Symmetry and Sliceness

A knot is called slice if it bounds a smoothly embedded disk in the four-ball. Much of low-dimensional topology revolves around obstructing this property of a knot. In this talk, we will introduce a method of obstructing sliceness which takes into account the actions of different symmetries on the Floer homology group. Using this, we will show that the (2,1)-cable of the figure-eight knot is not slice, answering a 42-year-old question posed by Kawauchi. Historically, this knot gained particular interest since it was regarded as one of the simplest "potential" counter-example to the slice-ribbon conjecture. This is joint work with Dai, Kang, Park, and Stoffregen.