Smooth Four-D Crosscap Number

Every knot bounds a once punctured connected sum of real projective planes, RP^2, in B^4. For slice knots there are differing defintion that have the value be 0 or 1. References are cited below.

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

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

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

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.

Viro, O., Positioning in codimension 2 and the boundary, Uspehi Mat. Nauk 30 (1975) 231–232.

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