The 4-holed sphere and tight contact structures

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 corresponding open book whose monodromy is a composition of positive Dehn twists. Given a page $S$, one can similarly ask for a classification of those monodromies $h$ for which $\xi_{S,h}$ is tight. The answer is only known when $S$ is a 1-holed torus (see here and here) or an $n$-holed sphere with $n\leq 3$. In these cases, $\xi_{S,h}$ is tight if and only if $h$ is right-veering if and only if the Heegaard Floer contact invariant of $\xi_{S,h}$ is nonzero. 

This result is trivial for the $n$-holed sphere with $n\leq 3$ since the mapping class group is so simple in this case (it is generated by Dehn twists about curves parallel to the boundary components). The next simplest case is the $4$-holed sphere (or the $2$-holed torus, though that seems harder). The mapping class group of the $4$-holed sphere is generated by Dehn twists about the four curves $d_1, d_2, d_3, d_4$ parallel to the boundary components shown in red below, and the three curves $a,b,c$ which encircle pairs of boundary components shown in blue below. 

The relations are generated by the fact that Dehn twists about disjoint curves commute together with the lantern relation, which says that by $$\tau_a\tau_b\tau_c =\tau_{d_1}\tau_{d_2}\tau_{d_3}\tau_{d_4}.$$ 

Ito-Kawamuro proved that $\xi_{S,h}$ is tight whenever $S$ has genus zero and the fractional Dehn twist coefficients of $h$ are greater than one for every boundary component of $S$. On the other hand, if any of the fractional Dehn twist coefficients are non-positive (and $h$ is pseudo-Anosov) then $\xi_{S,h}$ is overtwisted, by Honda-Kazez-Matic. So it more or less suffices to consider the case in which all fractional Dehn twist coefficients are positive but are not all greater than one. 

We note that unlike for the 1-holed torus and for the sphere with fewer than 4 holes, it is not true for $S$ the 4-holed sphere that $\xi_{S,h}$ is tight whenever $h$ is right-veering, as shown for example by Lekili. It is also not true that if $\xi_{S,h}$ is tight then the Heegaard Floer contact invariant is nonzero in this case. Indeed, consider the monodromy given by $$h = \tau_{d_1}^2\tau_{d_2}^2\tau_{d_3}^2\tau_{d_4}^2a^{-4}.$$ One can check that all fractional Dehn twist coefficients are greater than one and hence $\xi_{S,h}$ is tight by Ito-Kawamuro. On the other hand, let $(S',h')$ be the induced open book where $S'$ is the 3-holed sphere obtained by capping off one of the boundary components. It is easy to see that the induced monodromy $h'$ is not right-veering. Therefore, $\xi_{S',h'}$ is overtwisted, by Honda-Kazez-Matic, and so the contact invariant $$c\big(\xi_{S',h'}\big)\in\widehat{\mathit{HF}}(-Y_{S',h'})$$ is zero. Note that $Y_{S',h'}$ is obtained via 0-surgery on a curve in $Y_{S,h}$ corresponding to the boundary component that is capped off. The associated trace cobordism $W$ induces a map $$F_W:\widehat{\mathit{HF}}(-Y_{S',h'}) \to \widehat{\mathit{HF}}(-Y_{S,h})$$ and Baldwin proved that for some $\mathfrak{s}\in\mathrm{Spin}^c(W)$, the component $F_{W,s}$ of this map sends $$c\big(\xi_{S',h'}\big)\mapsto c\big(\xi_{S,h}\big),$$ which implies that the latter class vanishes.





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