Nonorientable link cobordisms and torsion order in Floer homologies

Abstract

We use unoriented versions of instanton and knot Floer homology to prove inequalities involving the Euler characteristic and the number of local maxima appearing in nonorientable cobordisms, which mirror results of a recent paper by Juhász, Miller and Zemke concerning orientable cobordisms. Most of the subtlety in our argument lies in the fact that maps for nonorientable cobordisms require more complicated decorations than their orientable counterparts. We introduce unoriented versions of the band unknotting number and the refined cobordism distance and apply our results to give bounds on these based on the torsion orders of the Floer homologies. Finally, we show that the difference between the unoriented refined cobordism distance of a knot K from the unknot and the nonorientable slice genus of K can be arbitrarily large.