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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 showthat this quantity is unbounded, unlike its topological counterpart. We alsoinvestigate the $\mathbb{CP}^2$-genus of torus knots. We apply these results toimprove the minimal genus bound for some homology classes in $\mathbb{CP}^2\#\mathbb{CP}^2$.

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