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

a punctured sphere Knot diagrams on string figures of a model as l # KEEFE ) ' is E¥u U 434 l '¥k¥E¥i¥E¥I k¥3 ) ) YEH # #t¥¥ A ( Th Aa ITT ' D Y

= $2 R2 → IR ' v - → § ÷ . .

⑦ i diagram of K DE $2 knot KE $3 : : underlying projection of D = PID ) P i regular neighbourhood of in $2 P N ( P ) ' Q = IR ' un ⑦ K E $3 ' t € 7 . :* . . •

{ R I R is ' l N ( P ) ) - RIP ) a component of $ R - crossings of D : the number of ( ( D ) SER FIS ) 511¥13 ' ' , D E on Fist c ( K ) c ( D ) f - - = min { ECE ) /EFD} CCD , s ) : -_ - C' ( D. 4) E CID , 5) E CID . R ) ( ( D ) -

Exc

thm-CCD.pe/=Cl First proved by Keiji Taga mi ( National Fisheries University ) . ÷i÷÷÷÷÷:"÷:¥÷::÷÷¥÷: : generic immersion ' → $2 Thm2_ y :$

⇒ @ Turaevcobracket ( arranged for our purpose ) : oriented surface F , get } : = { y :$ I 9 : generic immersion ' → F H - y is not null - homo topic on F - { YEH I 4=93 : = HI = , (9) : H - l y 9 ";Y y ye . :* u ) ( x F no . . F * A

2 ( H x H ) co bracket Tura ev i H I → [ Y ] → pff , y , U , [ Bp , i ] ) ) ( ( ( Yp , - ( Cyp , a ] C Yp , z ) ) , ] , Yp , TP , 2 y i of y i = the set of crossings C (9) i = { P , 9pm to } E C 19 ) / Yp . , ¥0 D l 9)

ftp.3-isawe/l-definedhomotopyinvarian- ) 0=0 A ) → " " " g- in % . . . y - ( LSP . 2 ] ( [ Yp , , ] , [ Bp , z ) ) , Cyp , i ] ) - ( Hai , Esq , D) , [ 9g . D ) O ( C Yoo = t . , ]

- Iq ,z 9131 5C y a JI Sgp ,z g ' v Too y , . - ( ftp.z ] ,C9p . , ] ) ( [ Yp , , ] , [ 9pm ] ) - ( Maid , g. D) O g. D ) ( [ Yq = 1- . , ] , [ Yp , ,j ] = [ 48 ; Pi > I ] # q , q , p#pz 5=1,2 ' 2=1.23 , 4%1 Y I Y E x K Yp , ,2 481,2 ¢

Z ( H X H ) Z E o → n - - I ah I . 2h ) ) : = I a , I t - t n f a , CK a k ( X h - t , Z , ) t - - - - i "i¥¥IY;""

¥¥:÷÷÷÷÷÷÷÷÷:÷÷:÷:÷÷i : generic ' → $2 immersion Prop.si y :$ , in ;;aaerize ' ) 41$ "

that Cheng informed Zhi yun us consequence of • immediate -1hm 2 is an . Intersections of on surfaces - Scott ] [ Hass curves Math Israel 1985 J . . it

⇒ ' l N ( P ) ) { RI R is - RIP ) a component of $ R - - E Sh ER $ ⇐ S , E - - Thml - ECID.se#ClD.R)=ClD ⇒ elk )= ( ( D. 4) E CID . SDE ) - - IRI = m -12 = ( ( P ) = ( ( D) m m -12 OE NE max { CCD .SI/lsl=n } ( max ( D , n ) : = - n } min { CCD , s ) / 1st ( min ( D , n ) - : =

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