  In 1982 Louis Kauffman conjectured that if a knot in the 3-sphere is a slice
knot then on any Seifert surface for that knot there exists a homologically
essential simple closed curve of self-linking zero which is itself a slice
knot, or at least has Arf invariant zero. Since that time, considerable
evidence has been amassed in support of this conjecture. In particular, many
invariants that obstruct a knot from being a slice knot have been explictly
expressed in terms of invariants of such curves on the Seifert surface. We give
counterexamples to Kauffman's conjecture, that is, we exhibit (smoothly) slice
knots that admit (unique minimal genus) Seifert surfaces on which every
homologically essential simple closed curve of self-linking zero has non-zero
Arf invariant and non-zero signatures.
