The slice-ribbon conjecture and derivative links

Contact for zoom details: Barthel, Ozornova, Ray, Teichner.

The slice-ribbon conjecture of Fox is one of the most well-known open problems in knot theory.  A knot K in the 3-sphere is (smoothly) slice if it bounds a disk in the 4-ball, and K is ribbon if it bounds an immersed disk in the 3-sphere with ribbon singularities.  Every ribbon knot is slice, but it is unknown if every slice knot is ribbon.  In this talk, we characterize ribbonness and sliceness by exhibiting special families of curves contained Seifert surfaces for K, called derivative links.  In addition, we discuss a family of knots potentially in between ribbon and slice knots, called handle-ribbon knots, and we characterize these knots with derivative links as well.  This is joint work with Maggie Miller.