If X is a simply-connected closed 4-manifold containing an oriented embedded surface S of genus g, is there always an immersed sphere S' which represents the same homology class and has only g transverse double-points? This is an open question, though a "relative" version of the question (concerning surfaces in the 4-ball bounding a given knot in the 3-sphere) is known to have a negative answer. This talk will ask whether there are additive invariants of knots which can be used to detect this distinction between genus and double-points. A previous talk in this seminar, by Tom Mrowka, introduced the tools from gauge theory that will be used, but I will try to make this talk largely independent.