A note on the concordance $\mathbb{Z}$-genus

Abstract

We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden-Livingston-Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.