Claim: every slice knot is a ribbon knot. A knot K ⊂ S³ is slice if it bounds a smoothly embedded 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 posed the question informally in his 1962 knot theory notes; Milnor highlighted it alongside the unknotting problem. The conjecture gained teeth when topological-vs-smooth distinctions emerged in dimension four: topologically slice knots (Freedman) include many that are not smoothly slice (Donaldson constraints), making the smooth ribbon criterion genuinely restrictive.
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