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Sutured manifolds and generalized Thurston norms. (English) Zbl 0673.57015
Recently, D. Gabai has developed the theory of codimension one foliations on sutured 3-manifolds [J. Differ. Geom. 18, 445-503 (1983; Zbl 0533.57013); ibid. 26, 461-478 and 479-536 (1987; Zbl 0627.57012 and Zbl 0639.57008)]. Using this theory, he has answered positively the Poenaru conjecture, the Property R conjecture, the superadditivity of knot genus under band connected sum, the Property P conjecture for sattelite knots. Eventually it became clear that the theory of sutured 3- manifolds could be developed from a different combinatorial perspective, without foliations. The author gives an account of these developments. The absence of foliations simplifies the proofs. A central role is played by a generalization of the Thurston norm, the so-called $$\beta$$-norm, where $$\beta$$ is a properly embedded 1-dimensional complex inside a 3- manifold.
Reviewer: S.V.Matveev

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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