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

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

Geometric Topology 2025-04-08 v2

Authors: Marco Marengon , Allison N. Miller , Arunima Ray , András I. Stipsicz

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 $\mathbb{CP}^2\# \mathbb{CP}^2$ .

Keywords

knot theory

Cite

@article{arxiv.2210.12486,
  title  = {A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$},
  author = {Marco Marengon and Allison N. Miller and Arunima Ray and András I. Stipsicz},
  journal= {arXiv preprint arXiv:2210.12486},
  year   = {2025}
}

Comments

7 pages, 3 figures; v2: minor modifications following a referee report, this is the version accepted to appear in Proc. AMS

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

Abstract

Keywords

Cite

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