SNACKs and slice disks
Here's a short post on a construction I really enjoy. Theorem: (Kawauchi) Let $K \subset S^3$ be a strongly negative amphichiral knot (SNACK). Then $K$ bounds a smoothly embedded disk in a rational homology ball. Let me first explain the statement and then give a proof. A knot $K$ is strongly negative amphichiral if there is an orientation-reversing diffeomorphism of $(S^3,K)$ which is also orientation-reversing on $K$ and the fixed point set is two points which sit on $K$. Sometimes $K$ can be slice, e.g. the square knot is a slice SNACK, but often it is not, e.g. the figure-eight knot. (The Fox-Milnor condition shows it is not even topologically slice.) Here is the proof, which is very quick. First, $X_0(K)$ denote a 0-framed 2-handle attachment to $B^4$ along $K$. Then $K$ is smoothly slice in this four-manifold - the slice disk is just the core of the 2-handle! (This applies even if $K$ is not SNACK...