Monday

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. At the same time, Cappell and Shaneson observed a similar obstruction implicit in a formula for the Rokhlin invariant of a 3-manifold presented as an irregular dihedral cover. More recently, Kjuchkova formulated this observation
into an invariant of Fox-p-colored knots which obstructs the knot being ribbon. We will introduce a similar invariant based on the same topological setup, but which is extremely computable, and also obstructs a knot from being ribbon. We will demonstrate how to compute it for some surprisingly large examples, and then we will survey some constructions in the literature of potential counter-examples of the Slice-Ribbon Conjecture. This is joint work with Sylvain Cappell.