A well known theorem of Lackenby ([Math. Ann. 308, No.4, 615-632 (1997; Zbl 0876.57015)]) relates Dehn surgery properties of a knot to the intersection between the knot and essential surfaces in the 3-manifold. In the paper under review, the author extends Lackenby’s Theorem to the case of 2-handles attached to a sutured 3-manifold along a suture, by determining the relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2-handle in the manifold after attaching a 2-handle along the suture. The author makes use of Scharlemann’s combinatorial version of sutured manifold theory ([J. Differ. Geom. 29, No.3, 557-614 (1989; Zbl 0673.57015)]) and takes inspiration from Gabai’s proof that, under suitable hypotheses, there is at most one way to fill a torus boundary component of a 3-manifold so that the Thurston norm decreases ([J. Differ. Geom. 26, 461-478 (1987; Zbl 0627.57012)]). On the other hand, in order to prove the theorem, band-taut sutured manifolds are introduced and band-taut sutured manifold hierarchies are proved to exist. As an application, the paper shows that tunnels for tunnel number one knots or links in any 3-manifold can be isotoped to lie on a branched surface corresponding to a certain taut sutured manifold hierarchy of the knot or link exterior. Other interesting applications are contained in [Trans. Am. Math. Soc. 366, No. 7, 3747-3769 (2014; Zbl 06303179)], where band sums are proved to satisfy the cabling conjecture, and new proofs that unknotting number one knots are prime and that genus is superadditive under band sum are obtained.

REVIEW OF: "Taylor Scott A., Band-taut sutured manifolds, Algebr. Geom. Topol. 14, No. 1, 157-215 (2014)". [DE062342080] / Casali, Maria Rita. - In: EXCERPTS FROM ZENTRALBLATT MATH. - ISSN 2190-3484. - ELETTRONICO. - Zbl pre06234208:(2014), pp. .-...

### REVIEW OF: "Taylor Scott A., Band-taut sutured manifolds, Algebr. Geom. Topol. 14, No. 1, 157-215 (2014)". [DE062342080]

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*CASALI, Maria Rita*

##### 2014

#### Abstract

A well known theorem of Lackenby ([Math. Ann. 308, No.4, 615-632 (1997; Zbl 0876.57015)]) relates Dehn surgery properties of a knot to the intersection between the knot and essential surfaces in the 3-manifold. In the paper under review, the author extends Lackenby’s Theorem to the case of 2-handles attached to a sutured 3-manifold along a suture, by determining the relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2-handle in the manifold after attaching a 2-handle along the suture. The author makes use of Scharlemann’s combinatorial version of sutured manifold theory ([J. Differ. Geom. 29, No.3, 557-614 (1989; Zbl 0673.57015)]) and takes inspiration from Gabai’s proof that, under suitable hypotheses, there is at most one way to fill a torus boundary component of a 3-manifold so that the Thurston norm decreases ([J. Differ. Geom. 26, 461-478 (1987; Zbl 0627.57012)]). On the other hand, in order to prove the theorem, band-taut sutured manifolds are introduced and band-taut sutured manifold hierarchies are proved to exist. As an application, the paper shows that tunnels for tunnel number one knots or links in any 3-manifold can be isotoped to lie on a branched surface corresponding to a certain taut sutured manifold hierarchy of the knot or link exterior. Other interesting applications are contained in [Trans. Am. Math. Soc. 366, No. 7, 3747-3769 (2014; Zbl 06303179)], where band sums are proved to satisfy the cabling conjecture, and new proofs that unknotting number one knots are prime and that genus is superadditive under band sum are obtained.##### Pubblicazioni consigliate

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