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

TL;DR

This paper provides a combinatorial re-derivation of Manolescu's unoriented skein exact triangle for knot Floer homology using grid diagrams, extends it to Z coefficients, and explores its implications for link homology.

Contribution

It offers a new combinatorial proof of the skein triangle, extends the theory to integer coefficients, and applies it to prove properties of quasi-alternating links.

Findings

Re-derivation of Manolescu's skein triangle using grid diagrams

Extension of the skein triangle to Z coefficients with sign refinements

Proof of homological sigma-thinness for quasi-alternating links

Abstract

We re-derive Manolescu's unoriented skein exact triangle for knot Floer homology over F_2 combinatorially using grid diagrams, and extend it to the case with 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 re-establish the homological sigma-thinness of quasi-alternating links.