  In an earlier work, we introduced a family of t-modified knot Floer
homologies, defined by modifying the construction of knot Floer homology
HFK-minus. The resulting groups were then used to define concordance
homomorphisms indexed by t in [0,2]. In the present work we elaborate on the
special case t=1, and call the corresponding modified knot Floer homology the
unoriented knot Floer homology. Using elementary methods (based on grid
diagrams and normal forms for surface cobordisms), we show that the resulting
concordance homomorphism gives a lower bound for the smooth 4-dimensional
crosscap number of a knot K --- the minimal first Betti number of a smooth
(possibly non-orientable) surface in the 4-disk that meets the boundary
3-sphere along the given knot K.
