Nonorientable four-ball genus can be arbitrarily large
A classical problem in low-dimensional topology is to find a surface of minimal genus bounding a given knot K in the 3-sphere. Of course, the minimal genus will depend on the class of surface allowed: must it lie in S3 as well, or can it bend into B4? must the embedding be smooth, or only locally flat? must the surface admit an orientation, or can it be nonorientable? Our ability to bound or compute these genera varies dramatically between classes. Orientable surfaces form homology classes, so are amenable to algebraic topology (cf Alexander polynomial), and they admit complex structures, so can be understood using gauge theory (cf Ozsvath-Szabo's \tau). In contrast, the largest lower bound on the genus of a nonorientable surface smoothly embedded in B4 bounding any knot K was, until recently, the integer 3. We will construct a better bound using Heegaard-Floer d-invariants and the Murasugi signature. In particular, we will show that the minimal b_1 of a smoothly embedded, nonorientable surface in B4 bounding the torus knot T(2k,2k-1) is k-1.