Hyperbolic volumes and pants distances for two-bridge knots

「リーマン面・不連続群論」研究集会, Feb.1.2007

Hyperbolic volumes and

pants distances

for two-bridge knots

市原一裕

Kazuhiro Ichihara

§ 1. Back ground

Classification of 3-manifolds Every closed orientable 3-manifold is;

• Reducible (containing essential 2-sphere),

• Toroidal (containing essential torus),

• Seifert fibered (foliated by circles), or

• Hyperbolic (admitting Riem.metric of curv. − 1) .

By the Geometrization Conjecture raised by Thurston, (late ’70s).

& Established(?) by Perelman (2002?).

What’s the NEXT?

• Attack the remaining Open Problems.

(e.g., Virtually Haken Conjecture,

“Heegaard genus VS rank of π

” problem, etc . . . )

• Study the Relationships between 3-manifolds.

(e.g., degree-one map, Dehn surgery, etc . . . )

• Relate Geometric & Topological invariants.

(e.g., Volume conjecture (for knots), etc . . . )

§ 2. Motivation

Let M be a closed orientable 3-manifold.

“Thm.” [Brock-Souto]

Suppose that M is hyperbolic.

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 .

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

Heegaard splitting

Definitions

• A Heegaard splitting of M means the de- composition M = U ∪ V into two handlebod- ies U and V .

• The surface appearing as U ∩ V is called the Heegaard surface.

• The genus of a Heegaard surface is called

the genus of the Heegaard splitting.

A Heegaard splitting M = U ∪ V is called;

• reducible if ^∃ essential disks D _U ⊂ U & D _V ⊂ V with ∂D _U = ∂D _V .

• weakly reducible if ^∃ essential disks D _U ⊂ U

& D _V ⊂ V with ∂D _U ∩ ∂D _V = ∅ .

• strongly irreducible if it is not weakly re-

ducible.

Pants complex

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 complex)

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

• vertex ←→ isotopy class

of a pants decomposition

• edge ←→ single “S-move” or “A-move”

Invariant δ _P ( U ∪ V )

Let M = U ∪ V be a Heegaard splitting of M with Heegaard surface S .

Then they define δ _P ( U ∪ V ) as min

½

d _{P ( S )} ([ P _U ] , [ P _V ]) | [ P _U ] ∈ H ( U ) , [ P _V ] ∈ H ( V )

¾

where, for the handlebody U , H ( U ) denotes;









[ P ] ∈ P ( ∂U )

¯¯¯¯

¯¯¯¯

∃ a union ∆ of disks in U with

∂ ∆ ⊂ P s.t. U − ∆ = ∼ solid tori









In the same way, H ( V ) is set for V .

§ 3. Result Thm. [I.]

Let K be a hyperbolic two-bridge knot in the 3-sphere S ³ . Then we have

v₈

2

(δ

( T

∪ T

) − 3) ≤ vol( S

− K ) ≤ 10 v

(2δ

( T

∪ T

) − 3)

for ^∀ two-bridge decomposition ( S ³ , K ) = T ₁ ∪ T ₂ . Here

v ₈ = vol (ideal regular octahedron) .

= 3 . 66 . . .

Let K be a knot in S ³

Definitions.

• A bridge decomposition for K means the decomposition ( S ³ , K ) = T ₁ ∪ T ₂ into two trivial tangles T ₁ = ( B ₁ , t ₁ ) & T ₂ = ( B ₂ , t ₂ ).

• The 2-sphere appearing as B ₁ ∩ B ₂ is called the bridge decomposing sphere.

• The knot is called a two-bridge knot if the

tangles T _i ’s are 2-string tangles.

Invariant δ _P ( T ₁ ∪ T ₂ )

Let ( S ³ , K ) = T ₁ ∪ T ₂ 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 ( T ₁ ∪ T ₂ ) as min

½

d _{P (Σ)} ([ P ₁ ] , [ P ₂ ]) | [ P ₁ ] ∈ H ( T ₁ ) , [ P ₂ ] ∈ H ( T ₂ )

¾

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



 ¯¯

¯¯

∃ −

§ 4. Outline of proof

Let K be a hyperbolic two-bridge knot in S ³ .

Remark:

It is known that which two-bridge knots are hyperbolic (Menasco, ’84).

Suppose that a two-bridge decomposition

( S ³ , K ) = T ₁ ∪ T ₂ for K is given.

Observation: ( δ _P ( T ₁ ∪ T ₂ ) for K ) For a two-bridge decomposition,

Σ ₄ = S ² − Exterior( K ) = four-punctured sphere

⇓

A pants decomposition on Σ ₄ consists of single circle.

⇓

Only consider single disk in B _i − t _i

Such curves on Σ ₄ are parametrized by irreducible fractions; “slopes” on Σ.

⇓

P (Σ ₄ ) is identified with

the modular diagram D ;

From ( S ³ , K ) = T ₁ ∪ T ₂ ,

⇓

Consider the “pillow-cased” diagram, called the Schubert form S ( α, β ).

⇓

Claim 1.

δ _P ( T ₁ ∪ T ₂ ) =

length of the shortest path in D

connecting h ¹ ₀ i to h _α ^β i

Claim 2.

∃ correspondence such that path λ in D

connecting h ¹ ₀ i to h ^β _α i ⇔

diagram of K _β/α ,

called Conway form C ( a ₁ , a ₂ , . . . , a _n )

Moreover,

length( λ ) − 1 = “twist number” n

holds.

By Claim 1 and Claim 2, we have;

Lemma 1.

Let D ₀ be the Conway form of K _β/α with minimal twist number t ( D ₀ ).

Then δ _P ( T ₁ ∪ T ₂ ) − 1 = t ( D ₀ ) holds.

On the other hand, we have Lemma 2.

∃ alternating diagram D _alt of K _β/α with twist number t ( D _alt )

such that t ( D ₀ ) ≤ t ( D _alt ) ≤ 2 t ( D ₀ ).

The following fact is due to

Lackenby, Agol-D.Thurston, Agol-Storm-W.Thurston

Theorem (L,A-D.T, A-S-W.T)

Let D be a prime alternating diagram of

a hyperbolic link L in S ³ with twist number t ( D ). Then

v

2 ( t ( D ) − 2) ≤ vol ( S ³ − K ) ≤ 10 v ₄ ( t ( D ) − 1)

holds.