  A knot \widetilde{K} \subset S^3 is q-periodic if there is a \mathbb
Z_q-action preserving \widetilde{K} whose fixed set is an unknot U. The
quotient of \widetilde{K} under the action is a second knot K. We construct
equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's
classical condition on the Alexander polynomials of periodic knots is a quick
consequence of these diagrams. For \widetilde{K} a two-periodic knot, we show
there is a spectral sequence whose E^1 page is
\hat{\mathit{HFL}}(S^3,\widetilde{K}\cup U)\otimes V^{\otimes (2n-1)})\otimes
\mathbb Z_2((\theta)) and whose E^{\infty} pages is isomorphic to
(\hat{\mathit{HFL}}(S^3,K\cup U)\otimes V^{\otimes (n-1)})\otimes \mathbb
Z_2((\theta)), as \mathbb Z_2((\theta))-modules, and a related spectral
sequence whose E^1 page is (\hat{\mathit{HFK}}(S^3,\widetilde{K})\otimes
V^{\otimes (2n-1)}\otimes W)\otimes \mathbb Z_2((\theta)) and whose E^{\infty}
page is isomorphic to (\hat{\mathit{HFK}}(S^3,K)\otimes V^{\otimes (n-1)}
\otimes W)\otimes \mathbb Z_2((\theta)). As a consequence, we use these
spectral sequences to recover a classical lower bound of Edmonds on the genus
of \widetilde{K}, along with a weak version of a classical fibredness result of
Edmonds and Livingston.
