Concordance crosscap number of a knot

We define the concordance crosscap number γ c ( K ) of a knot K as the minimum crosscap number among all the knots concordant to K . The four‐dimensional crosscap number γ *( K ) is the minimum first Betti number of non‐orientable surfaces smoothly embedded in the four‐dimensional ball, bounding the knot K . Clearly, γ *( K ) ⩽ γ c ( K ). We construct two infinite sequences of knots for which γ* ( K ) < γ c ( K ). In particular, the knot 7 4 is one of the examples.