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.
|Data di pubblicazione:||2014|
|Titolo:||REVIEW OF: "Taylor Scott A., Band-taut sutured manifolds, Algebr. Geom. Topol. 14, No. 1, 157-215 (2014)". [DE062342080]|
|Autori:||Casali, Maria Rita|
|Autori del volume:||Taylor, Scott A.|
|Appare nelle tipologie:||Recensione in Rivista|
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