In an earlier work, we introduced a family of t-modified knot Floerhomologies, defined by modifying the construction of knot Floer homologyHFK-minus. The resulting groups were then used to define concordancehomomorphisms indexed by t in [0,2]. In the present work we elaborate on thespecial case t=1, and call the corresponding modified knot Floer homology theunoriented knot Floer homology. Using elementary methods (based on griddiagrams and normal forms for surface cobordisms), we show that the resultingconcordance homomorphism gives a lower bound for the smooth 4-dimensionalcrosscap number of a knot K --- the minimal first Betti number of a smooth(possibly non-orientable) surface in the 4-disk that meets the boundary3-sphere along the given knot K.
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