) YEH # #t A ( Th Aa ITT ' D Y = $2 R2 IR ' v - . . i - - PowerPoint PPT Presentation

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yeh t a th aa itt d y 2 r2 ir v i diagram of k de 2
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) YEH # #t A ( Th Aa ITT ' D Y = $2 R2 IR ' v - . . i - - PowerPoint PPT Presentation

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a punctured sphere Knot diagrams on string figures of a model as l # KEEFE ) ' is Eu U 434 l 'kEiEI k3 ) ) YEH # #t A ( Th Aa ITT ' D Y = $2 R2 IR ' v - . . i diagram of K DE $2 knot KE $3 : :

slide-1
SLIDE 1

Knot diagrams

  • n

a punctured sphere

as

a model

  • f

string figures

434

'is E¥u U

l # KEEFE )

YEH # #t¥¥ A

l '¥k¥E¥i¥E¥I k¥3 )

(

Y

D Th Aa ITT

'

)

slide-2
SLIDE 2

R2 → IR

'

v - = $2

÷

.

. → §

slide-3
SLIDE 3

KE $3

:

knot

DE $2

i diagram of K

P

= PID )

: underlying projection of D

N ( P)

i regular neighbourhood of

P

in $2

⑦ K E $3

= IR

' un

t

€7.:* . .

⑦

'

' Q

slide-4
SLIDE 4

R

  • RIP)

{R I R is

a component of $ ' l N ( P))

( ( D )

: the number of

crossings of

D

SER

,

FIS )

511¥13

E

'

'

D

c ( D )

  • =

c ( K )

f

  • n Fist

CCD , s ) : -_

min { ECE ) /EFD}

( ( D )

  • C' ( D. 4) E CID , 5) E CID . R)
slide-5
SLIDE 5

Exc

slide-6
SLIDE 6

thm-CCD.pe/=Cl

First proved

by Keiji Taga mi (National Fisheries University)

.

Thm2_

y :$

' → $2 : generic

immersion

÷i÷÷÷÷÷:"÷:¥÷::÷÷¥÷:

slide-7
SLIDE 7

@ Turaevcobracket ( arranged for

  • ur purpose )

F

: oriented surface

H

: = { y :$ ' → F

I 9 : generic immersion

,get}
  • y is not null
  • homo topic
  • n F

H

: = HI = , (9) :
  • { YEH I 4=93

ye

u

no

.:* . .

⇒

9 ";Y y

l

y

F

) (

x

F

A

*

slide-8
SLIDE 8

I

i H

→

2 ( H x H )

Tura ev

cobracket

U

[ Y ] → pff ,y,

(( ( Yp ,

, ] ,

C Yp , z))

  • ( Cyp, a ]
, [ Bp , i ] ))

y

TP, 2

Yp ,

i

C (9)

i = the set of crossings
  • f y

D l 9)

i = { P

E C 19 )

/ Yp. , ¥0 , 9pm to}
slide-9
SLIDE 9

ftp.3-isawe/l-definedhomotopyinvarian-

)

A

→

) 0=0

g- in

"

""

.

%

. .

y ( [ Yp , , ] , [ Bp , z ) )

  • ( LSP . 2 ]
, Cyp , i ])

t

( C Yoo

. ,] , [9g .D)
  • ( Hai , Esq , D)

=

O

slide-10
SLIDE 10

9131

  • Iq,z

JI

5C

a

y

v

g'

Sgp,z

Too

. ,

y

( [ Yp , , ] , [ 9pm ] )

  • ( ftp.z ] ,C9p . , ])

1-

( [ Yq

. ,] ,

g.D)

  • ( Maid ,
  • g. D)

=

O

Pi

q, q ,

[ Yp,,j]=[ 48;

>I]

p#pz

#

'

2=1.23

,

5=1,2

Y

Y

E

I

4%1

x K

Yp, ,2 481,2

¢

slide-11
SLIDE 11

n

  • Z ( H

X H )

→

Z E o

n f a , CK

i , Z , ) t
  • t

a k ( Xh

. 2h )) : =

I a , I t

  • t

I ah I

"i¥¥IY;""

slide-12
SLIDE 12

Prop.si

y :$

' → $2 : generic

immersion

,

¥¥:÷÷÷÷÷÷÷÷÷:÷÷:÷:÷÷i

in;;aaerize

41$

' )

"

slide-13
SLIDE 13

Zhi yun

Cheng informed

us

that

  • 1hm 2

is

an

immediate

consequence of

  • .

[ Hass

  • Scott ]

Intersections of

curves

  • n surfaces

Israel

J

.

Math

.

1985

it

slide-14
SLIDE 14

R

  • RIP)

{RI R is

a component of $ ' l N ( P))

$ ⇐

S

, E
  • E Sh ER

Thml

⇒ elk )= ( (D. 4) E CID . SDE

  • ECID.se#ClD.R)=ClD

) ( ( D)

= ( ( P ) =

m

⇒

IRI

= m -12

OE NE

m -12

( max ( D , n )

: =

max { CCD.SI/lsl=n}

( min ( D , n )

: =

min{ CCD , s)

/ 1st

  • n}
slide-15
SLIDE 15
slide-16
SLIDE 16

Thank

you

for your listening !