Sutured manifolds and generalized Thurston norms
Recently, \textit{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.
scientific article; zbMATH DE number 4102087
scientific article