Additivity of tunnel number for small knots

Schultens, Jennifer

doi:10.1007/s000140050131

Cited by 55 publications

(34 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A strongly irreducible Heegaard splitting can be isotoped so that its splitting surface, S, intersects an incompressible surface, P, only in curves essential in both S and P. This is a deep fact and is proven, for instance, in [14,Lemma 6]. This fact, together with the fact that incompressible surfaces can be isotoped to meet only in essential curves, establishes the following:…”

Section: Definition 14mentioning

confidence: 65%

Heegaard Genus Formula for Haken Manifolds

Schultens

2006

Geom Dedicata

View full text Add to dashboard Cite

Suppose M is a compact orientable 3-manifold and Q ⊂ M a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of M -Q with respect to the path metric by M 1 , . . . , M k . Denote the smallest possible genus of a Heegaard splitting of M, or M j respectively, for which ∂ M, or ∂ M j respectively, is contained in one compression body by g(M, ∂ M), or g(M j , ∂ M j ) respectively. Denote the maximal number of non-parallel essential annuli that can be simultaneously embedded in M j by n j . ThenKeywords Heegaard genus · Essential surface · Genus formula Mathematics Subject Classification (2000) 57N10Heegaard splittings have long been used in the study of 3-manifolds. One reason for their continued importance in this study is that the Heegaard genus of a compact 3-manifold has proven to capture the topology of the 3-manifold more accurately than many other invariants. In particular, it provides an upper bound for the rank of the fundamental group of the 3-manifold, and this upper bound need not be sharp, as seen in the examples provided by Boileau and Zieschang [1]. We here prove the following: Let M be a compact orientable 3-manifold and Q ⊂ M an orientable boundary incompressible essential surface. Denote the completions of the components of M − Q with respect to the path metric by M 1 , . . . , M k . Denote the smallest possible genus of a Heegaard splitting of M, or M j respectively, for which ∂ M, or ∂ M j respectively, is contained in one compression body by g(M, ∂ M) or g(M j , ∂ M j ) respectively. Here g(M, ∂ M) is called the relative genus of M. Denote the maximal number of non parallel essential annuli that can be simultaneously embedded in M j by n j . Then

show abstract

Section: Definition 14mentioning

confidence: 65%

Heegaard Genus Formula for Haken Manifolds

Schultens

2006

Geom Dedicata

View full text Add to dashboard Cite

show abstract

“…See [14]. Now we give an extension of Schultens's lemma as follow: Proof By Haken's lemma, M is irreducible.…”

Section: Definition 22mentioning

confidence: 99%

The amalgamation of high distance Heegaard splittings is always efficient

Kobayashi

Qiu

2008

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…Since c > 0, X .c/ admits an essential torus T that gives the decomposition X .c/ D X 0 [ T Q .c/ , where X 0 Š X and Q .c/ is a c -times punctured annulus cross S 1 . Since T is incompressible and † is strongly irreducible, we may isotope † so that every component of † \ T is essential in both surfaces (see, for example, Schultens [26,Lemma 6]). Isotope † to minimize j † \ T j subject to this constraint.…”

Section: Decomposing Xmentioning

confidence: 99%