Characteristic Knots and Ribbon Disks

In the 1970's dihedral representations of knot groups were used to define twisted signature-type invariants which generalize the older invariants of Levine and Tristram. The most prominent examples are the Casson-Gordon invariants, which provide obstructions to being topologically slice as well as more sensitive obstructions to being ribbon. Around the same time, Cappell and Shaneson noticed that a different-yet-similar obstruction to being ribbon was implicit in a certain formula for the Rokhlin invariant of a 3-manifold presented as an irregular dihedral cover of a knot in the 3-sphere. More recently, Kjuchkova formulated a version of this invariant and has been able to compute it for some examples in joint work with Cahn. We will introduce a third knot invariant, based on a slightly different topological setup, which is easily computable and obstructs being ribbon. We will demonstrate how to compute it for some explicit examples, and we will then survey some constructions in the literature of potential counter-examples of the Slice-Ribbon Conjecture. This is joint work with Sylvain Cappell.