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For closed 3–manifolds, Heegaard Floer homology is related to the Thurston norm through
results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3–manifold
, there is a bijection between
vertices of the
polytope
carrying the group
and the faces of the Thurston norm unit ball that correspond to fibrations of
over the unit circle. Moreover, the Thurston norm unit ball of
is dual to the
polytope of
.
We prove a similar bijection and duality result for a class of 3–manifolds
with boundary called sutured manifolds. A sutured manifold is essentially
a cobordism between two possibly disconnected surfaces with boundary
and
. We
show that there is a bijection between vertices of the sutured Floer polytope carrying the
group
and equivalence classes of taut depth-one foliations that form the foliation cones of
Cantwell and Conlon. Moreover, we show that a function defined by Juhász, which
we call the geometric sutured function, is analogous to the Thurston norm in this
context. In some cases, this function is an asymmetric norm and our duality
result is that appropriate faces of this norm’s unit ball subtend the foliation
cones.
An important step in our work is the following fact: a sutured manifold admits a
fibration or a taut depth-one foliation whose sole compact leaves are exactly the connected
components of
and
,
if and only if, there is a surface decomposition of the sutured manifold resulting in a
product manifold.
