For a knot \(K\in S^3\), a non-orientable genus \(\gamma(K)\), a slice genus \(\gamma^*(K)\), and a concordance crosscap number \(\gamma_c(K)\) are defined, the latter being the minimal non-orientable genus (crosscap number) in the concordance class of \(K\). A necessary condition is given for certain classes of Pretzel knots \(K\) to satisfy \(\gamma_c(K)= 1\), and an infinite sequence of knots \(K\) is given with \(\Gamma^*(K)<\gamma_c(K)\) – generally one has \(\gamma^*(K)< \gamma_c(K)\leq \gamma(K)\).
The proof uses the Gordon-Litherland formula for the signature, and the properties of the Alexander polynomial of a slice knot.