A note on sign conventions in link Floer homology

Abstract

For knots in $S^3$, the bi-graded hat version of knot Floer homology is defined over $\mathbb{Z}$; however, for an $l$-component link $L$ in $S^3$ with $l>1$, there are $2^{l-1}$ bi-graded hat versions of link Floer homology defined over $\mathbb{Z}$; the multi-graded hat version of link Floer homology, defined from holomorphic considerations, is only defined over ${\mathbb{F}}_2$; and there is a multi-graded version of link Floer homology defined over $\mathbb{Z}$ using grid diagrams. In this short note, we try to address this issue, by extending the ${\mathbb{F}}_2$-valued multi-graded link Floer homology theory to $2^{l-1}$ $\mathbb{Z}$-valued theories. A grid diagram representing a link gives rise to a chain complex over ${\mathbb{F}}_2$, whose homology is related to the multi-graded hat version of link Floer homology of that link over ${\mathbb{F}}_2$. A sign refinement of the chain complex exists, and for knots, we establish that the sign refinement does indeed correspond to the sign assignment for the hat version of the knot Floer homology. For links, we create $2^{l-1}$ sign assignments on the grid diagrams, and show that they are related to the $2^{l-1}$ multi-graded hat versions of link Floer homology over $\mathbb{Z}$, and one of them corresponds to the existing sign refinement of the grid chain complex.