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

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

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