Cobordisms of knots, braid index, and the Upsilon invariant
If two knots occur as intersections of a given algebraic curve in C^2 with concentric three-spheres, they are connected by a cobordism whose genus is the diﬀerence of the slice genera of the two knots. A cobordism with this genus is called optimal. We use the Upsilon invariant of Ozsvath-Stipsicz-Szabo to obstruct the existence of optimal cobordisms. In doing so, we generalize a result of Morton-Franks-Williams on the minimal braid index of knots with a "full twist", and show that in some cases Upsilon can provide a lower bound on the braid index of any representative in a concordance class of knots. This is joint work with Peter Feller.