Slicing knots in definite 4-manifolds

Kjuchukova, Alexandra; Miller, Allison; Ray, Arunima; Sakallı, Sümeyra

doi:10.1090/tran/9151

Slicing knots in definite $4$-manifolds
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by Alexandra Kjuchukova, Allison N. Miller, Arunima Ray and Sümeyra Sakallı;

Trans. Amer. Math. Soc. 377 (2024), 5905-5946

DOI: https://doi.org/10.1090/tran/9151

Published electronically: June 11, 2024

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Abstract:

We study the $\mathbb {CP}^2$-slicing number of knots, i.e. the smallest $m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded, null-homologous disk in a punctured connected sum $(\#^m\mathbb {CP}^2)^{\times }$. We find knots for which the smooth and topological $\mathbb {CP}^2$-slicing numbers are both finite, nonzero, and distinct. To do this, we give a lower bound on the smooth $\mathbb {CP}^2$-slicing number of a knot in terms of its double branched cover and an upper bound on the topological $\mathbb {CP}^2$-slicing number in terms of the Seifert form.

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Slicing knots in definite $4$-manifolds
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Abstract: