Concordance Crosscap Number of a Knot

Details Concordance Crosscap Number of a Knot Concordance Crosscap Number of a Knot

2006-08-16

arxiv.org/abs/math/0608421v1

Mathematics

Geometric Topology

9 pages, 7 figures

Scientific paper

We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap number is smaller than or equal to the concordance crosscap number. We construct two infinite sequences of knots to explain the gap between the two. In particular, the knot $7_4$ is one of the examples.

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