4D Crosscap Number

Every knot bounds a once punctured connected sum of real projective planes, RP2, in B4. The minumum number required is the 4D crosscap number. A knot is slice if and only if this number is 0. A tool for ruling out 4d-crosscap number ≤ 1 is a result of Yasuhara: If K bounds a punctured Mobius band in B4, then 4Arf(K) - signature(K) = 0, 2, or -2 mod 8. This applies in both the smooth and topological categories.

References

[1] Baston, J., Nonorientable four-ball genus can be arbitrarily large, Math. Res. Lett. 21 (2014), no. 3, 423-436.

[2] Gilmer, P. and Livingston, C., The nonorientable four-genus of knots, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 559-577.

[3] Jabuka, S. and Kelly, T., The nonorientable 4-genus for knots with 8 or 9 crossings, Algebraic and Geometric Topology 18 (2018), 1823-1856.

[4] 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.

[5] Viro, O., "Positioning in codimension 2 and the boundary," Uspehi Mat. Nauk 30 (1975), 231-232.

[6] Yasuhara, A., "Connecting lemmas and representing homology classes of simply connected 4-manifolds," Tokyo J. Math. 19 (1996), no. 1, 245-261.