Heegaard splittings and the pants complex

Johnson, Jesse

doi:10.2140/agt.2006.6.853

Cited by 13 publications

(32 citation statements)

References 7 publications

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“…The concepts presented may be extended to manifolds with boundary or non-seperating 2-spheres, but we omit this extension for simplicity. For further reference, see [10].…”

Section: The Distance Of a Bridge Splittingmentioning

confidence: 99%

“…Similarly, the pants distance D P (Σ) is the minimum distance D P (v, w) between vertices v, w ∈ P (Σ K ) defining (V, α) and (W, β), respectively. These definitions resemble the definitions of the distance and pants distance of a Heegaard splitting of a manifold introduced by Johnson [10].…”

Section: The Distance Of a Bridge Splittingmentioning

confidence: 99%

“…In [10], Johnson invokes the pants complex P (Σ) and defines the related dual curve complex C * (Σ) of a Heegaard surface Σ for a 3-manifold M . He proves that these complexes can be used to assign an integer complexity to Σ and that this complexity converges to an integer A(M ) or A P (M ) (depending on the complex used) upon taking a sequence of stabilizations of Σ.…”

Section: Introductionmentioning

confidence: 99%

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Bridge and pants complexities of knots

Zupan

2012

Journal of the London Mathematical Society

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We modify an approach of Johnson to define the distance of a bridge splitting of a knot K in a 3-manifold M using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of (M, K). We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of K. Mathematics Subject Classification 57M25, 57M27, 57M50 (primary). 44 ALEXANDER ZUPAN Theorem 1.1. Let K be a knot in a closed, orientable 3-manifold M, where M has no S 2 × S 1 summands, let Σ be a splitting surface for (M, K), and let Σ h c be an (h, c)-stabilization of Σ. Then the limits lim h,c→∞ B(Σ) and lim h,c→∞ B P (Σ) exist. Moreover, they do not depend on Σ and thus define invariants B(M, K) and B P (M, K) of the pair (M, K).The pants complex P (Σ) of a surface Σ is itself an interesting object of study; for instance, Brock has shown that it is quasi-isometric to the Teichmüller space T (Σ) equipped with the Weil-Petersson metric [2], and Souto has revealed other surprising connections between pants distance and the geometry of certain 3-manifolds [23]. We exploit a theorem of Guéritaud and Futer [3] to prove the following theorem.Theorem 1.2. Suppose that K is a hyperbolic 2-bridge knot, Σ is a (0, 2)-splitting surface for K, v 3 is the volume of a regular ideal tetrahedron, and v 8 is the volume of a regular ideal octahedron. ThenThe paper proceeds in the following manner: in Section 2, we provide background information and include a proof of the analog of the Reidemeister-Singer theorem for bridge splittings in arbitrary manifolds. In Section 3, we introduce the curve, dual-curve, and pants complexes, and in Section 4 we define the distance of a bridge splitting and prove several basic facts about this distance. In Section 5, we use the distance of the previous section to define bridge and pants complexity, and prove the main theorem. In Section 6, we demonstrate several properties of these new invariants and, in Section 7, we define the concept of a critical splitting to provide explicit calculations in Section 8. Finally, in Section 9, we discuss connections between pants distance and hyperbolic volume, and in Section 10 we include several interesting open questions.

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“…The concepts presented may be extended to manifolds with boundary or non-seperating 2-spheres, but we omit this extension for simplicity. For further reference, see [10].…”

Section: The Distance Of a Bridge Splittingmentioning

confidence: 99%

Section: The Distance Of a Bridge Splittingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bridge and pants complexities of knots

Zupan

2012

Journal of the London Mathematical Society

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“…In [96], the curve complex C(S g,n ) is replaced by the pants complex of S g,n and a similar distance function is introduced. The paper [97] contains some results relating the bridge number of hyperbolic knots and an invariant defined in terms of the distance function on C(S g,n ).…”

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

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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.

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“…This appears to be no simple task. In particular, the number of crossings in the graphic is related to the pants distance of each of the two Heegaard splittings (see [7]) and therefore can be arbitrarily high.…”

Section: Theorem There Is a Common Stabilization Of Genus Less Than mentioning

confidence: 99%