Existence of a topologically slice, not smoothly slice knot

exists_topologically_slice_not_smoothly_slice

Submitter: Kim Morrison.

Notes: The smooth/topological gap in dimension four. Casson, Akbulut-Matveyev, and Hedden-Kirk-Livingston gave explicit witnesses; Piccirillo later resolved the Conway knot 11n34 — a much smaller, celebrated example of the same phenomenon. The solver may choose any witness. Topological sliceness here means bounding a *locally flat* topological 2-disk in R^3 x [0, infty); the locally flat clause is essential since without it the cone over any knot is a topological disk.

Source: Casson, 1980s (unpublished); Akbulut-Matveyev, *A convex decomposition theorem for 4-manifolds*, IMRN 1998. See also Hedden-Kirk-Livingston, *Non-slice linear combinations of algebraic knots*, J. Eur. Math. Soc. 14 (2012).

Informal solution: Take K = positive Whitehead double of a knot K_0 with tau(K_0) > 0 (e.g. the right-handed trefoil; Akbulut-Matveyev). Whitehead doubles always have trivial Alexander polynomial, so K is topologically slice by Freedman's theorem. The Ozsvath-Szabo tau-invariant of K equals tau(K_0) > 0, and tau is a smooth slice obstruction (tau != 0 implies not smoothly slice), so K is not smoothly slice.

theorem declaration uses `sorry`exists_topologically_slice_not_smoothly_slice :
    ∃ K : LeanEval.KnotTheory.PLKnot, K.TopologicallySlice ∧ ¬ K.SmoothlySlice := by⊢ ∃ K, K.TopologicallySlice ∧ ¬K.SmoothlySlice
  sorryAll goals completed! 🐙

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