Abstract: | 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. |