We say a knot in the 3-sphere is slice if it bounds a smooth disk into the 4-ball. If the disk can be obtained from a special kind of immersion into S^3 (pushed into B^4), then we say the knot is ribbon. A long-standing open conjecture of Fox posits that every slice knot is ribbon. I will talk about this open problem and discuss why it would be so difficult to disprove this conjecture, if it is false (and one obstruction that might work). I’ll relate sliceness of knots to lower bounds on the genus of some surfaces embedded in nontrivial 4-manifolds, and use some old (difficult) such lower bounds to recover a recent interesting result of Dai—Kang—Mallick—Park—Stoffregen that a certain knot is not slice (which is unfortunate, because it’s much easier to prove that it isn’t ribbon). This is joint work with Paolo Aceto, Nickolas A. Castro, JungHwan Park and András Stipsicz.