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. In particular we prove that the sign refinement comes from a 2–cohomological class corresponding to the spin extension of the permutation group.
"Sign refinement for combinatorial link Floer homology." Algebr. Geom. Topol. 8 (3) 1581 - 1592, 2008. https://doi.org/10.2140/agt.2008.8.1581