Introduction

The purpose of this site is to record forward-looking problems of interest around Floer homology as used in low-dimensional topology. The focus of this list is meant to be narrower than that of the Kirby list, the problems more approachable.

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Knot Floer homology and pseudo-Anosov flows

Problem:  How is the knot Floer homology of a hyperbolic knot in its next-to-top Alexander grading related to the orbits of pseudo-Anosov flows 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 This follows from an unpublished result of Gabai, a proof of which was outlined by Mosher; see this paper of Landry--Tsang for some of the history.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 problem has been answered . 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-Anosov repr...
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The 4-holed sphere and tight contact structures

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Problem: Classify open books with page the 4-holed sphere which correspond to tight contact structures. Recall that an open book is a pair $(S,h)$ where $S$ is a compact, oriented surface with boundary, and $h$ is a diffeomorphism of $S$ which restricts to the identity on $\partial S$. We will refer to $h$ as the monodromy  of the open book and $S$ as the page. A construction of Thurston-Winklenkemper assigns a contact 3-manifold $(M_{S,h},\xi_{S,h})$ to this open book, and Giroux proved that this assignment defines a bijective correspondence between open books up to positive stabilization and contact 3-manifolds up to contactomorphism (see here and here for recent proofs of Giroux's result). This problem is concerned with understanding how topological features of the monodromy are reflected in geometric features of the contact structure in Giroux's correspondence. For instance, Giroux proved that a contact 3-manifold is Stein fillable if and only if there is a correspondin...
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Bordered HF and Fox-Milnor for slice fibered knots

Problem: Categorify the Fox-Milnor criterion for slice fibered knots. If $K\subset S^3$ is a slice knot (either topologically or smoothly) then its Alexander polynomial satisfies $$\Delta_K(t) = f(t)f(t^{-1})$$ for some $f\in \mathbb{Z}[t,t^{-1}]$. This is called the Fox-Milnor criterion . Knot Floer homology is said to categorify  the Alexander polynomial; more precisely, its graded Euler characteristic satisfies $$\chi(\widehat{\mathit{HFK}}(K)) = \bigoplus_{m,a}(-1)^m t^a \dim \widehat{\mathit{HFK}}(K)  = \Delta_K(t).$$ It is therefore natural to ask (the admittedly vague question of) whether there are extra features of the knot Floer homology of slice knots which recover the Fox-Milnor criterion for the Alexander polynomial. One would hope moreover that these features provide a stronger obstruction to slicing a knot than the classical Fox-Milnor criterion. This is one thing that could be meant by the charge to categorify  Fox-Milnor.  It's not at all clear howev...
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