Knot Floer homology and twist coefficients

Question: What information does the knot Floer complex of a fibered knot record about the fractional Dehn twist coefficient of its monodromy?

Let $\phi$ be a diffeomorphism of a surface $S$ with connected boundary, which restricts to the identity on the boundary. The fractional Dehn twist coefficient of $\phi$, denoted by $c(\phi)$, is a measure of the twisting near the boundary of the surface in the free isotopy between $\phi$ and its Nielsen--Thurston representative. It was defined by Honda-Kazez-Matic as a quantification of their notion of right-veering surface diffeomorphisms, which in turn came from their study of contact structures on 3-manifolds (a contact structure is tight if and only if all of its supporting open books have right-veering monodromy). It is also closely related to and inspired by Gabai's notion of degeneracy slope.

Suppose for example that $\phi$ is freely-isotopic to a pseudo-Anosov map $\phi_0$. The latter fixes a pair of transverse, singular, transversely measured foliations $\mathcal{F}_s \pitchfork \mathcal{F}_u$ on the surface, contracting the measure of the stable foliation $\mathcal{F}_s$ by some real number $\lambda >1$ and expanding the measure of the unstable foliation $\mathcal{F}_u$ by the same factor. The boundary of $S$ intersects a finite number of singular leaves of each foliation called prongs. If $\mathcal{F}_s$ has $n$ prongs meeting the boundary then $$c(\phi) = k/n$$ for some integer $k$. Honda--Kazez--Matic showed that $\phi$ is right-veering in this case if and only if $k>0$. Moreover, if $\phi$ is the monodromy of a hyperbolic fibered knot in the 3-sphere (or any manifold with finite $\pi_1$) then work of Gabai-Oertel implies that $k \in \{-1,0,1\}$.

Foundational work of Ghiggini and Ni shows that knot Floer homology detects whether a knot is fibered. More recent work of Baldwin-Hu-Sivek, Ghiggini-Spano, and Ni (see also this) shows that the knot Floer homology of a fibered knot contains information about the fixed points of its monodromy. This was crucial in the proofs by Baldwin--Hu--Sivek and Farber-Reinoso-Wang that Khovanov and knot Floer homology detect the cinquefoil knot. In a related vein, Baldwin-Ni-Sivek recently proved that the knot Floer complex of a fibered knot detects whether its monodromy is right-veering. Together the surgery dual formula of Hedden-Levine, this shows that the $UV=0$ version of the knot Floer homology of a fibered knot determines the fractional Dehn twist coefficient of its monodromy to within an integer.

The question at the top is asking for a refinement of the latter work. Here is some more motivation: if the knot Floer complex detects fractional Dehn twist coefficients then knot Floer homology detects each $T(2,2g+1)$ torus knot. This could have consequences for important problems in Dehn surgery like the chirally cosmetic surgery conjecture, which predicts in particular that these torus knots are the only knots with chirally cosmetic surgeries of the same sign.

The reasoning is as follows: suppose $$\widehat{\mathit{HFK}}(K) \cong \widehat{\mathit{HFK}}(T(2,2g+1)).$$ Then $K$ is a genus $g$ fibered knot with right-veering monodromy. It suffices to prove that $K$ is a torus knot. One can use the Alexander polynomial to show that $K$ isn't a satellite knot, so it remains to rule out the possibility that $K$ is hyperbolic. If $K$ is hyperbolic then its monodromy $\phi$ is freely-isotopic to a pseudo-Anosov map whose stable foliation meets the boundary of the fiber surface in $n$ prongs. Euler characteristic considerations imply that $n \leq 4g-2$. From the results cited above, we therefore have that $$c(\phi) >1/(4g-2).$$ On the other hand, the fractional Dehn twist coefficient of the monodromy of $T(2,2g+1)$ is $1/(4g+2).$ So an affirmative answer to the question above would yield a contradiction.

We note that knot Floer homology can't detect fractional Dehn twist coefficient on the nose in all cases, as the torus knot $T(3,4)$ has the same knot Floer complex as the $(2,3)$-cable of the right-handed trefoil, while the monodromies of these two knots have twist coefficients $1/12$ and $1/6$, respectively, but it might contain more refined information about twist coefficients in the pseudo-Anosov case. Along these lines, it is conjectured that knot Floer homology distinguishes hyperbolic knots from torus knots and cables thereof.

The fixed point Floer homology of the monodromy $\phi$ of a fibered knot $K$ is equal to its degree-1 periodic Floer homology, which is encoded in the next-to-top Alexander grading of knot Floer homology, $$\widehat{\mathit{HFK}}(K,g-1),$$ as made precise by Ghiggini and Ni. More generally, the degree-k periodic Floer homology, suitably interpreted, should satisfy $$\mathit{PFH}_k(\phi) \cong \widehat{\mathit{HFK}}(K,g-k)$$ (an analogous result is known to hold for closed mapping tori, by work of Lee-Taubes and the equivalence of Heegaard and monopole Floer homology). On possible route to answering the questions above is to understand how to compute $\mathit{PFH}_k(\phi)$ for higher $k$, and how it is related to fixed points of powers of $\phi$.

Although not directly related to the problem above, we note for the reader's benefit that Hedden-Mark proved a different connection between Floer homology and twist coefficients; namely, that the dimension of the reduced Heegaard Floer homology of a closed 3-manifold $Y$ bounds the fractional Dehn twist coefficient of every fibered knot in $Y$. 


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