Concordance crosscap number of a knot

Abstract

AbstractWe 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.