A note on surfaces in ℂℙ² and ℂℙ²#ℂℙ²

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