  Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove
a rank inequality between the knot Floer homology of $\widetilde{K}$ and the
knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and
Sarkar. We also conjecture a filtered refinement of this inequality, for which
we give computational evidence, and produce applications to the Alexander
polynomials of $\widetilde{K}$ and $K$.
