A note on sign conventions in link Floer homology
Abstract
For knots in S3, the bi-graded hat version of knot Floer homology is defined over Z; however, for a link L in S3 with #|L|=l>1, there are 2l-1 bi-graded hat versions of link Floer homology defined over Z, the multi-graded hat version of link Floer homology is only defined over F2 from holomorphic considerations, and there is a multi-graded version of link Floer homology defined over Z using grid diagrams. In this short note, we try to address this issue, by extending the F2-valued multi-graded link Floer homology theory to 2l-1 Z-valued theories. A grid diagram representing a link gives rise to a chain complex over F2, whose homology is related to the multi-graded hat version of link Floer homology of that link over F2. 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 2l-1 sign assignments on the grid diagrams, and show that they are related to the 2l-1 multi-graded hat versions of link Floer homology over Z, and one of them corresponds to the existing sign refinement of the grid chain complex.