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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...
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Absolutely exotic contractible four-manifolds

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Since exotic four-manifolds are always "in", I wanted to talk about a cool construction of exotica I like due to Akbulut-Ruberman.  In their paper, they construct compact contractible four-manifolds which are "absolutely exotic", which means they are homeomorphic, but there is no diffeomorphism between them.  To clarify the terminology, often times, for manifolds with boundary, one talks about "relatively exotic" manifolds, which means that you have a homeomorphism from one four-manifold to another, but the boundary homeomorphism doesn't extend to a diffeomorphism of the four-manifolds.  [For context, it's worth pointing out that long before, we know how to construct lots of non-compact, contractible exotica: namely exotic $\mathbb{R}^4$'s.  In fact, Taubes proved there are uncountably many exotic $\mathbb{R}^4$'s.] I want to describe their work (but with a bit of a Floer homology lean).   For simplicity, let's construct a pair of n...
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