KazuhiroIchihara Hyperbolicvolumesandpantsdistancesfortwo-bridgeknots

3rd East Asian School of Knots and Related Topics, Feb.8.2007

Hyperbolic volumes and

pants distances

for two-bridge knots

市原一裕

Kazuhiro Ichihara

大阪産業大学 Osaka Sangyo University

§1. Motivation

“Thm.” [Brock-Souto, unpublished]

For ^∀g, ^∃ constant L_g > 0 s.t.

L⁻_g ¹ · δ_P(U ∪ V ) ≤ vol(M) ≤ L_g · δ_P(U ∪ V ) for ^∀strongly irreducible Heegaard splitting M = U ∪ V of genus g for ^∀closed orientable hyperbolic 3-manifold M.

This is included in Note:

Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds.

by Juan Souto as Theorem 8.4 (Brock-Souto) . The note is available at Souto’s web page;

http://www.math.uchicago.edu/~juan/papers.html

§2. Pants graph;

(by using which δ_P(U ∪ V ) is defined) Let Σ be a compact orientable surface.

Def. (Pants decomposition)

A pants decomposition of Σ means a set of disjoint simple closed curves on Σ cutting it into pairs of pants.

Def. (Pants graph)

The graph constructed as follows is called the pants graph P(Σ) of Σ.

• vertex ←→ isotopy class

of a pants decomposition

• edge ←→ single “S-move” or “A-move”

§3. Result

Thm. [I.]

Let K be a hyperbolic two-bridge knot in S³. Then we have

v₈

2 (δ_P(T₁ ∪ T₂) − 3) ≤ vol(S³ − K) ≤ 2v₈ (δ_P(T₁ ∪ T₂) − 2)

for ^∀two-bridge decomposition (S³, K) = T₁ ∪ T₂. Here

v = vol(ideal regular octahedron) .

= 3.66 . . .

Invariant δ_P(T₁ ∪ T₂)

Let (S³, K) = T₁ ∪ T₂ be a bridge decomposition for K with bridge decomposing sphere S.

Let Σ = S ∩ Exterior(K) and T_i = (B_i, t_i).

Then we define δ_P(T₁ ∪ T₂) as min

½

d_P(Σ)([P₁], [P₂]) | [P₁] ∈ H(T₁), [P₂] ∈ H(T₂)

¾

where, for i = 1, 2, H(T_i) denotes;







[P ] ∈ P(Σ)

¯¯¯¯

¯¯¯¯

∃a union ∆ of disks in B_i − t_i with

∂∆ ⊂ P s.t. (B_i − t_i) − ∆ =∼ solid tori







