Concordance crosscap number of a knot

We define the concordance crosscap number _c(K) of a knot K asthe minimum crosscap number among all the knots concordant toK. The four-dimensional crosscap number *(K) is the minimumfirst Betti number of non-orientable surfaces smoothly embeddedin the four-dimensional ball, bounding the knot K. Clearly,*(K) _c(K). We construct two infinite sequences of knots forwhich *(K) < _c(K). In particular, the knot 7₄ is one of theexamples.