An unoriented skein exact triangle in unoriented link Floer homology

TL;DR

This paper introduces band maps in unoriented link Floer homology, establishing an unoriented skein exact triangle, and employs a Heegaard Floer analogue of a 2-surgery exact triangle to advance the understanding of unoriented knot invariants.

Contribution

It defines new band maps in unoriented link Floer homology and proves they form an unoriented skein exact triangle, using a novel Heegaard Floer analogue of 2-surgery exact triangle.

Findings

Established an unoriented skein exact triangle in link Floer homology.

Defined band maps analogous to those in equivariant Khovanov homology.

Connected unoriented knot Floer homology to $I^{lat}$ via a 2-surgery exact triangle.

Abstract

We define band maps in unoriented link Floer homology and show that they form an unoriented skein exact triangle. These band maps are similar to the band maps in equivariant Khovanov homology given by the Lee deformation. As a key tool, we use a Heegaard Floer analogue of Bhat's recent 2-surgery exact triangle in instanton Floer homology, which may be of independent interest. Unoriented knot Floer homology corresponds to $I^{\sharp}$ of the knot in our 2-surgery exact triangle.