The 4d-crosscap number of a knot K is the minimum first Betti number among all nonorientable surfaces
bounded by K in B4. In particular, for a slice knot K, the 4d-crosscap number is 1.
Every knot bounds a once punctured connected sum of real projective planes, RP2, in B4. If the minumum number required is postive, then this minimum is the 4d-crosscap number. If the minimum is 0, the 4d-crosscap number is 1.
A tool for ruling out 4d-crosscap number ≤ 1 is a result of Yasuhara: If K bounds a Mobius band in B4, then 4Arf(K) - signature(K) = 0, 2, or -2 mod 8. This applies in both the smooth and topological categories.
Slaven Jabuka and Tynan Kelly  provided the complete data for 8 and 9 crossing knots. Nakisa Ghabarian  provided the data for 10 crossing knots.
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