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講演資料 今井 淳 Jun O'Hara

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(1)

kS\NtX

#f!_ sTgXl~

(2)

\!

kS\H=NgV?AHtX

8hNfNkS\"m_\"H_3 tXG7&kS\

2DNkS\,18HO) 6J1JrK

kS\TQL

1MX8HTQL 8g<s:?`0

>N,nHN8oj

v?"J3!5K?MN@ 8*JDNAK"b,RJX

(3)

\!

kS\H=NgV?AHtX

8hNfNkS\"m_\"H_3 tXG7&kS\

2DNkS\,18HO) 6J1JrK

kS\TQL

1MX8HTQL 8g<s:?`0

>N,nHN8oj

v?"J3!5K?MN@ 8*JDNAK"b,RJX

(4)

\!

kS\H=NgV?AHtX

8hNfNkS\"m_\"H_3 tXG7&kS\

2DNkS\,18HO) 6J1JrK

kS\TQL

1MX8HTQL 8g<s:?`0

>N,nHN8oj

v?"J3!5K?MN@ 8*JDNAK"b,RJX

(5)

8hNfNkS\JNgVK

BQ' u~Q'

@U'

(6)

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK u~Q'

@U'

(7)

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK u~Q'

@U'

(8)

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK

u~Q'M/?$"ez"j\s"fO"

shoulder knot

"

!!!!H_3

@U'

(9)

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK

u~Q'M/?$"ez"j\s"fO"

shoulder knot

"

!!!!H_3

@U'

(10)

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK

u~Q'M/?$"ez"j\s"fO"

shoulder knot

"

!!!!H_3

@U'kSNlV"NCHJ%N.5K"

!!!

cut the Gordian knot, nodo d’amore

(11)

kS\NcA>0ND$?kS\

half hitch

(12)

kS\NcA>0ND$?kS\

half hitch square kont,

reef knot

(

^kS

)

(13)

kS\NcA>0ND$?kS\

half hitch square kont,

reef knot

(

^kS

)

running knot

(14)

kS\NcABQ"*hSu~Q

sheep shank

(15)

kS\NcABQ"*hSu~Q

sheep shank

"oSkS

(16)

kS\NcABQ"*hSu~Q

sheep shank

"oSkS Z#kS

(17)

kS\NgV?A

\m_"s&j s/

(18)

kS\NgV?A

\m_"s&j s/

H_3

(19)

kS\NgV?A

\m_"s&j s/

H_3

(pure) braid

(20)

tXG7&kS\

ksG$k

<@rjK}CF%5J1lP[I1J$#

jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~

(21)

tXG7&kS\

ksG$k

)

&<r;/7F$/

<@rjK}CF%5J1lP[I1J$#

jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~

(22)

tXG7&kS\

ksG$k

)

&<r;/7F$/

)

ksG$J$

<@rjK}CF%5J1lP[I1J$#

jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~

(23)

tXG7&kS\

ksG$k

)

&<r;/7F$/

)

ksG$J$

<@rjK}CF%5J1lP[I1J$# jA !tXG7&kS\a3!5uVK"+,+HH

roiJ$h&Kda~^l?_~

(24)

tXG7&kS\

ksG$k

)

&<r;/7F$/

)

ksG$J$

<@rjK}CF%5J1lP[I1J$#

jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~

(25)

J1JkS\Nc

Trefoil

J0UkS\K

=)

!J8NzkS\K

=)

(26)

J1JkS\Nc

Trefoil

J0UkS\K

=)

Figure eight

!J8NzkS\K

=)

(27)

sDNkS\,18"HO)

jA !sDNkS\,V18WJ1?"$=HTC/K G"kHO"sD,4`3GG-F$kH7F"4`3 rZi:K"3*KQA7F"l}r>}KEMgo; ilkh&KG-k3H#

J1JcJV+@JkS\WH$&K



3kS\O?H18)

(28)

sDNkS\,18"HO)

jA !sDNkS\,V18WJ1?"$=HTC/K G"kHO"sD,4`3GG-F$kH7F"4`3 rZi:K"3*KQA7F"l}r>}KEMgo; ilkh&KG-k3H#

J1JcJV+@JkS\WH$&K



3kS\O?H18)

(29)

sDNkS\,18"HO)

jA !sDNkS\,V18WJ1?"$=HTC/K G"kHO"sD,4`3GG-F$kH7F"4`3 rZi:K"3*KQA7F"l}r>}KEMgo; ilkh&KG-k3H#

J1JcJV+@JkS\WH$&K



3kS\O?H18)

(30)

3kSOJK+)

3kSNQA

(31)

3kSOJK+)

3kSNQA

(32)

3kSOJK+)

3kSNQA

(33)

3kSOJK+)

3kSNQA

(34)

3kSOJK+)

3kSNQA

(35)

3kSOJK+)

3kSNQA

(36)

kS\N&f6,rA*}+i

,&9

(1777 – 1855)

10^k/% m_\Nm_tNQ,=(r?(?#JE'$XK

S*&5P<kN!' js/G ()1

(37)

kS\N&f6,rA*}+i

,&9

(1777 – 1855)

10^k/%

100

^k/%N/ii& 7e<^s

m_\Nm_tNQ,=(r?(?#JE'$XK S*&5P<kN!'

js/G ()1

(38)

kS\N&f6,rA*}+i

,&9

(1777 – 1855)

10^k/%

100

^k/%N/ii& 7e<^s

m_\Nm_tNQ,=(r?(?#JE'$XK S*&5P<kN!'

Hopf

js/G ()1

(39)

kS\N&f6,rA*}+i

,&9NoRNj9F#s0 1kt#s*J&#j"`&H`=sK

26R@ '6Ra(<FkN2X

1kt#s*

P%G%F$H kS\Nlw=

(40)

kS\N&f6,rA*}+i

,&9NoRNj9F#s0

1kt#s*J&#j"`&H`=sK

(1824 – 1907)

26R@

(1867)

'6Ra(<FkN2X

1kt#s*

P%G%F$H kS\Nlw=

(41)

kS\N&f6,rA*}+i

,&9NoRNj9F#s0

1kt#s*J&#j"`&H`=sK

(1824 – 1907)

26R@

(1867)

'6Ra(<FkN2X

1kt#s* P%G%F$H

P%G%F$H

(1831 – 1901)

kS\Nlw=

(42)

Intermezzo AkQs III $

4ZGm_gC?OsI+UJj{Krnj^7?J8

^K#4ZrZC?j7J$G""3*KQA7?@1 G"&^Nh&KsDNXC+r0;kG7g&+#

(43)

kS\N&f6,rA3F3FT

"l/5s@<

(1888 – 1971)

'

"l/5s@<?`0!JkS\TQLK

(1920’s)

i$G^$9?< '

2DNkS\^0,18kS\r=o9

(44)

kS\N&f6,rA3F3FT

"l/5s@<

(1888 – 1971)

'

"l/5s@<?`0!JkS\TQLK

(1920’s)

i$G^$9?<

(1893 – 1971)

'

2DNkS\^0,18kS\r=o9

(45)

kS\N&f6,rA3F3FT

"l/5s@<

(1888 – 1971)

'

"l/5s@<?`0!JkS\TQLK

(1920’s)

i$G^$9?<

(1893 – 1971)

'

2DNkS\^0,18kS\r=o9

() \A*K!NIjQAr-Bs\9H\j"&

(Reidemeister moves (1932))

(46)

kS\N&f6,rAVl$/9k<

t)<s&8g<s:

(1952 – )

nQGD@ 8g<s:?`0JkS\TQLK

U#<k:^u^

(47)

kS\N&f6,rAVl$/9k<

t)<s&8g<s:

(1952 – )

nQGD@

8g<s:?`0JkS\TQLK U#<k:^u^

(48)

kS\N&f6,rAVl$/9k<

t)<s&8g<s:

(1952 – )

nQGD@

A H_32 ; 8g<s:?`0JkS\TQLK U#<k:^u^

(49)

kS\N&f6,rAVl$/9k<

t)<s&8g<s:

(1952 – )

nQGD@

A H_32 ; 8g<s:?`0JkS\TQLK U#<k:^u^!

(1990)

(50)

kS\TQLA=N&O\*+)

dj ?BN&NkS\H6jN&NkS\Oc&N+)

?BN&NkS\ dj !3NsDOL?7F\vKc&N+)

(51)

kS\TQLA=N&O\*+)

dj ?BN&NkS\H6jN&NkS\Oc&N+)

6jN&NkS\ dj !3NsDOL?7F\vKc&N+)

(52)

kS\TQLA=N&O\*+)

dj ?BN&NkS\H6jN&NkS\Oc&N+)

?BN&NkS\ 6jN&NkS\ dj !3NsDOL?7F\vKc&N+)

(53)

kS\TQLA=N&O\*+)

dj ?BN&NkS\H6jN&NkS\Oc&N+)

?BN&NkS\ 6jN&NkS\ dj !3NsDOL?7F\vKc&N+)

(54)

kS\,18+I&+N=j

|,'2DNkS\^0N=o9kS\,18

, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$

183HrZ@9kKO"B]KQA7F+;lP h$#

8NzkS\a=N@|

(55)

kS\,18+I&+N=j

|,'2DNkS\^0N=o9kS\,18

, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$

183HrZ@9kKO"B]KQA7F+;lP h$#

8NzkS\a=N@|

(56)

kS\,18+I&+N=j

|,'2DNkS\^0N=o9kS\,18

, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$

183HrZ@9kKO"B]KQA7F+;lP h$#

8NzkS\a=N@|

(57)

kS\,18+I&+N=j

|,'2DNkS\^0N=o9kS\,18

, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$

183HrZ@9kKO"B]KQA7F+;lPh

$#JJ1HOBiJ$K 8NzkS\a=N@| )



!

(58)

kS\,18+I&+N=j

|,'2DNkS\^0N=o9kS\,18

, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$

183HrZ@9kKO"B]KQA7F+;lPh

$#JJ1HOBiJ$K 8NzkS\a=N@|*



q

(59)

kS\,18+I&+N=j

RH(kS\a=N@|)



!

$/idCFbG-J$+iH$CF"2D,[Jk" HO@(J$#

[JkbN,[Jk3Hr(9;q'JkS\KTQL TQLHO"

PkS\Q! "k8gJc'PtQ"P?`0Q"2K

JkL|G"18kS\KO18MrP~5;kbN#

(60)

kS\,18+I&+N=j

RH(kS\a=N@|)



!

$/idCFbG-J$+iH$CF"2D,[Jk" HO@(J$#

[JkbN,[Jk3Hr(9;q'JkS\KTQL TQLHO"

PkS\Q! "k8gJc'PtQ"P?`0Q"2K

JkL|G"18kS\KO18MrP~5;kbN#

(61)

kS\,18+I&+N=j

RH(kS\a=N@|)



!

$/idCFbG-J$+iH$CF"2D,[Jk" HO@(J$#

[JkbN,[Jk3Hr(9;q'JkS\KTQL TQLHO"

PkS\Q! "k8gJc'PtQ"P?`0Q"2K

JkL|G"18kS\KO18MrP~5;kbN#

(62)

kS\,18+I&+N=j

RH(kS\a=N@|)



!

$/idCFbG-J$+iH$CF"2D,[Jk" HO@(J$#

[JkbN,[Jk3Hr(9;q'JkS\KTQL TQLHO"

PkS\Q! "k8gJc'PtQ"P?`0Q"2K

JkL|G"18kS\KO18MrP~5;kbN#

(63)

TQLHO

P]HJk8g T "V18WG"kH$&X8 "

"k8g S"T +i S XNL| f G"V18WbNNT /h,y7$bN'a  b ) f(a) = f(b)#

PvrHkH f (a) 6= f (b) ) a 6 b

c'T = Z = f 0t g"'4GdC?>j,18"

S = f0; 1g"f'2GdC?>jrP~5;k#

T K1MX8  r~lk () T N,dr?(k

eNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK

?@7 [i℄ = f4n + ijn 2 Zg

=?

=?

=?

=?

<

<

f ,TQL () f O [i℄ e18MrHk

(64)

TQLHO

P]HJk8g T "V18WG"kH$&X8 "

"k8g S"T +i S XNL| f G"V18WbNNT /h,y7$bN'a  b ) f(a) = f(b)#

PvrHkH f (a) 6= f (b) ) a 6 b c'T = Z = f 0t g"'4GdC?>j,18"

S = f0; 1g"f'2GdC?>jrP~5;k#

T K1MX8  r~lk () T N,dr?(k

eNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK

?@7 [i℄ = f4n + ijn 2 Zg

=?

=?

=?

=?

<

<

f ,TQL () f O [i℄ e18MrHk

(65)

TQLHO

P]HJk8g T "V18WG"kH$&X8 "

"k8g S"T +i S XNL| f G"V18WbNNT /h,y7$bN'a  b ) f(a) = f(b)#

PvrHkH f (a) 6= f (b) ) a 6 b

c'T = Z = f 0t g"'4GdC?>j,18"

S = f0; 1g"f'2GdC?>jrP~5;k#

T K1MX8  r~lk () T N,dr?(k

eNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK

?@7 [i℄ = f4n + ijn 2 Zg

=?

=?

=?

=?

<

<

f ,TQL () f O [i℄ e18MrHk

(66)

TQLHO

P]HJk8g T "V18WG"kH$&X8 "

"k8g S"T +i S XNL| f G"V18WbNNT /h,y7$bN'a  b ) f(a) = f(b)#

PvrHkH f (a) 6= f (b) ) a 6 b

c'T = Z = f 0t g"'4GdC?>j,18"

S = f0; 1g"f'2GdC?>jrP~5;k#

T K1MX8  r~lk () T N,dr?(k

eNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK

?@7 [i℄ = f4n + ijn 2 Zg

=?

=?

=?

=?

<

<

f ,TQL () f O [i℄ e18MrHk

(67)

TQLHO

P]HJk8g T "V18WG"kH$&X8 "

"k8g S"T +i S XNL| f G"V18WbNNT /h,y7$bN'a  b ) f(a) = f(b)#

PvrHkH f (a) 6= f (b) ) a 6 b

c'T = Z = f 0t g"'4GdC?>j,18"

S = f0; 1g"f'2GdC?>jrP~5;k#

T K1MX8  r~lk () T N,dr?(k

eNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK

,

?@7 [i℄ = f4n + ijn 2 Zg

=?

=?

=?

=?

<

<

f ,TQL () f O [i℄ e18MrHk

(68)

TQLHO

P]HJk8g T "V18WG"kH$&X8 "

"k8g S"T +i S XNL| f G"V18WbNNT /h,y7$bN'a  b ) f(a) = f(b)#

PvrHkH f (a) 6= f (b) ) a 6 b

c'T = Z = f 0t g"'4GdC?>j,18"

S = f0; 1g"f'2GdC?>jrP~5;k#

T K1MX8  r~lk () T N,dr?(k

eNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK

,

?@7 [i℄ = f4n + ijn 2 Zg

=?

=?

=?

=?

<

<

,TQL O e18MrHk

(69)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

   

   

   

  

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V H NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(70)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

   

   

  

   

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V H NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(71)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

   

   

  

   

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V H NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(72)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

   

   

  

   

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V H NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(73)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V H NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(74)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V H NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(75)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V

14

H

15

NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(76)

15Q:kNTQL

T '

15

Q:kNuV4FJ

16!

DN5hjJkK"

'T<9r9i$I5;F\j"&"

S = f+1; 1g"f'V9Ndf  "kU!

1MX8  O"

V1DNT<9r#K0+9W HV1DNT<9rDK0+9W

N2DN`nr-Bs+jV;P@ilk#

f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"

V

14

H

15

NT<9@1~lX(k3HOG-J$W 3H,,+k

kS\TQLJ8g<s:?`0KKaj^7g&#

(77)

H_32A Jones ?`0XNM16

n \N3+iJkH_34N Bn O



GQJi;Krjak3HKhj"V2WKJk# et*JHj7$,G-k*

3NH_32O! !NANbNH=NU5G8. 5l"!NX80JH_3X80Kr~?9#

  

(78)

H_32A Jones ?`0XNM16

n \N3+iJkH_34N Bn O



GQJi;Krjak3HKhj"V2WKJk# et*JHj7$,G-k*

3NH_32O! !NANbNH=NU5G8. 5l"!NX80JH_3X80Kr~?9#

  

(79)

H_32A Jones ?`0XNM16

n \N3+iJkH_34N Bn O



GQJi;Krjak3HKhj"V2WKJk# et*JHj7$,G-k*

3NH_32O! !NANbNH=NU5G8. 5l"!NX80JH_3X80Kr~?9#

  

(80)

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk

(81)

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk



(82)

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk



(83)

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W" S = ft (t

1

2 ) NJiNQbv7?K0t8t?`0 g kS\^?Om_\ L N8g<s:?`0 VL(t) O"

!N2DN-AGC'U1ilk#

+@JkS\r{G=9H VÆ(t) = 1 91$sX80

t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)

?@7"L+; L ; L0 O"k^0N1DN r@ Nt ,@1r!GV-9(FG-kbN'





(84)

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W" S = ft (t

1

2 ) NJiNQbv7?K0t8t?`0 g kS\^?Om_\ L N8g<s:?`0 VL(t) O"

!N2DN-AGC'U1ilk# +@JkS\r{G=9H VÆ(t) = 1

91$sX80

t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)

?@7"L+; L ; L0 O"k^0N1DN r@ Nt ,@1r!GV-9(FG-kbN'





(85)

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W" S = ft (t

1

2 ) NJiNQbv7?K0t8t?`0 g kS\^?Om_\ L N8g<s:?`0 VL(t) O"

!N2DN-AGC'U1ilk#

+@JkS\r{G=9H VÆ(t) = 1 91$sX80

t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)

?@7"L+; L ; L0 O"k^0N1DN r@ Nt ,@1r!GV-9(FG-kbN'





(86)

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W" S = ft (t

1

2 ) NJiNQbv7?K0t8t?`0 g kS\^?Om_\ L N8g<s:?`0 VL(t) O"

!N2DN-AGC'U1ilk#

+@JkS\r{G=9H VÆ(t) = 1 91$sX80

t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)

?@7"L+; L ; L0 O"k^0N1DN

r@

Nt ,@1r!GV-9(FG-kbN'

(87)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)

(88)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)



(89)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)





 

(90)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)













(91)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)

















(92)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)



















(93)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)























(94)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)























(95)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)























1 1

(96)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)























n

1 1 1 1

o

(97)

RH(kS\N8g<s:?`0

V

Æ

(t) = 1 u t 1

V

L

+

(t) tV

L

(t) = (t 1

2

t 1

2

)V

L

0

(t)























n

1 1 1 5

o

(98)

8g<s:?`0GNkS\=j

RH(kS\H=N@|



K K

V

K

(t) = t + t + t 6=V

K

+

(t) = t + t + t

?BN&NkS\H6jN&NkS\

?BN&NkS\ 6jN&NkS\

V

TL

(t) = t 3

+ t 5

t 8

6=V

FL

(t) = 1 t+3t 2

3t 3

+3t 4

4t 5

+3t 6

2t 7

+t 8

(99)

8g<s:?`0GNkS\=j

RH(kS\H=N@|Oc&*



K K



V

K

(t) = t 4

+ t 3

+ t 1

6=V

K

+

(t) = t 4

+ t 3

+ t

?BN&NkS\H6jN&NkS\

?BN&NkS\ 6jN&NkS\

V

TL

(t) = t 3

+ t 5

t 8

6=V

FL

(t) = 1 t+3t 2

3t 3

+3t 4

4t 5

+3t 6

2t 7

+t 8

(100)

8g<s:?`0GNkS\=j

RH(kS\H=N@|Oc&*



K K



V

K

(t) = t 4

+ t 3

+ t 1

6=V

K

+

(t) = t 4

+ t 3

+ t

?BN&NkS\H6jN&NkS\

?BN&NkS\

TL

6jN&NkS\

FL

V

TL

(t) = t + t t

6=V

FL

(t) = 1 t+3t 2

3t 3

+3t 4

4t 5

+3t 6

2t 7

+t 8

(101)

8g<s:?`0GNkS\=j

RH(kS\H=N@|Oc&*



K K



V

K

(t) = t 4

+ t 3

+ t 1

6=V

K

+

(t) = t 4

+ t 3

+ t

?BN&NkS\H6jN&NkS\Oc&*

?BN&NkS\

TL

6=

6jN&NkS\

FL

V

TL

(t) = t 3

+ t 5

t 8

6=V (t) = 1 t+3t 2

3t 3

+3t 4

4t 5

+3t 6

2t 7

+t 8

(102)

>,nHNX"A3!5?MN

|\tXqbGNkS\}@NLV  XJH]m8<K JIAXfK

J"kG-&JK2!5?MNJJLKNc'

~-U1D= ~-U1TD= -&

J7 -&

" j

(103)

>,nHNX"A3!5?MN

|\tXqbGNkS\}@NLV  XJH]m8<K JIAXfK

3!5?MN@    H]m8<"v?X

?MN'Jv?XTKHCFNKVuVWJNlDK# Ij*K_kHI3b

(

f</jCI

(

>

)

uVH

)

18 J"kG-&JK2!5?MNJJLKNc'

~-U1D= ~-U1TD= -&

J 7 -&

" j

(104)

>,nHNX"A3!5?MN

?MN'Jv?XTKHCFNKVuVWJNlDK# Ij*K_kHI3b

(

f</jCI

(

>

)

uVH

)

18 J"kG-&JK2!5?MNJJLKNc'

~-U1D= ~-U1TD= -&

J 7 -&

" j

(105)

>,nHNX"A3!5?MN

?MN'Jv?XTKHCFNKVuVWJNlDK# Ij*K_kHI3b

(

f</jCI

(

>

)

uVH

)

18 J"kG-&JK2!5?MNJJLKNc'

~-U1D= ~-U1TD= -&

J 7 -&

" j

(106)

>,nHNX"A3!5?MN

?MN'Jv?XTKHCFNKVuVWJNlDK# Ij*K_kHI3b

(

f</jCI

(

>

)

uVH

)

18 J"kG-&JK2!5?MNJJLKNc'

~-U1D= ~-U1TD= -&

J 7 -&

" j

(107)

>,nHNX"A3!5?MN

?MN'Jv?XTKHCFNKVuVWJNlDK# Ij*K_kHI3b

(

f</jCI

(

>

)

uVH

)

18 J"kG-&JK2!5?MNJJLKNc'

~-U1D= ~-U1TD= -&

J 7 -&

" j

(108)

?MNHkS\

?MNGJ$c'

ߟ߱ࠇߡ޿ࠆ ੤ࠊߞߡ޿ࠆ

߇߁

?MNHkS\NDJ,j

?MNN=o7}

(!    gN2!5^G 0!    eLH=NQJI yAuVJI

ZC?=C? "jQ

(109)

?MNHkS\

?MNGJ$c'

ߟ߱ࠇߡ޿ࠆ ੤ࠊߞߡ޿ࠆ

߇߁

?MNHkS\NDJ,j

?MNN=o7}

(!    gN2!5^G 0!    eLH=NQJI yAuVJI

ZC?=C? "jQ

(110)

?MNHkS\

?MNGJ$c'

ߟ߱ࠇߡ޿ࠆ ੤ࠊߞߡ޿ࠆ

߇߁

?MNHkS\NDJ,j

?MNN=o7}

(!    gN2!5^G 0!    eLH=NQJI

yAuVJI

ZC?=C? "jQ

(111)

?MNHkS\

?MNGJ$c'

ߟ߱ࠇߡ޿ࠆ ੤ࠊߞߡ޿ࠆ

߇߁

?MNHkS\NDJ,j

?MNN=o7}

(!    gN2!5^G 0!    eLH=NQJI yAuVJI

ZC?=C? "jQ

(112)

?MNHkS\

?MNGJ$c'

ߟ߱ࠇߡ޿ࠆ ੤ࠊߞߡ޿ࠆ

߇߁

?MNHkS\NDJ,j

?MNN=o7}

(!    gN2!5^G 0!    eLH=NQJI yAuVJI

ZC?=C? "jQ

(113)

?MNHkS\

?MNGJ$c'

ߟ߱ࠇߡ޿ࠆ ੤ࠊߞߡ޿ࠆ

߇߁

?MNHkS\NDJ,j

?MNN=o7}

(!    gN2!5^G 0!    eLH=NQJI yAuVJI

ZC?=C? (cut and paste)"jQ

(114)

ZC?=C?G?MNrD/k

3!5eL S3 NkS\ K KhC?jQ

S

3 +i K NIua5Nbtrhj|$?bNHLN I<JCDr"=LNH<i9G/CD1k#

(115)

ZC?=C?G?MNrD/k

3!5eL S3 NkS\ K KhC?jQ

K NIua5

Nbtrhj|/# 3NIua5O

Lj*KO

=jCIH<i9#

S

3 +i K NIua5Nbtrhj|$?bNHLN I<JCDr"=LNH<i9G/CD1k#

(116)

ZC?=C?G?MNrD/k

3!5eL S3 NkS\ K KhC?jQ

K NIua5

Nbtrhj|/# 3NIua5O

Lj*KO

=jCIH<i9#

S

3 +i K NIua5Nbtrhj|$?bNHLN I<JCDr"=LNH<i9G/CD1k#

(117)

kS\KhC?3!5eLNjQ

S

3 +i K NIua5Nbtrhj|$?bNHLN I<JCDr"=LNH<i9G"V$~1N,TC? j+5Jkh&K"=jgo;k#

[

LN?MN,G-k#

3!5eL+i"kS\"m_\KhC?jQG" J"kror~?9K$UN?MN,@ilk#

(118)

kS\KhC?3!5eLNjQ

S

3 +i K NIua5Nbtrhj|$?bNHLN I<JCDr"=LNH<i9G"V$~1N,TC? j+5Jkh&K"=jgo;k#

[

LN?MN,G-k# 3!5eL+i"kS\"m_\KhC?jQG"

J"kror~?9K$UN?MN,@ilk#

(119)

kS\KhC?3!5eLNjQ

S

3 +i K NIua5Nbtrhj|$?bNHLN I<JCDr"=LNH<i9G"V$~1N,TC? j+5Jkh&K"=jgo;k#

[

LN?MN,G-k#

3!5eL+i"kS\"m_\KhC?jQG" J"kror~?9K$UN?MN,@ilk#

(120)

>,nHNX"

nQGDJ8g<s:K }WOXJ,[Xt"ds&P/9?<X80JIK

kS\HGtJ9< ;*aJe#gXKK

v?X*kS\}@'kS\NAN#(5rO+j"- l$JAraakJkS\N(Mk.<"&Av?X"

"?Qr@t" K b,RJX"DNA

(121)

>,nHNX"

nQGDJ8g<s:K

}WOXJ,[Xt"ds&P/9?<X80JIK kS\HGtJ9< ;*aJe#gXKK

v?X*kS\}@'kS\NAN#(5rO+j"- l$JAraakJkS\N(Mk.<"&Av?X"

"?Qr@t" K b,RJX"DNA

(122)

>,nHNX"

nQGDJ8g<s:K

}WOXJ,[Xt"ds&P/9?<X80JIK kS\HGtJ9< ;*aJe#gXKK

v?X*kS\}@'kS\NAN#(5rO+j"- l$JAraakJkS\N(Mk.<"&Av?X"

"?Qr@t" K b,RJX"DNA

(123)

>,nHNX"

nQGDJ8g<s:K

}WOXJ,[Xt"ds&P/9?<X80JIK kS\HGtJ9< ;*aJe#gXKK

v?X*kS\}@'kS\NAN#(5rO+j"- l$JAraakJkS\N(Mk.<"&Av?X"

ideal knots

"?Qr@t"

etc

K

b,RJX"DNA

(124)

>,nHNX"

nQGDJ8g<s:K

}WOXJ,[Xt"ds&P/9?<X80JIK kS\HGtJ9< ;*aJe#gXKK

v?X*kS\}@'kS\NAN#(5rO+j"- l$JAraakJkS\N(Mk.<"&Av?X"

ideal knots

"?Qr@t"

etc

K

b,RJX"DNA

(125)

b,RJX

]j^<,kS\KJCF$kH"*AN-AJE5J IK,Qok3H,"ki7$#

=[ Du]j^<,9/

JkH"Js+@JKkS\KJkN(, 1 KJk#

tMB3G"is@`KkS\r/85;"INkS\

?,INh&JN(G=lk+"H$&&f,"k# )}JReNis@`&)</

+Jr57J$^l~

(126)

b,RJX

]j^<,kS\KJCF$kH"*AN-AJE5J IK,Qok3H,"ki7$#

=[

(Frisch-Wasserman-Delbruck)

Du]j^<,9/ JkH"Js+@JKkS\KJkN(, 1 KJk#

tMB3G"is@`KkS\r/85;"INkS\

?,INh&JN(G=lk+"H$&&f,"k# )}JReNis@`&)</

+Jr57J$^l~

(127)

b,RJX

]j^<,kS\KJCF$kH"*AN-AJE5J IK,Qok3H,"ki7$#

=[

(Frisch-Wasserman-Delbruck)

Du]j^<,9/ JkH"Js+@JKkS\KJkN(, 1 KJk#

tMB3G"is@`KkS\r/85;"INkS\

?,INh&JN(G=lk+"H$&&f,"k# )}JReNis@`&)</

+Jr57J$^l~

(128)

DNA

G*-7j\K@ sEi;s=$

Du"D^jkS\KJCF$k3H,"k#

(129)

DNA

G*-7j\K@ sEi;s=$ Du"D^jkS\KJCF$k3H,"k#

(130)

DNA

G*-7j\K@ sEi;s=$

Du"D^jkS\KJCF$k3H,"k#

(131)

sEi;s

2\NkS\r K1; K2 3"r K H9k'

K

1 H

K

2 Nm_tr

Lk"M8ltr T w" K Ni$8s0t r W r H9kH

Lk = W r + T w

Jky0,.)#

(132)

sEi;s

2\NkS\r K1; K2 3"r K H9k'

K

1 H

K

2 Nm_tr

Lk"M8ltr T w" K Ni$8s0t

(writhe)

r W r H9kH

Lk = W r + T w

(133)

m_tN,&9NQ,x0

V$k<W C2 KE.,.lF$kH9k# S*&5P<kN!''@ y GN'l dH = j dl  r

4jrj 3

jdl'@ x GNE.WG"r = y x u$k<W C1 eQ,9kH"

1

4 Z

C

1 Z

C

2

(y x)  (dx  dy)

jy xj 3

C

1 H C

2 Nm_t

Lk(C

1

; C

2

)#

(134)

m_tN,&9NQ,x0

V$k<W C2 KE.,.lF$kH9k#

y x

S*&5P<kN!''@ y GN'l dH = j dl  r

4jrj 3

jdl'@ x GNE.WG"r = y x u$k<W C1 eQ,9kH"

1

4 Z

C

1 Z

C

2

(y x)  (dx  dy)

jy xj 3

C

1 H C

2 Nm_t

Lk(C

1

; C

2

)#

(135)

m_tN,&9NQ,x0

V$k<W C2 KE.,.lF$kH9k#

y x

S*&5P<kN!''@ y GN'l dH = j dl  r

4jrj 3

jdl'@ x GNE.WG"r = y x u$k<W C1 eQ,9kH"

1

4 Z

C

1 Z

C

2

(y x)  (dx  dy)

jy xj 3

Figure eight !J8NzkS\K

参照

Outline

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