Band-taut sutured manifolds

Taylor, Scott A¹

¹Department of Mathematics and Statistics, Colby College, 5832 Mayflower Hill, Waterville, ME 04901, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 1, pp. 157-215

Résumé

Attaching a 2–handle to a genus two or greater boundary component of a 3–manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting 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. We use this result to show 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. In a subsequent paper, we use the theorem to prove that band sums satisfy the cabling conjecture, and to give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. To prove the theorem, we introduce band-taut sutured manifolds and prove the existence of band-taut sutured manifold hierarchies.

Détail

DOI : 10.2140/agt.2014.14.157

Classification : 57M50, 57M25
Keywords: $3$–manifold, sutured manifold, $2$–handle addition, tunnel

Bibliographie

[1] J Cantwell, L Conlon, The sutured Thurston norm

[2] D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445

[3] D Gabai, Foliations and the topology of $3$–manifolds, II, J. Differential Geom. 26 (1987) 461

[4] D Gabai, Foliations and the topology of $3$–manifolds, III, J. Differential Geom. 26 (1987) 479

[5] E Kalfagianni, Surgery $n$–triviality and companion tori, J. Knot Theory Ramifications 13 (2004) 441

[6] M Lackenby, Dehn surgery on knots in $3$–manifolds, J. Amer. Math. Soc. 10 (1997) 835

[7] M Lackenby, Surfaces, surgery and unknotting operations, Math. Ann. 308 (1997) 615

[8] M Lackenby, Upper bounds in the theory of unknotting operations, Topology 37 (1998) 63

[9] M Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom. 29 (1989) 557

[10] M Scharlemann, Producing reducible $3$–manifolds by surgery on a knot, Topology 29 (1990) 481

[11] M Scharlemann, A Thompson, Unknotting tunnels and Seifert surfaces, Proc. London Math. Soc. 87 (2003) 523

[12] S A Taylor, Comparing $2$–handle additions to a genus $2$ boundary component, to appear in Trans. AMS

[13] S A Taylor, Boring split links and unknots, PhD thesis, University of California, Santa Barbara (2008)