The Conway knot is topologically slice but not smoothly slice

Given: Full AC and the Conway knot $C$ fixed above.

Full AC, written AC, is The Axiom of Choice.

AC supplies countable choice through AC implies DC implies countable choice, which is used by the handle and Lee branches.

The Conway knot has normalized Alexander polynomial one gives the actual normalized equality ${\mathrm{\Delta }}_C(t)=1$.

Alexander polynomial one implies topological sliceness: under full AC, every smooth oriented Alexander-one knot bounds a locally flat proper disk in standard $B^4$. This is its smooth-input application; no broader TOP-generic disk-embedding assertion is used.

Locally flat proper surfaces and concordance collars fixes the proper local-flatness and boundary-collar conclusion.

A knot is smoothly slice exactly when its zero-trace embeds in $S^4$: a smooth knot is smoothly slice if and only if its zero-trace smoothly embeds in standard $S^4$.

Piccirillo’s partner has the same zero-trace and Rasmussen invariant two: under full AC, the Figure 2 partner $K_P$ has $X_0(C)\cong X_0(K_P)$ and $s(K_P)=2$.

Rasmussen's s-invariant bounds smooth four-ball genus: under countable choice, $\mathrm{\mid }s(K)\mathrm{\mid }\le 2g_4^{\mathrm{s}\mathrm{m}}(K)$, with its collared properly embedded cobordism hypotheses satisfied by an actual smooth slice disk.