A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$

Marco Marengon; Allison N. Miller; Arunima Ray; András I. Stipsicz

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arXiv · 2210.12486

A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$

Marco Marengon·Allison N. Miller·Arunima Ray·András I. Stipsicz

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Abstract

In this brief note, we investigate the $\mathbb{CP}^2$-genus of knots, i.e. the least genus of a smooth, compact, orientable surface in $\mathbb{CP}^2\setminus \mathring{B^4}$ bounded by a knot in $S^3$. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the $\mathbb{CP}^2$-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in $\mathbb{CP}^2\# \mathbb{CP}^2$.

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Marco Marengon, Allison N. Miller, Arunima Ray, András I. Stipsicz. 2022-10-22. A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$. https://arxiv.org/abs/2210.12486

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