Euclidean length on a horotorus and the Culler-Shalen norm of slopes

Euclidean length on a horotorus

and

the Culler-Shalen norm of slopes

市原一裕

Kazuhiro Ichihara

Osaka Sangyo Univ.

1

Contents

§ 1. Backgrounds

§ 2. Definitions and Results

§ 3. Example and Experiments

§ 4. Outline of Proof

Notations

Throughout this talk, M denotes

a compact, connected, orientable 3-mfd with single torus boundary ∂M .

Suppose that M is hyperbolic i.e. int M admits a complete Riem metric with constant curv − 1.

A slope on ∂M is the isotopy class of

an unoriented simple closed curve on ∂M .

2

Notations

Throughout this talk, M denotes

a compact, connected, orientable 3-mfd with single torus boundary ∂M .

Suppose that M is hyperbolic i.e. int M admits a complete Riem metric with constant curv − 1.

A slope on ∂M is the isotopy class of

Notations

Throughout this talk, M denotes

a compact, connected, orientable 3-mfd with single torus boundary ∂M .

Suppose that M is hyperbolic i.e. int M admits a complete Riem metric with constant curv − 1.

A slope on ∂M is the isotopy class of

an unoriented simple closed curve on ∂M .

2-b

§ 1. Backgrounds

Geometrization Conj

All compact 3-manifolds are classified as

· Reducible

· Toroidal

· Seifert fibered

⊃ { 3-mfds, Finite π ₁ } ⊃ { 3-mfds, Cyclic π ₁ }

§ 1. Backgrounds

Geometrization Conj

All compact 3-manifolds are classified as

· Reducible

· Toroidal

· Seifert fibered

⊃ { 3-mfds, Finite π ₁ } ⊃ { 3-mfds, Cyclic π ₁ }

· Hyperbolic

3-a

§ 1. Backgrounds

Geometrization Conj

All compact 3-manifolds are classified as

· Reducible

· Toroidal

· Seifert fibered

⊃ { 3-mfds, Finite π ₁ } ⊃ { 3-mfds, Cyclic π ₁ }

§ 1. Backgrounds

Geometrization Conj

All compact 3-manifolds are classified as

· Reducible

· Toroidal

· Seifert fibered

⊃ { 3-mfds, Finite π ₁ } ⊃ { 3-mfds, Cyclic π ₁ }

· Hyperbolic

3-c

§ 1. Backgrounds

Geometrization Conj

All compact 3-manifolds are classified as

· Reducible

· Toroidal

· Seifert fibered

⊃ { 3-mfds, Finite π ₁ } ⊃ { 3-mfds, Cyclic π ₁ }

Dehn filling = gluing a solid torus V to M ;

M

V

The slope r on ∂M represented by the meridian of V determines the homeo type of the resultant.

So it is denoted by M ( r ).

4

Dehn filling = gluing a solid torus V to M ;

M

r

V

The slope r on ∂M represented by the meridian

of V determines the homeo type of the resultant.

Theorem (Thurston)

If M is hyperbolic, then all but finitely many Dehn fillings yield hyperbolic 3-manifolds.

In view of this, the finitely many exceptions are called exceptional fillings.

Problem

When, how many, what kind of exceptional fillings can occur?

5

Theorem (Thurston)

If M is hyperbolic, then all but finitely many Dehn fillings yield hyperbolic 3-manifolds.

In view of this, the finitely many exceptions are called exceptional fillings.

Ploblem

When, how many, what kind of

Theorem (Thurston)

If M is hyperbolic, then all but finitely many Dehn fillings yield hyperbolic 3-manifolds.

In view of this, the finitely many exceptions are called exceptional fillings.

Ploblem

When, how many, what kind of exceptional fillings can occur?

5-b

Recall that:

Geometrization Conj

All compact 3-manifolds are classified as

· Reducible

· Toroidal

· Seifert fibered

⊃ { 3-mfds, Finite π ₁ } ⊃ { 3-mfds, Cyclic π ₁ }

· Hyperbolic

The Euclidean length L _T ( · ) on a horotorus T and the Culler-Shalen norm k · k

have played very important roles.

Facts. (rough statement)

· If M ( r ) is type C , then k r k = min k r k .

· If M ( r ) is type F , then k r k ≤ 3min k r k .

· If M ( r ) is type R , T , S , then L _T ( r ) ≤ 6.

7

The Euclidean length L _T ( · ) on a horotorus T and the Culler-Shalen norm k · k

have played very important roles.

Facts. (rough statement)

· If M ( r ) is type C , then k r k = min k r k .

· If M ( r ) is type F , then k r k ≤ 3min k r k .

· If M ( r ) is type R , T , S , then L _T ( r ) ≤ 6.

Definitions of L _T ( · ) and k · k are quite different.

However:

These results have somehow similar flavors;

If M ( r ) is non-hyp, then the value for r is small.

Question.

Is there a relationship between L _T ( · ) and k · k ?

8

Definitions of L _T ( · ) and k · k are quite different.

However:

These results have somehow similar flavors;

If M ( r ) is non-hyp, then the value for r is small.

Question.

Is there a relationship between L _T ( · ) and k · k ?

Definitions of L _T ( · ) and k · k are quite different.

However:

These results have somehow similar flavors;

If M ( r ) is non-hyp, then the value for r is small.

Question.

Is there a relationship between L _T ( · ) and k · k ?

8-b

§ 2. Definitions and Results

2.1 Length on a horotorus

int M is a complete hyp 3-mfd of finite vol.

⇓

The universal cover is H ^{3 = (} R ³ ₊ ^{, dx 2 + dy} _z 2 ^{2 + dz 2}

)

,

simply conn, open Riem 3-mfd of const curv − 1,

called the hyperbolic 3-space.

§ 2. Definitions and Results

2.1 Length on a horotorus

int M is a complete hyp 3-mfd of finite vol.

⇓

The universal cover is H ^{3 = (} R ³ ₊ ^{, dx 2 + dy} _z 2 ^{2 + dz 2}

)

, simply conn, open Riem 3-mfd of const curv − 1, called the hyperbolic 3-space.

9-a

A horosphere appears as { z = const } ⊂ H ^{3 ,}

equivariant under the action of π ₁ ( M ).

By covering projection, we have

a ∂ -parallel torus in M , called a horotorus.

int M

∂ -parallel torus

~

A horosphere appears as { z = const } ⊂ H ^{3 ,}

equivariant under the action of π ₁ ( M ).

By covering projection, we have

a ∂ -parallel torus in M , called a horotorus.

int M

∂ -parallel torus

~

10-a

The induced metric on a horosphere in H ³

is a Euclidean metric.

⇓ (the covering is local isometry)

We have a Euclidean metric on a horotorus in M .

⇓

The length of a loop on a horotorus is defined.

⇓ ( ∂M & horotorus are parallel)

The length of a slope on ∂M is defined

The induced metric on a horosphere in H ³

is a Euclidean metric.

⇓ (the covering is local isometry)

We have a Euclidean metric on a horotorus in M .

⇓

The length of a loop on a horotorus is defined.

⇓ ( ∂M & horotorus are parallel) The length of a slope on ∂M is defined

as the minimal length of the representatives.

11-a

The induced metric on a horosphere in H ³

is a Euclidean metric.

⇓ (the covering is local isometry)

We have a Euclidean metric on a horotorus in M .

⇓

The length of a loop on a horotorus is defined.

⇓ ( ∂M & horotorus are parallel)

The length of a slope on ∂M is defined

The induced metric on a horosphere in H ³

is a Euclidean metric.

⇓ (the covering is local isometry)

We have a Euclidean metric on a horotorus in M .

⇓

The length of a loop on a horotorus is defined.

⇓ ( ∂M & horotorus are parallel)

The Euclidean length of a slope on ∂M is defined as the minimal length of the representatives.

11-c

2.2 Culler-Shalen norm

Let Γ denote π ₁ ( M ) for brevity.

For a representation ρ : Γ → SL ₂ ( C ^),

the character χ _ρ : Γ → C

is defined by χ _ρ ( γ ) := trace ( ρ ( γ )).

The set of all characters of SL ₂ ( C )-representations

(after taking the closure and desingularizarions)

2.2 Culler-Shalen norm

Let Γ denote π ₁ ( M ) for brevity.

For a representation ρ : Γ → SL ₂ ( C ^),

the character χ _ρ : Γ → C

is defined by χ _ρ ( γ ) := trace ( ρ ( γ )).

The set of all characters of SL ₂ ( C )-representations (after taking the closure and desingularizarions) is called the character variety, denoted by X (Γ).

12-a

Recall (holonomy representation)

π ₁ ( M ) is a discrete subgroup of Isom ⁺ ( H ^{3 ).}

(regarded as covering transformations)

Isom ⁺ ( H ^{3 ) = ∼ { pz} _rz ⁺ ₊ ^q _s ^y C ^{} /} ± I = PSL ∼ ₂ ( C ⁾

(identify ∂ H ^{3 with} C ∪ {∞} )

We obtain a faithful discrete representation;

π ₁ ( M ) −→ ^{= ∼} Γ < PSL ₂ ( C ⁾

This can be lifted to π ₁ ( M ) → SL ₂ ( C ^),

Recall (holonomy representation)

π ₁ ( M ) is a discrete subgroup of Isom ⁺ ( H ^{3 ).}

(regarded as covering transformations)

Isom ⁺ ( H ^{3 ) = ∼ { pz} _rz ⁺ ₊ ^q _s ^y C ^{} /} ± I = PSL ∼ ₂ ( C ⁾

(identify ∂ H ^{3 with} C ∪ {∞} )

We obtain a faithful discrete representation;

π ₁ ( M ) −→ ^{= ∼} Γ < PSL ₂ ( C ⁾

This can be lifted to π ₁ ( M ) → SL ₂ ( C ^),

called the holonomy representation ρ _hol .

13-a

Recall (holonomy representation)

π ₁ ( M ) is a discrete subgroup of Isom ⁺ ( H ^{3 ).}

(regarded as covering transformations)

Isom ⁺ ( H ^{3 ) = ∼ { pz} _rz ⁺ ₊ ^q _s ^y C ^{} /} ± I = ∼ PSL ₂ ( C ⁾

(identify ∂ H ^{3 with} C ∪ {∞} )

We obtain a faithful discrete representation;

π ₁ ( M ) −→ ^{= ∼} Γ < PSL ₂ ( C ⁾

This can be lifted to π ₁ ( M ) → SL ₂ ( C ^),

Recall (holonomy representation)

π ₁ ( M ) is a discrete subgroup of Isom ⁺ ( H ^{3 ).}

(regarded as covering transformations)

Isom ⁺ ( H ^{3 ) = ∼ { pz} _rz ⁺ ₊ ^q _s ^y C ^{} /} ± I = ∼ PSL ₂ ( C ⁾

(identify ∂ H ^{3 with} C ∪ {∞} )

We obtain a faithful discrete representation;

π ₁ ( M ) −→ ^{= ∼} Γ < PSL ₂ ( C ⁾

This can be lifted to π ₁ ( M ) → SL ₂ ( C ^),

called the holonomy representation ρ _hol .

13-c

The principal component X ₀ (Γ) of X (Γ) is

the irreducible compo 3 the character of ρ _hol . Using a trace function, for a slope r on ∂M , we have a rational map I _r : X ₀ (Γ) → C

by I _r ( χ _ρ ) := χ _ρ ( r ) = trace ( ρ ( r )).

Fact: dim _C X ₀ (Γ) = 1, i.e., X ₀ (Γ) is alg curve.

For a slope r on ∂M ,

The principal component X ₀ (Γ) of X (Γ) is

the irreducible compo 3 the character of ρ _hol . Using a trace function, for a slope r on ∂M , we have a rational map I _r : X ₀ (Γ) → C

by I _r ( χ _ρ ) := χ _ρ ( r ) = trace ( ρ ( r )).

Fact: dim _C X ₀ (Γ) = 1, i.e., X ₀ (Γ) is alg curve.

For a slope r on ∂M ,

the Culler-Shalen norm k r k is defined by 2deg( I _r ).

14-a

The principal component X ₀ (Γ) of X (Γ) is

the irreducible compo 3 the character of ρ _hol . Using a trace function, for a slope r on ∂M , we have a rational map I _r : X ₀ (Γ) → C

by I _r ( χ _ρ ) := χ _ρ ( r ) = trace ( ρ ( r )).

Fact: dim _C X ₀ (Γ) = 1, i.e., X ₀ (Γ) is alg curve.

For a slope r on ∂M ,

2.3 Results

Theorem 1.

Suppose that M is the exterior of

· a hyperbolic two-bridge knot, or

· a ( − 2 , 3 , n )-pretzel knot ( n ≥ 7, odd ) in S ³ . Then

k r k ≥ 2

3 L _T ( r )

holds for ^∀ slope r and for ^∀ horotorus T .

15

Remark: _@ C s.t. L _T ( r ) > C k r k holds in general.

Theorem 2 Let m be the fixed merid- ional slope,

and r ₁ , r ₂ integral slopes on ∂M . If r ₁ ≥ the maximal ∂ -slope and

r ₂ ≤ the minimal ∂ -slope for M , then L _T ( r ₁ ) + L _T ( r ₂ ) > k r ₁ k

k m k + k r ₂ k

k m k

Remark: _@ C s.t. L _T ( r ) > C k r k holds in general.

Theorem 2

Let m be the fixed meridional slope, and r ₁ , r ₂ integral slopes on ∂M .

If r ₁ ≥ the maximal ∂ -slope and

r ₂ ≤ the minimal ∂ -slope for M , then L _T ( r ₁ ) + L _T ( r ₂ ) > k r ₁ k

k m k + k r ₂ k k m k holds for the maximal horotorus T .

16-a

Definition ( ∂ -slope) A slope on ∂M determined by the boundary components of

an essential embedded surface in M is called the boundary slope ( ∂ -slope).

Note:

The Culler-Shalen theory detects boundary slopes.

Definition ( ∂ -slope) A slope on ∂M determined by the boundary components of

an essential embedded surface in M is called the boundary slope ( ∂ -slope).

Note:

The Culler-Shalen theory detects boundary slopes.

17-a

§ . Break (Advertisement)

Workshop

“Topology and Computers 2005”

November 28 – November 30, 2005 Osaka Sangyo University,

Umeda Satellite

(Osaka Ekimae Build. 4, 22F)

§ 3. Example and Experiments

3.1 Example: the Figure-eight knot in S ³

M := the exterior of the knot.

This is hyperbolic.

We take

the maximal horotorus T .

§ 3. Example and Experiments

3.1 Example: the Figure-eight knot in S ³

M := the exterior of the knot.

This is hyperbolic.

We take

the maximal horotorus T .

18-a

Modulus of the maximal horotorus:

On the univ cover of T ;

O y

x µ

λ

µ, λ : lift of

the meridian, longitude.

µ ⊥ λ

L _T ( µ ) = 1, L _T ( λ ) = 2 √

3

L _T ( r _{(1 , 4)} ) = 2 √

7

Computation of the Culler-Shalen norm:

The distance ∆( r, r ⁰ ) of two slopes r, r ⁰ is

the minimal geometric intersection number of the representatives of r, r ⁰ .

Proposition 1.

Let s ₁ , · · · .s _n be ∂ -slopes fon ∂M . There exist non-negative even constants a ₁ , · · · , a _n s.t.

k r k = ^∑ a _i ∆( r, s _i ) hold for ^∀ r .

Moreover, at least two a _i ’s are positive ( ≥ 2).

20

Computation of the Culler-Shalen norm:

The distance ∆( r, r ⁰ ) of two slopes r, r ⁰ is

the minimal geometric intersection number of the representatives of r, r ⁰ .

Proposition 1.

Let s ₁ , · · · , s _n be ∂ -slopes on ∂M . There exist non-negative even constants a ₁ , · · · , a _n s.t.

k r k = ^∑ a _i ∆( r, s _i ) hold for ^∀ r .

Computation of the Culler-Shalen norm:

The distance ∆( r, r ⁰ ) of two slopes r, r ⁰ is

the minimal geometric intersection number of the representatives of r, r ⁰ .

Proposition 1.

Let s ₁ , · · · , s _n be ∂ -slopes on ∂M . There exist non-negative even constants a ₁ , · · · , a _n s.t.

k r k = ^∑ a _i ∆( r, s _i ) hold for ^∀ r .

Moreover, at least two a _i ’s are positive ( ≥ 2).

20-b

For the figure-eight knot case,

k r k = 2 ∆( r, r _{(1 , 4)} ) + 2 ∆( r, r _{(1 , − 4)} ) (Ohtsuki)

Inequality (for Theorem 1) Thus, we obtain

√ 8

3 L _T ( r ) ≥ k r k ≥ 8

√ 7 L _T ( r )



 > 2

3 L _T ( r ) , by Thm 1





For the figure-eight knot case,

k r k = 2 ∆( r, r _{(1 , 4)} ) + 2 ∆( r, r _{(1 , − 4)} ) (Ohtsuki)

Inequality (for Theorem 1) Thus, we obtain

√ 8

3 L _T ( r ) ≥ k r k ≥ 8

√ 7 L _T ( r )



 > 2

3 L _T ( r ) , by Thm 1





21-a

For the figure-eight knot case,

k r k = 2 ∆( r, r _{(1 , 4)} ) + 2 ∆( r, r _{(1 , − 4)} ) (Ohtsuki)

Inequality (for Theorem 1) Thus, we obtain

√ 8

3 L _T ( r ) ≥ k r k ≥ 8

√ 7 L _T ( r )



 > 2

3 L _T ( r ) , by Thm 1





Inequality (for Thm 2) In this case, k m k = 4.

So we have L _T ( r ) ≥

√ 3 2

k r k k m k

(The equality holds for the longitude r _{(1 , 0)} .) If r _{(1 ,n )} , n ≥ 4 or n ≤ − 4, then we have

L _T ( r _{(1 ,n )} ) > k r _{(1 ,n )} k k m k

(

L _T ( r ₁ ) + L _T ( r ₂ ) > ^k _k ^{r 1 k}

m k + ^{k r 2 k}

k m k by Thm 2

)

22

Inequality (for Thm 2) In this case, k m k = 4.

So we have L _T ( r ) ≥

√ 3 2

k r k k m k

(The equality holds for the longitude r _{(1 , 0)} .) If r _{(1 ,n )} , n ≥ 4 or n ≤ − 4, then we have

L _T ( r _{(1 ,n )} ) > k r _{(1 ,n )} k k m k

( )

3.2 Computer Experiments

(This part is supported by S.Mizushima (TITECH)) We have observed L _T ( r ) & _k ^{k r k}

m k are related.

On the other hand, it is known that Proposition 2.

For ess surf F with ∂ -slope r , 6 ^{− χ ( F )}

]∂F ≥ L _T ( r ).

Question.

Is there a relationship between ^{− χ ( F )}

]∂F and _k ^{k r k}

m k ?

23

3.2 Computer Experiments

(This part is supported by S.Mizushima (TITECH)) We have observed L _T ( r ) & _k ^{k r k}

m k are related.

On the other hand, it is known that Proposition 2.

For ess surf F with ∂ -slope r , 6 ^{− χ ( F )}

]∂F ≥ L _T ( r ).

Question.

− χ ( F ) k r k

3.2 Computer Experiments

(This part is supported by S.Mizushima (TITECH)) We have observed L _T ( r ) & _k ^{k r k}

m k are related.

On the other hand, it is known that Proposition 2.

For ess surf F with ∂ -slope r , 6 ^{− χ ( F )}

]∂F ≥ L _T ( r ).

Question.

Is there a relationship between ^{− χ ( F )}

]∂F and _k ^{k r k}

m k ?

23-b

3.2 Computer Experiments

(This part is supported by S.Mizushima (TITECH)) We have observed L _T ( r ) & _k ^{k r k}

m k are related.

On the other hand, it is known that Proposition 2.

For ess surf F with ∂ -slope r , 6 ^{− χ ( F )}

]∂F ≥ L _T ( r ).

Question.

− χ ( F ) k r k

We performed a computer-aided experiments for two-bridge knots in S ³ .

We used

Dunfield’s computer program, available at http://www.its.caltech.edu/ ˜ dunfield/

to find and list up essential surfaces, and Ohtsuki’s formula (corrected by Mattman) to compute the Culler-Shalen norm.

24

We performed a computer-aided experiments for two-bridge knots in S ³ .

We used

Dunfield’s computer program, available at http://www.its.caltech.edu/ ˜ dunfield/

to find and list up essential surfaces, and

Ohtsuki’s formula (corrected by Mattman)

We performed a computer-aided experiments for two-bridge knots in S ³ .

We used

Dunfield’s computer program, available at http://www.its.caltech.edu/ ˜ dunfield/

to find and list up essential surfaces, and Ohtsuki’s formula (corrected by Mattman) to compute the Culler-Shalen norm.

24-b

25-a

25-c

§ 4. Outline of Proof (Theorem 1) Theorem 3

Suppose that ^∃ two ∂ -slopes s ₁ , s ₂ on ∂M detected by the C-S theory s.t.

∃ ess surf S ₁ , S ₂ with ∂ -slopes s ₁ , s ₂ satisfy

∆( s ₁ , s ₂ ) ≥ 2 − χ ( S _i )

]∂S _i for i = 1 , 2.

Then k r k ≥ ² ₃ L _T ( r ) holds for ^∀ slope r and

Theorem 3 implies Theorem 1

For a hyperbolic two-bridge knot exterior and a ( − 2 , 3 , n )-pretzel knot exterior ( n ≥ 7, odd ), it is known by Ohtsuki and Mattman

that the assumption of Thm 3. is satisfied _¤

27

Remark

· The ∂ -slopes undetected by the C-S theory are found very recently.

· An alternating knot exterior always contains

ess surfaces S ₁ , S ₂ satisfying ∆( s ₁ , s ₂ ) ≥ 2 ^{− χ ( S i )}

]∂S _i . (That is, the checher-board surfaces)

However it is unknown whether the ∂ -slopes

Remark

· The ∂ -slopes undetected by the C-S theory are found very recently.

· An alternating knot exterior always contains

ess surfaces S ₁ , S ₂ satisfying ∆( s ₁ , s ₂ ) ≥ 2 ^{− χ ( S i )}

]∂S _i . (That is, the checher-board surfaces)

However it is unknown whether the ∂ -slopes are detected or not.

28-a

Lemma (distance & length) On any horotorus T ,

∆( r, r ⁰ ) = L _T ( r ) · L _T ( r ⁰ ) · sin | θ _r − θ _{r 0} | Area( T )

Here θ _r denotes the angle between

the geodesic representatives of r and m on T .

Outline of Proof of Thm 3.

Let s ₁ , · · · , s _n be boundary slopes on ∂M .

By Proposition 1, we have k r k = ^{∑ n} _i=1 a _i ∆(r, s _i )

By assumption, there exist two ∂ -slopes, say s ₁ , s ₂ , on ∂M detected by the C-S theory, i.e., a ₁ , a ₂ are non-zero.

Then, by Lemma (distance & length), we have;

k r k ≥ a ₁ ∆(r, s ₁ ) + a ₂ ∆(r, s ₂ )

= a ₁ L _T (r ) · L _T (s ₁ ) · sin | θ _r − θ _{s 1} |

Area(T ) + a ₂ L _T (r) · L _T (s ₂ ) · sin | θ _r − θ _{s 2} | Area(T )

= L _T (r) Area(T )

( a ₁ L _T (s ₁ ) sin | θ _r − θ _{s 1} | + a ₂ L _T (s ₂ ) sin | θ _r − θ _{s 2} | ⁾ .

30

Now, without loss of generality, we assume that a ₁ L _T (s ₁ ) ≥ a ₂ L _T (s ₂ ).

Then we obtain:

k r k ≥ L _T (r)

Area(T ) · a ₂ L _T (s ₂ ) · ⁽ sin | θ _r − θ _{s 1} | + sin | θ _r − θ _{s 2} | ⁾ (1)

Claim. sin | θ _r − θ _{s 1} | + sin | θ _r − θ _{s 2} | > sin | θ _{s 1} − θ _{s 2} | holds.

It follows from this claim and Equation (1) k r k > L _T (r)

Area(T ) · a ₂ L _T (s ₂ ) · sin | θ _{s 1} − θ _{s 2} |

= a ₂ · L _T (r)

L _T (s ₁ ) · L _T (s ₁ )L _T (s ₂ ) sin | θ _{s 1} − θ _{s 2} | Area(T )

L (r) · ∆(s ₁ , s ₂ )

By Proposition 2, we obtain

k r k ≥ a ₂ · L _T (r) · ∆(s ₁ , s ₂ )

L _T (s ₁ ) ≥ a ₂

6 · ∆(s ₁ , s ₂ )

− χ(S ₁ )/]∂S ₁ · L _T (r) By the assumption that

∆(s ₁ , s ₂ ) ≥ 2 − χ(S ₁ )

]∂S ₁ and a ₂ ≥ 2 , we conclude that

k r k ≥ 2

3 L _T (r) .

¤

32