Smooth Four-D Crosscap Number


Every knot bounds a once punctured connected sum of real projective planes, RP^2, in B^4. The minumum number required is the Four-D 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 B^4, then 4Arf(K) - signature(K) = 0, 2, or -2 mod 8. This applies in both the smooth and topological category.

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