If a knot K bounds a
genus-one Seifert surface F ⊂ S3 and F contains an essential simple closed curve
α that has induced framing 0 and is smoothly slice, then K is smoothly
slice. Conjecturally, the converse holds. It is known that if K is slice and the
determinant of K is not 1, then there are strong constraints on the algebraic
concordance class of such α, and it was thought that these constraints might imply
that α is at least algebraically slice. We present a counterexample; in the
process we answer negatively a question of Cooper and relate the result
to a problem of Kauffman. Results of this paper depend on the interplay
between the Casson–Gordon invariants of K and algebraic invariants of
α.