  If a knot K bounds a genus one Seifert surface F in the 3-sphere and F
contains an essential simple closed curve alpha 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, then there are strong constraints on the
algebraic concordance class of such alpha, and it was thought that these
constraints might imply that alpha 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
alpha.
