Slicing Knots In Definite 4-Manifolds

Document Type

Article

Publication Date

6-11-2024

Published In

Transactions Of The American Mathematical Society

Abstract

We study the ℂℙ²-slicing number of knots, i.e. the smallest 𝑚 ≥ 0 such that a knot 𝐾 ⊆ 𝑆³ bounds a properly embedded, null-homologous disk in a punctured connected sum (#𝑚ℂℙ²)⨉. We find knots for which the smooth and topological ℂℙ²-slicing numbers are both finite, nonzero, and distinct. To do this, we give a lower bound on the smooth ℂℙ²-slicing number of a knot in terms of its double branched cover and an upper bound on the topological ℂℙ²-slicing number in terms of the Seifert form.

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