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Is there a result relating sutured manifolds and surfaces of minimal genus? perhaps someone has a very clever point of view of these two notions that can share.

In other matters, do we know how to construct minimal genus surfaces out of (weakly) incompressible surfaces? I have been searching for a construction but not lucky so far.

Thanks a lot! C

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Sutured manifolds and sutured manifold hierarchies were defined for the very purpose of studying surfaces of minimal genus within a homology class. See the original papers of Gabai on this topic, starting with

  • "Foliations and the topology of 3-manifolds." Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 77–80.

The main theorem of this paper is that if $M$ is a compact oriented 3-manifold and if $c \in H_2(M;\partial M;Z)$ then for any minimal genus properly embedded surface $S \subset M$ representing $c$ (e.g. a minimal genus Seifert surface in a knot complement) the surface $S$ is a leaf of some taut, transversely oriented foliation of $M$. The method of proof is to construct a sutured manifold hierarchy starting by cutting $M$ open along $S$.

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  • $\begingroup$ Thanks a lot. I am revising that bibliography. Everything points out that Gabai's paper has the answer to my trivial question. Thanks also for summarizing a bit the work by Gabai, I really appreciate it. $\endgroup$
    – Carlos Barrod
    Commented May 14, 2014 at 20:55

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