Knot Floer homology and pseudo-Anosov flows
Problem: Is the knot Floer homology of a hyperbolic knot in its next-to-top Alexander grading related to the number of orbits of the pseudo-Anosov flow on its complement which generate the first homology of the complement? If $K\subset Y$ is a hyperbolic knot then its complement admits a pseudo-Anosov flow. This follows from an unpublished result of Gabai, a proof of which was outlined by Mosher; see this paper of Landry--Tsang for some of the history. If $K$ is fibered, then the suspension flow of its monodromy is pseudo-Anosov, and in this case, the answer to the problem is yes. Indeed, work of Ni and independently Ghiggini--Spano , which relies on (1) a relationship between periodic Floer homology and monopole Floer homology due to Lee--Taubes , and (2) an isomorphism between monopole and Heegaard Floer homology due to Kutluhan--Lee--Taubes , shows that if $K$ is fibered then $$\widehat{\mathit{HFK}}(Y,K,g-1)-1$$ is an upper bound for the number of fixed points of the pseudo-...