Unoriented knot Floer homology and the unoriented four-ball genus

TL;DR

This paper introduces the unoriented knot Floer homology, a new invariant that provides lower bounds for the smooth 4-dimensional crosscap number of knots, expanding the tools for studying non-orientable surfaces in four-dimensional topology.

Contribution

It develops the unoriented knot Floer homology for t=1 and demonstrates its application in bounding the smooth 4-dimensional crosscap number of knots.

Findings

Unoriented knot Floer homology is defined using elementary methods.

The homology provides a lower bound for the smooth 4-dimensional crosscap number.

The approach links knot invariants to non-orientable surface topology in 4D.

Abstract

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.