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Eliashberg and Thurston showed that a taut foliation may be deformed to tight (positive and negative) contact structures. Vogel proved that for a taut foliation without torus leaves, the associated contact structures are unique up to isotopy. My question is roughly: if a taut foliation is deformed through taut foliations without torus leaves, are the associated contact structures unique up to isotopy?

To be more specific, suppose one has a taut foliation carried by an oriented branched surface not carrying any tori, so that the complement is a product taut sutured manifold. Any oriented lamination carried by the branched surface extends to a taut foliation, which is presumably deformation equivalent to any other one. Examples of such laminar branched surfaces without vertices occur for veering triangulations by deforming the oriented triangulation to be generic (double along the torus boundary to get a closed manifold). enter image description here Then can one associate unique (up to isotopy) positive and negative tight contact structures (in the sense of Eliashberg-Thurston) to such branched surfaces?

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  • $\begingroup$ Suppose that $M$ has Thurston norm ball with adjacent fibered faces. Suppose that $F$ is a taut surface on the boundary of both. So $F$ is the compact leaf of two distinct depth one foliations. Is there a way to interpolate between these? (Say by using “shifts” along annuli?) If so that kind of “connectedness” result, plus your desired uniqueness result would combine in a way that seems too strong… $\endgroup$
    – Sam Nead
    Commented Jul 9, 2022 at 8:32
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    $\begingroup$ I think these two depth one foliations will be carried by distinct branched surfaces. $\endgroup$
    – Ian Agol
    Commented Jul 9, 2022 at 12:40
  • $\begingroup$ I think that a pair of depth one foliations may be deformation equivalent without both being carried by a single branched surface. One tool to produce such is "shifts in annuli". Now, by definition, annulus shifts only effect the foliation in an $I$--bundle region. But every finite depth foliation ends with a product, so perhaps there is some hope that (a) changing weights in branched surfaces and (b) annulus shifts suffice to realise all deformations. $\endgroup$
    – Sam Nead
    Commented Jul 10, 2022 at 8:43
  • $\begingroup$ Okay, so maybe my “rough” question might have a negative answer. So then the question is what deformations of taut foliations preserve the contact structure. But the question still stands as stated for families of taut foliations carried by a single branched surface. $\endgroup$
    – Ian Agol
    Commented Jul 10, 2022 at 17:54

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