Claim: every smoothly slice knot is a ribbon knot. A knot K ⊂ S³ is slice if it bounds a properly embedded smooth disc in B⁴; K is ribbon if it bounds an immersed disc in S³ whose only singularities are ribbon intersections (arcs of double points). Ribbon ⟹ slice is immediate.
Fox raised the question in his 1962 knot theory notes; Milnor highlighted it as one of the most important problems in low-dimensional topology. Topologically slice knots (Freedman) include non-ribbon examples, making the smooth category genuinely restrictive. Akbulut–Ruberman, Gompf and others built the 4D toolkit; recent gauge-theoretic and Heegaard Floer obstructions (Hom, Ozsváth–Szabó) have ruled out many potential approaches without resolving the core question.
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