Localization and the link Floer homology of doubly-periodic knots

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 $\widetilde{U}$. 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 $\widehat{HFL}( S^3, \widetilde{K} \cup \widetilde{U}) \otimes V^{\otimes (2n-1)} ) \otimes \mathbb{Z}_2((\theta))$ and whose $E^{\infty}$ page is isomorphic to $(\widehat{HFL}( S^3, K \cup U) \otimes V^{\otimes (n−1)}) \otimes \mathbb{Z}_2((\theta))$, as $\mathbb{Z}_2((\theta))$-modules, and a related spectral sequence whose $E^1$ page is $(\widehat{HFK}(S^3, \widetilde{K}) \otimes V^{\otimes (2n-1)} \otimes W) \otimes Z_2((θ))$ and whose $E^{\infty}$ page is isomorphic to $(\widehat{HFK}( S^3,K) \otimes V^{\otimes (n-1)} \otimes W) \otimes \mathbb{Z}_2((\theta))$. 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. We give an example of a knot $\widetilde{K}$ which is not obstructed from being two-periodic with a particular quotient knot $K$ by Edmonds’ and Murasugi’s conditions, but for which our spectral sequence cannot exist.

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Published 5 October 2015