We prove that the (2,1)-cable of the figure-eight knot is not smoothly slice, thereby solving a question posed by Kawauchi in 1980. While doing so, we will review the involutive and equivariant actions in Heegaard Floer homology, and how one can use those symmetries to tackle various knot slicing problems in the smooth category.

This is a joint work with Irving Dai, Abhishek Mallick, JungHwan Park, and Matthew Stoffregen.