I would like to ask if there is a reference which carries out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{HFK}(K;\mathbb{Z})$, where the differentials $\partial$ are not all zero.
In particular, I am looking for a hands-on assignment of orientations to moduli spaces leading to a computation of homology with $\mathbb{Z}$ coefficients.
Such orientation assignment is at least defined in Definition 3.11/3.12, p.29 of https://arxiv.org/pdf/math/0101206.pdf (for Heegaard Floer homology, though I think it should apply to know floer without much change).