Alternate Heegaard genus bounds distance

Scharlemann, Martin; Tomova, Maggy

doi:10.2140/gt.2006.10.593

Cited by 103 publications

(107 citation statements)

References 10 publications

(28 reference statements)

Supporting

Mentioning

104

Contrasting

Order By: Relevance

“…Distance inequalities analogous to Theorem 1.2, in the setting of Heegaard splittings rather than surface bundles, appear in Hartshorn [Har02], and then more fully in Scharlemann-Tomova [ST06]. Bachman-Schleimer [BS05] use Heegaard surfaces to give bounds on the curve-complex translation distance of the monodromy of a fibering.…”

Section: Hierarchies Of Pocketsmentioning

confidence: 99%

Fibered faces, veering triangulations, and the arc complex

Minsky

Taylor

2017

Geom. Funct. Anal.

View full text Add to dashboard Cite

show abstract

Section: Hierarchies Of Pocketsmentioning

confidence: 99%

Fibered faces, veering triangulations, and the arc complex

Minsky

Taylor

2017

Geom. Funct. Anal.

View full text Add to dashboard Cite

show abstract

“…To show that the 1-tunnel complexity of S 3 is unbounded, we use the following very strong result given by Scharlemann and Tomova [25] and Johnson [15]. (Note that Scharlemann and Tomova [25] proved a much more general case and Johnson [15] deduced the following from it.)…”

Section: Unboundedness Of Tunnel Complexitymentioning

confidence: 99%

“…(Note that Scharlemann and Tomova [25] proved a much more general case and Johnson [15] deduced the following from it.) Theorem 4.6 [25; 15] Let be an unknotting tunnel of a tunnel number one knot K S 3 .…”

Section: Unboundedness Of Tunnel Complexitymentioning

confidence: 99%

Tunnel complexes of 3–manifolds

Koda

2011

Algebr. Geom. Topol.

View full text Add to dashboard Cite

For each closed 3-manifold M and natural number t , we define a simplicial complex T t .M /, the t -tunnel complex, whose vertices are knots of tunnel number at most t . These complexes have a strong relation to disk complexes of handlebodies. We show that the complex T t .M / is connected for M the 3-sphere or a lens space. Using this complex, we define an invariant, the t -tunnel complexity, for tunnel number t knots. These invariants are shown to have a strong relation to toroidal bridge numbers and the hyperbolic structures.

show abstract

“…The paper [68] shows that for a closed orientable 3-manifold containing an incompressible surface of genus g, any Heegaard splitting has Hempel distance at most 2g. See also [158] …”

Section: Heegaard Splittings and Hempel Distance Of 3-manifoldsmentioning

confidence: 99%

Curve Complexes Versus Tits Buildings: Structures and Applications

2011

Buildings, Finite Geometries and Groups

View full text Add to dashboard Cite

Abstract. Tits buildings ∆ Q (G) of linear algebraic groups G defined over the field of rational numbers Q have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of G(Q). Curve complexes C(Sg,n) of surfaces Sg,n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical building. Another important class of buildings consists of Euclidean buildings, for example, the BruhatTits buildings of linear algebraic groups defined over local fields. In this chapter, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and at the same time suggest methods to solve them. Furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.

show abstract

Alternate Heegaard genus bounds distance

Cited by 103 publications

References 10 publications

Fibered faces, veering triangulations, and the arc complex

Fibered faces, veering triangulations, and the arc complex

Tunnel complexes of 3–manifolds

Curve Complexes Versus Tits Buildings: Structures and Applications

Contact Info

Product

Resources

About