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 [4] provided the complete data for 8 and 9 crossing knots. Nakisa Ghabarian [2] provided the data for 10 crossing knots. Megan Fairchild [8] provided the results for non-alternating 11 crossing knots, leaving all but 11n_{17, 40, 159, 166, 177, 178} undetermined, all in the range [1,2].
[1] Baston, J., Nonorientable four-ball genus can be arbitrarily large, Math. Res. Lett. 21 (2014), no. 3, 423-436.
[2] Ghanbarian, N., "The non-orientable 4-genus for knots with 10 crossings," Arxiv preprint.
[3] Gilmer, P. and Livingston, C., The nonorientable four-genus of knots, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 559-577.
[4] Jabuka, S. and Kelly, T., The nonorientable 4-genus for knots with 8 or 9 crossings, Algebraic and Geometric Topology 18 (2018), 1823-1856.
[5] Ozsvath, P., Stipsicz, A., and Szabo, Z., Unoriented knot Floer homology and the unoriented four-ball genus, Algebraic and Geometric Topology 18 (2018), 1823-1856.
[6] Viro, O., "Positioning in codimension 2 and the boundary," Uspehi Mat. Nauk 30 (1975), 231-232.
[7] Yasuhara, A., "Connecting lemmas and representing homology classes of simply connected 4-manifolds," Tokyo J. Math. 19 (1996), no. 1, 245-261.
[8] Fairchild, M., The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots. Arxiv preprint.