PDF Hirota - Keio

(1)

From

one ^- cone

tori Hirota ka

^to

Akiyoshi

^two ^-

bridge

^cone

manifolds

( Osaka City University )

Topology

^and

Geometry of

^Low ^- dimensional

Manifolds

at Nara Women 's

University

October ₃₁ . 2018

(2)

Outline

Part

I ^.

From

once ^-

punctured

torus

to two ^-

bridge

^knot

complements

(

^with ^Sakuma , Wada

Yamashita

_,

)

Part

² ^.

From

^one ^- cone torus

to two ^-

bridge

^cone

manifolds

( joint Partly

^with

Yamashita )

Kst

(3)

2-

bridge

^knots ^C ^and ^links ⁾

÷

^E ^D

s

:c iii. :*:: :

^" ^.

with leg 's ^P

't

y & Scp ^. g) ^i.

hyperbolic

⇐ =3

^⇐

sat

SC ^5. ²⁾ ⁼ ^C ⁽ ^2.2 ⁾ SC 17,77=42,213 )

2%

(4)

Hyperbolic

^str ^of ² ^- ^br ^knot

complement

^- ^ASWY

picture

quasi

fuchsia

^double ^cusp ^gp ^cone ^sing ^upper

tunnel

TA To

^-

E

_{' '}

ii :#

^Riley. ^slice ^me ^.

Le

*

^The ^Ford ^domains ^are

characterized

^!^!

3%

(5)

Riley slice

Yamashita

^'s

output

^of

Riley

^slice

* There

practical

is ^a ^{' '}

algorithm

^" ^to ^check

if

a

given

^two ^-

parabolic

_group ^is ^discrete ^or ^{not !} ^!

gig

(6)

Demonstration I

(7)

The ongoing project

Goal ^:

Replace

^the ^" ^main ^cusp ^" ^with ^cone

singularity

^.

:÷÷÷ ÷

^for ^figure^. ⁸^.

them

^The double

covering space

^of

S3

branched

along Scp

^. ^so ⁾ ^is ^the ^lens _space ^LCP ^. ⁸⁾ ^.

5%

(8)

Farey

tessellation

.

:

to ^•

Too ÷'s ÷ =

^, ^→^• ^÷^-

%

(9)

Description of

^comb ^str ^due

to Jorgensen

÷÷÷÷ ÷

÷

f-

I

^dual

¥#¥Ft

_-

w

^-

(10)

Idea of proof

^- ^based ^on

Jorgensen

^'s

argument

*

Jorgensen

characterized ^the

combinatorial

^structures of ^the ^Ford ^domains

for punctured

^torus

groups

^.

→ s :

⇒

^⇒ ^a

I ±u

_, ^a

/

Oo

%

(11)

Part 2

From

^one ^- ^cone

torus to

two

^-

bridge

^cone

manifolds

(12)

Steps

^on

the way

cone sing quasi fuchsia ^double ^cusp ^gp

at

TT

GO

Ft "

Is

0<10 ① ②

^, ^Quasi

③ Additional fuchsia

^Today ^. ^IT ^- ^cone

OILS singularity

^za ²⁼² ^'T

- - - -

Something happens

^!^!

%f

(13)

Demonstration 2

(14)

Quasi fuchsia

ⁿ

structure

[ Moroianu ^-

Schlenker

, Lemire ^- Schlenker ]

"

Quasi

fuchsia manifold

^with

particle

^"

( ^M ^, ^-2 ⁾ ⁼ ⁽ surface ^, ^I ^fin ^. pts^t ) ^x ^R

• all ^cone

angles

^< ^The

• admits

compact

^convex ^core

Thin ^[ ^L ^- ^S ^] ÎF ^C ^M ⁾ Î ^Teich ^C ÔM ⁾

To oczo ^a ^2T ^s :

.

" "

. ← o

↳ ①

z

JJ ?I

Oo^, ^. ^← ^cone ângle ^< ÎÔ ^soo ^ca

(15)

Ford domain for

^cone

hyp

^Str

Suppose

^⇒

CCM

sit ^. C = glue ^. ^.

Choro

^ball ^with ^cone

)

The Ford ^domain ^w ^. ^r ^. ^t ^.

C

is

-

in ^-

Gl is

^cut

:*

^mmmm^locus ^want ^. ^C

I " %

(16)

Partial results

Suppose

ÔC Ô ît

(

^⇒ ^oszos ^za

) °

^T20^"

Than ^I ^[ ^A ^. ²⁰¹⁵³

Ford

^ThmThe^•^•ôn ²^The^Theôn^T20^[^{Too HR}Â ^,^XR^space^Ford²⁰¹⁸^domain^has^has^]

of

^domain^"^"

good good

^thin^for

for

âny^"^"^str^comb^combânyîs^" ^.

fuchsia

^. ^"

parametrized

^Str^Str^thin^.^. ^" ^" ^structure^structure

by

the

hyp

^Str

of

^O ⁽

compact

^convex ^core ⁾ ^.

,yj

(17)

Explosion of

^isometric

hemispheres

In Demo

² , r

(

isom ^hemi ) ^→ ^is as a → 2h ^- o .

• 0=0 ( cusp ^case ⁾

Explosion

^• Ô ^> ^weÔ ⁽ ôbtainôur ôccurs^new^the

complete

^case^at ⁾ ^D=

hyp

^2K ^Str^, ^where

of

² ^- ^bn ^knot ^.

Explosion

^occurs

before

4=21

^⇒ ^We

only

^cannotvia Ford^reach domains^the ^desired ^Str

③

(18)

Employing

^Dirichlet ^domain

Series ^- ^Tan ^- Yamashita studied the extended

Riley slices

^, ^where ^Dirichlet ^domain ^is

employed

^.

"

Surely

^, ^the ^comb

Str _are same ^as

those

of

^Ford ^dom for ^the

Riley

^slice ^!^! ^"

- said Yamashita

④

(19)

Demonstration 3

(20)

Dirichlet domain

The

^Dirichlet ^domain ^w ^. ^r^. ^t ^. _p _is

-

i.

^p

M ^-

{

^x ^/ ^⇒ ² ^shortest

paths

^•

In

[email protected]

. p

Prop

⁽ ^M ^. ^{I )} ⁼ ⁽ ^S ^. ^I ^finite

pts

^{} )} ^x ^R ^, ^cone

angle

^STG

Suppose

^. ^M ^contains ^horo ^balls ^with ^cone ^at

both ends

of

^I

•

M

^contains ^a

compact

^convex ^core

{

}

^ca

⇒ # {

^comb^w ^. ^r ^. ^t ^.^Str_p ^E

of

I ^Dp

(21)

Key Lemma for Prop

Lemma ^Let X ^be ^a ^CAT ^C- ^I⁾ ^space ^.

Then ^we ^have ^the

following cemeteries

^:

XJ

^Z

IHJ

^at ^I

W x ⁱⁿ e

i

Y

Et It

^I

in

_e

i

5

a _, p

ZE

^⇒

d×

^C ^w ^. ^Z ⁾ ^z ^d # ^a Cui ^. E ⁾

%

(22)

Choice of

^base

points

Schematic

sectional

^Views

of

^Ford

/

^Dirichlet ^domains ^:

-2¥

^Ls^, ^-^21T ^-

Eoµ

^O ^→ ^E ²⁼²¹⁷

my

^- ^m ^- ^O ^→

!

^2K ^- ^{O C}^- ^L ^E ^2K

N symbolize

^the

of

^"

generators elliptic axes

^"

of slope to

^, top for

Kcp

^. ^so ⁾ ^.

Conj tp

^E ^Zo ^, ^the ^Dirichlet ^domain ^w ^. ^rt ^. ^p

has

good

^comb ^Str ^.

(23)

Demonstration 4

(24)

Some comments

④ The ^cone

hyp

^str

for

^the

figure

⁸ ^knot ⁱⁿ

Demo ⁴ ^are

studied by Hilden

^- ^Lozano ^- ^Montesinos

in detail

by using

^Dirichlet ^domains ^w ^. ^r^. ^t ^.

other base

points

^.

④ Another

family

^-

for

^bundle ^over ^S ^' ^-

is studied

by Jorgensen

^, ^and ^Heusen ^er ^-

-

Ponti Suarez

^- ^.

④

The families for figure

⁸ ^knot ^are

also

studied by

^Thurston ^.

¥