being smoothly slice that comes from minimum-genus bounds on smoothly embedded surfaces in definite 4-manifolds. As an example, we provide an alternate proof of the fact that the (2,1)-cable of the figure eight knot is not smoothly slice, as shown by Dai--Kang--Mallick--Park--Stoffregen in 2022. The main technical input of our argument consists of gauge-theoretic obstructions to smooth small-genus surfaces representing certain homology classes in \mathbbCP²#\mathbbCP² proved by Bryan in the 1990s.

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