Unoriented Cobordism Maps on Link Floer Homology

We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion of index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a disoriented link cobordism. Furthermore, we give a comparison with Oszv'ath-Stipsicz-Szab'o’s and Manolescu’s constructions of link cobordism maps for an unoriented band move.