Grid diagrams and Manolescu's unoriented skein exact triangle for knot Floer homology

Abstract

We rederive Manolescu’s unoriented skein exact triangle for knot Floer homology over ${\mathbb{F}}_2$ combinatorially using grid diagrams, and extend it to the case with $\mathbb{Z}$ coefficients by sign refinements. Iteration of the triangle gives a cube of resolutions that converges to the knot Floer homology of an oriented link. Finally, we reestablish the homological $\sigma$–thinness of quasialternating links.