Slice obstructions from genus bounds in definite 4-manifolds

Paolo Aceto; Nickolas A. Castro; Maggie Miller; JungHwan Park; András Stipsicz

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arXiv · 2303.10587

Slice obstructions from genus bounds in definite 4-manifolds

Paolo Aceto·Nickolas A. Castro·Maggie Miller·JungHwan Park·András Stipsicz

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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 $\mathbb{CP}^2\#\mathbb{CP}^2$ proved by Bryan in the 1990s.

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Paolo Aceto, Nickolas A. Castro, Maggie Miller, JungHwan Park, András Stipsicz. 2023-03-19. Slice obstructions from genus bounds in definite 4-manifolds. https://arxiv.org/abs/2303.10587

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