Sign refinement for combinatorial link Floer homology

TL;DR

This paper introduces an alternative combinatorial construction of link Floer homology with integer coefficients, utilizing the spin extension of the permutation group, and proves its invariance and relation to previous sign refinements.

Contribution

It provides a new combinatorial approach to link Floer homology with integer coefficients using the spin extension, establishing its invariance and connection to existing sign refinements.

Findings

The filtered homology is an invariant for links.

The new construction recovers the previous sign refinement.

The approach uses the spin extension of the permutation group.

Abstract

Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alternative construction of the combinatorial link Floer chain complex associated to a grid diagram with integer coefficients. We prove that the filtered homology of this complex is an invariant for the link and that it gives the previous sign refinement by means of a 2-cohomological class corresponding to the spin extension of the permutation group.