Chern-Simons functional, singular instantons, and the four-dimensional clasp number
Any knot in the 3-sphere is the boundary of a smoothly immersed disk in the 4-ball with transverse double points. The four-dimensional clasp number of a knot is the minimum number of double points that can occur. It is bounded below by the slice genus, but not much is known about their difference. Kronheimer and Mrowka have suggested that instantons may help show that the difference can be arbitrarily large. In this talk we will see that this is true, using equivariant instanton homology constructions. Other topological applications will also be discussed. This is joint work with Ali Daemi.