Unoriented Knot Floer Homology and the Unoriented Four-Ball Genus

Abstract

In an earlier work, we introduced a family tHFK(K) of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK- (K). The resulting groups were then used to define concordance homomorphisms gamma(t) indexed by t is an element of [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 of K. The corresponding concordance homomorphism when t = 1 is denoted by nu. Using elementary methods (based on grid diagrams and normal forms for surface cobordisms), we show that. gives a lower bound for the smooth four-dimensional crosscap number of K-the minimal first Betti number of a smooth (possibly non-orientable) surface in D-4 that meets the boundary S-3 along the given knot K.