Nonorientable four-ball genus can be arbitrarily large
  The nonorientable four-ball genus of a knot K is the smallest first Betti number of any smoothly embedded, nonorientable surface F in B^4 bounding K. In contrast to the orientable four-ball genus, which is bounded below by the Murasugi signature, the Ozsvath-Szabo tau-invariant, the Rasmussen s-invariant, the best lower bound in the literature on the nonorientable four-ball genus for any K is 3. We find a lower bound in terms of the signature of K and the Heegaard-Floer d-invariant of the integer homology sphere given by -1 surgery on K. In particular, we prove that the nonorientable four-ball genus of the torus knot T(2k,2k-1) is k-1. 
Joshua Batson
math.GT