kS\NtX#f!_sTgXl~|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.1/38

\!

kS\H=NgV?AHtX
8hNfNkS\"m_\"H_3
tXG7&kS\
2DNkS\,18HO)
6J1JrK|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.2/38

\!

kS\H=NgV?AHtX
8hNfNkS\"m_\"H_3
tXG7&kS\
2DNkS\,18HO)
6J1JrK
kS\TQL
1MX8HTQL
8g<s:?`0|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.2/38

\!

kS\H=NgV?AHtX
8hNfNkS\"m_\"H_3
tXG7&kS\
2DNkS\,18HO)
6J1JrK
kS\TQL
1MX8HTQL
8g<s:?`0>N,nHN8ojv?"J3!5K?MN@
8*JDNAK"b,RJX|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.2/38

8hNfNkS\JNgVK

BQ'|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.3/38

8hNfNkS\JNgVK

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

2008/Y)(mg,Jq&T1Viq

 p.3/38

8hNfNkS\JNgVK

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

2008/Y)(mg,Jq&T1Viq

 p.3/38

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK
u~Q'M/?$"ez"j\s"fO"!!!!H_3"

shoulder knot|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.3/38

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK
u~Q'M/?$"ez"j\s"fO"!!!!H_3@U'"shoulder knot|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.3/38

8hNfNkS\JNgVK

BQ'%hj""&HI""J>t`NdT/pK
u~Q'M/?$"ez"j\s"fO"!!!!H_3@U'kSNlV"NCHJ%N.5K"!!!

"

shoulder knot

ut the Gordian knot, nodo d'amore|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.3/38

kS\NcA>0ND$?kS\

half hit h|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.4/38

kS\NcA>0ND$?kS\

half hit hsquare kont,^kSreef knot()|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.4/38

kS\NcA>0ND$?kS\

half hit hsquare kont,^kSrunning knot

reef knot()|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.4/38

kS\NcABQ"*hSu~Q

sheep shank|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.5/38

kS\NcABQ"*hSu~Q"oSkS
sheep shank|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.5/38

kS\NcABQ"*hSu~Q

"oSkS

Z#kS

sheep shank|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.5/38

kS\NgV?A\m_"s&js/|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.6/38

kS\NgV?A\m_"s&js/

H_3|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.6/38

kS\NgV?A\m_"s&js/

H_3(pure) braid|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.6/38

tXG7&kS\
ksG$k|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.7/38

tXG7&kS\)

ksG$k &<r;/7F$/|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.7/38

tXG7&kS\))

ksG$k &<r;/7F$/ ksG$J$|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.7/38

tXG7&kS\))

ksG$k &<r;/7F$/ ksG$J$<@rjK}CF%5J1lP[I1J$#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.7/38

tXG7&kS\))

ksG$k &<r;/7F$/ ksG$J$<@rjK}CF%5J1lP[I1J$#

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

2008/Y)(mg,Jq&T1Viq

 p.7/38

J1JkS\Nc
Trefoil

J0UkS\K=)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.8/38

J1JkS\Nc
Trefoil

J0UkS\K=)Figure eight!J8NzkS\K=)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.8/38

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#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.9/38

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
|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.9/38

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)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.9/38

3kSOJK+)

3kSNQA|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.10/38

3kSOJK+)

3kSNQA|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.10/38

3kSOJK+)

3kSNQA|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.10/38

3kSOJK+)

3kSNQA|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.10/38

3kSOJK+)

3kSNQA|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.10/38

3kSOJK+)

3kSNQA

3kSaRH(kS|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.10/38

kS\N&f6,rA*}+i,&9(1777  1855)10^k/%|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.11/38

kS\N&f6,rA*}+i,&9(1777  1855)10^k/%

100^k/%N/ii&7e<^s|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.11/38

kS\N&f6,rA*}+i,&9(1777  1855)^k/%N/ii&
10^k/%
7e<^sm_\Nm_tNQ,=(r?(?#JE'$XK
S*&5P<kN!'
js/G ()1
100

Hopf

|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.11/38

kS\N&f6,rA*}+i,&9NoRNj9F#s0|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.12/38

kS\N&f6,rA*}+i,&9NoRNj9F#s0
1kt#s*J&#j"`&H`=sK
26R@'6Ra(<FkN2X(1824  1907)(1867)

1kt#s*|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.12/38

kS\N&f6,rA*}+i,&9NoRNj9F#s0
1kt#s*J&#j"`&H`=sK
26R@'6Ra(<FkN2X(1824  1907)(1867)

1kt#s*
P%G%F$H(1831  1901)

P%G%F$H
kS\Nlw=|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.12/38

Intermezzo

AkQs III $4ZGm_gC?OsI+UJj{Krnj^7?J8^K#4ZrZC?j7J$G""3*KQA7?@1
G"&^Nh&KsDNXC+r0;kG7g&+#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.13/38

kS\N&f6,rA3F3FT"l/5s@<'"l/5s@<?`0!JkS\TQLK(1888  1971)(1920's)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.14/38

kS\N&f6,rA3F3FT"l/5s@<'"l/5s@<?`0!JkS\TQLK
i$G^$9?<'2DNkS\^0,18kS\r=o9(1888  1971)(1920's)(1893  1971)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.14/38

kS\N&f6,rA3F3FT"l/5s@<'"l/5s@<?`0!JkS\TQLK
i$G^$9?<'2DNkS\^0,18kS\r=o9() \A*K!NIjQAr-Bs\9H\j"&(1888  1971)(1920's)(1893  1971)(Reidemeister moves (1932))|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.14/38

kS\N&f6,rAVl$/9k<t)<s&8g<s:(1952  )nQGD@|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.15/38

kS\N&f6,rAVl$/9k<t)<s&8g<s:(1952  )nQGD@

8g<s:?`0JkS\TQLK|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.15/38

kS\N&f6,rAVl$/9k<t)<s&8g<s:(1952  )nQGD@

A H_32 ; 8g<s:?`0JkS\TQLK|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.15/38

kS\N&f6,rAVl$/9k<t)<s&8g<s:(1952  )nQGD@

A H_32 ; 8g<s:?`0JkS\TQLK
U#<k:^u^! (1990)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.15/38

kS\TQLA=N&O\*+)

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

2008/Y)(mg,Jq&T1Viq

 p.16/38

kS\TQLA=N&O\*+)

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

6jN&NkS\|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.16/38

kS\TQLA=N&O\*+)

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

6jN&NkS\|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.16/38

kS\TQLA=N&O\*+)

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

6jN&NkS\

dj !3NsDOL?7F\vKc&N+)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.16/38

kS\,18+I&+N=j|,'2DNkS\^0N=o9kS\,18, i$G^$9?<&`<Vr?s+9lP\j"&|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.17/38

kS\,18+I&+N=j|,'2DNkS\^0N=o9kS\,18, i$G^$9?<&`<Vr?s+9lP\j"&3N"k4j:`OJ$|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.17/38

kS\,18+I&+N=j|,'2DNkS\^0N=o9kS\,18, i$G^$9?<&`<Vr?s+9lP\j"&3N"k4j:`OJ$
183HrZ@9kKO"B]KQA7F+;lPh$#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.17/38

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@| )!

|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.17/38

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|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.17/38

kS\,18+I&+N=j

RH(kS\a=N@|)!

|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.18/38

kS\,18+I&+N=j

RH(kS\a=N@|)!



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

2008/Y)(mg,Jq&T1Viq

 p.18/38

kS\,18+I&+N=j

RH(kS\a=N@|)!



$/idCFbG-J$+iH$CF"2D,[Jk"HO@(J$#[JkbN,[Jk3Hr(9;q'JkS\KTQL|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.18/38

kS\,18+I&+N=j

RH(kS\a=N@|)!



$/idCFbG-J$+iH$CF"2D,[Jk"HO@(J$#[JkbN,[Jk3Hr(9;q'JkS\KTQLTQLHO"
PkS\Q! "k8gJc'PtQ"P?`0Q"2K
JkL|G"18kS\KO18MrP~5;kbN#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.18/38

TQLHOP]HJk8g T "V18WG"kH$&X8 ""k8g S"T +i S XNL| f G"V18WbNNT/h,y7$bN'a  b ) f (a) = f (b)#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.19/38

TQLHOP]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|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.19/38

TQLHOP]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 bc'T = Z = f 0t g"'4GdC?>j,18"
S = f0; 1g"f '2GdC?>jrP~5;k#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.19/38

TQLHOP]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 bc'T = Z = f 0t g"'4GdC?>j,18"
S = f0; 1g"f '2GdC?>jrP~5;k#T K1MX8  r~lk () T N,dr?(k|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.19/38

TQLHOP]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 bc'T = Z = f 0t g"'4GdC?>j,18"
S = f0; 1g"f '2GdC?>jrP~5;k#T K1MX8  r~lk () T N,dr?(keNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK?@7 [i℄ = f4n + ijn 2 Z g
,

<<

=?
=?
=?

=?

|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.19/38

TQLHOP]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 bc'T = Z = f 0t g"'4GdC?>j,18"
S = f0; 1g"f '2GdC?>jrP~5;k#T K1MX8  r~lk () T N,dr?(keNc@H"Z = [0℄ [ [1℄ [ [2℄ [ [3℄JsrBK?@7 [i℄ = f4n + ijn 2 Z g
,

<<

=?
=?
=?

=?

f ,TQL () f O [i℄ e18MrHk|\tXq& /Y)(mg,Jq&T1Viq
2008 p.19/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
         |\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
         |\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
 
 
 

     |\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
 
 

 
     |\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
1MX8  O"

V1DNT<9r#K0+9W
H
V1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
1MX8  O"

V1DNT<9r#K0+9W
H
V1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
1MX8  O"

V1DNT<9r#K0+9W
H
V1DNT<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,,+k14

15|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
1MX8  O"

V1DNT<9r#K0+9W
H
V1DNT<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,,+kkS\TQLJ8g<s:?`0KKaj^7g&#
14

15|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.20/38

H_32AJones ?`0XNM16

n \N3+iJkH_34N Bn OGQJi;Krjak3HKhj"V2WKJk#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.21/38

H_32AJones ?`0XNM16

n \N3+iJkH_34N Bn OGQJi;Krjak3HKhj"V2WKJk#
et*JHj7$,G-k*|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.21/38

H_32AJones ?`0XNM16

n \N3+iJkH_34N Bn OGQJi;Krjak3HKhj"V2WKJk#
et*JHj7$,G-k*
3NH_32O! !NANbNH=NU5G8.
5l"!NX80JH_3X80Kr~?9#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.21/38

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.22/38

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk
|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.22/38

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk
|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.22/38

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W"
1
S = ft (t 2 ) NJiNQbv7?K0t8t?`0 g|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.23/38

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W"
1
S = ft (t 2 ) NJiNQbv7?K0t8t?`0 g
kS\^?Om_\ L N8g<s:?`0 VL(t) O"!N2DN-AGC'U1ilk#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.23/38

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W"
1
S = ft (t 2 ) NJiNQbv7?K0t8t?`0 g
kS\^?Om_\ L N8g<s:?`0 VL(t) O"!N2DN-AGC'U1ilk#+@JkS\r{G=9H VÆ(t) = 1|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.23/38

8g<s:?`0NC'E1

T = f kS\JHm_\Kg"'V18W"
1
S = ft (t 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
11
1
t VL+ (t) tVL (t) = (t 2 t 2 )VL0 (t)?@7"L+; L ; L0 O"k^0N1DN r@ Nt,@1r!GV-9(FG-kbN'|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.23/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2|\tXq&

2008

t

) ()

1
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 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

t

) ()

1
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2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

t

) ()

1
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|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

t

) ()

1
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

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2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

t

) ()

1
2 VL0 t



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|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

t

) ()

1
2 VL0 t



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



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



|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2


t

) ()

1
2 VL0 t



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



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
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|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2


t

) ()

1
2 VL0 t



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



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|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2


t

) ()

1
2 VL0 t



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

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(

1
t2

t

)

1
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=t 1

t|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

t



) ()

1
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V (t) = t 1 t 1(

1
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t
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1
|\tXq& /Y)(mg,Jq&T1Viq
1
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1
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2008

 p.24/38

RH(kS\N8g<s:?`011
VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

t



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1
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V trefoil(t) = t 1 t 1  1

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 p.24/38

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

RH(kS\H=N@|K

K|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.25/38

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

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

VK (t) = t 4 + t 3 + t 16=VK+ (t) = t4 + t3 + t
KK|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.25/38

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

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

VK (t) = t 4 + t 3 + t 16=VK+ (t) = t4 + t3 + t
K

K?BN&NkS\H6jN&NkS\?BN&NkS\

TL

6jN&NkS\|\tXq&

2008

FL/Y)(mg,Jq&T1Viq

 p.25/38

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

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

VK (t) = t 4 + t 3 + t 16=VK+ (t) = t4 + t3 + t
K

K?BN&NkS\H6jN&NkS\Oc&*
6=?BN&NkS\
6jN&NkS\
VTL(t) = t3 + t5 t8
6=VFL(t) = 1 t +3t2 3t3 +3t4 4t5 +3t6 2t7 + t8
TL

FL|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.25/38

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JIAXfK|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.26/38

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2008)/Y)(mg,Jq&T1Viq

 p.26/38

>,nHNX"A3!5?MN?MN'Jv?XTKHCFNKVuVWJNlDK#
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 p.26/38

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2008)/Y)(mg,Jq&T1Viq

 p.26/38

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2008)/Y)(mg,Jq&T1Viq

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Ij*K_kHI3b f</jCI > uVH 18
J"kG-&JK2!5?MNJJLKNc'~-U1D= ~-U1TD=
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 p.27/38

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 p.27/38

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 p.27/38

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2008/Y)(mg,Jq&T1Viq

 p.27/38

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 p.27/38

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2008/Y)(mg,Jq&T1Viq

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3!5eL S NkS\ K KhC?jQ|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.28/38

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3!5eL S NkS\ K KhC?jQ
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2008/Y)(mg,Jq&T1Viq

 p.28/38

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2008/Y)(mg,Jq&T1Viq

 p.28/38

kS\KhC?3!5eLNjQ
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S +i K NIua5Nbtrhj|$?bNHLN

I<JCDr"=LNH<i9G"V$~1N,TC?
j+5Jkh&K"=jgo;k#[|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.29/38

kS\KhC?3!5eLNjQ
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S +i K NIua5Nbtrhj|$?bNHLN

I<JCDr"=LNH<i9G"V$~1N,TC?
j+5Jkh&K"=jgo;k#[
LN?MN,G-k#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.29/38

kS\KhC?3!5eLNjQ
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S +i K NIua5Nbtrhj|$?bNHLN

I<JCDr"=LNH<i9G"V$~1N,TC?
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LN?MN,G-k#
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J"kror~?9K$UN?MN,@ilk#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.29/38

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2008/Y)(mg,Jq&T1Viq

 p.30/38

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2008/Y)(mg,Jq&T1Viq

 p.30/38

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2008/Y)(mg,Jq&T1Viq

 p.30/38

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et|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.30/38

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b,RJX"DNAideal knots

et|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.30/38

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2008/Y)(mg,Jq&T1Viq

 p.31/38

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IK,Qok3H,"ki7$#=[Du]j^<,9/
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2008/Y)(mg,Jq&T1Viq

 p.31/38

b,RJX]j^<,kS\KJCF$kH"*AN-AJE5J
IK,Qok3H,"ki7$#=[Du]j^<,9/
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2008/Y)(mg,Jq&T1Viq

 p.31/38

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G*-7j\K@|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.32/38

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

2008/Y)(mg,Jq&T1Viq

 p.32/38

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

2008/Y)(mg,Jq&T1Viq

 p.32/38

sEi;s2\NkS\r K1; K2 3"r K H9k'|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.33/38

sEi;s2\NkS\r K1; K2 3"r K H9k'

K1 H K2 Nm_tr Lk"M8ltr T w"
K Ni$8s0t (writhe) r W r H9kH
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Jky0,.)#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.33/38

m_tN,&9NQ,x0

V$k<W C2 KE.,.lF$kH9k#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.34/38

m_tN,&9NQ,x0

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

j dl  r
S*&5P<kN!''@ y GN'l dH = 4jrj3
jdl'@ x GNE.WG"r = y x|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.34/38

m_tN,&9NQ,x0

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

j dl  r
S*&5P<kN!''@ y GN'l dH = 4jrj3
jdl'@ x GNE.WG"r = y x
u$k<W C1 eQ,9kH"
Z Z
1(y x)  (dx  dy)
4 C1 C2
jy xj3C1 H C2 Nm_t Lk(C1; C2)# |\tXq& /Y)(mg,Jq&T1Viq
2008 p.34/38

,&9Q,x0+iI89kbNZ Z(y x)  (dx  dy)1
Lk(C1; C2) =4 C1 C2
jy xj3|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.35/38

,&9Q,x0+iI89kbNZ Z(y x)  (dx  dy)1
Lk(C1; C2) =4 C1 C2
jy xj3eN0G C1 = C2 H7?bN,
C1 Ni$8s0t'ZZ1(y x)  (dx  dy)
W r(C1) =4 C1C1n
jy xj3?@7  = f(x; x)jx 2 C1g|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.35/38

,&9Q,x0+iI89kbNZ Z(y x)  (dx  dy)1
Lk(C1; C2) =4 C1 C2
jy xj3eN0G C1 = C2 H7?bN,
C1 Ni$8s0t'ZZ1(y x)  (dx  dy)
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9KeN0GdPMrD1?bN,?Qr@t'
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jy xj3C1 N#(5rO+kL#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.35/38

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H]$=ai<<'DNAZG7Fkg9kZG|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.36/38

DNAkS\

H]$=ai<<'DNAZG7Fkg9kZG^0Nr@Ne<,UKJk3H,"k|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.36/38

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H]$=ai<<'DNAZG7Fkg9kZG^0Nr@Ne<,UKJk3H,"k

kS\,Qok3H,"k|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.36/38

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H]$=ai<<'DNAZG7Fkg9kZG^0Nr@Ne<,UKJk3H,"k

kS\,Qok3H,"kIsJkS\b"=N^0Nr@Ne<r$/D+Q(
lP"+@JkS\K9k3H,G-k#|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.36/38

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H]$=ai<<'DNAZG7Fkg9kZG^0Nr@Ne<,UKJk3H,"k

kS\,Qok3H,"kIsJkS\b"=N^0Nr@Ne<r$/D+Q(
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2008/Y)(mg,Jq&T1Viq

 p.36/38

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kS\}@O3l+iI3XT/N+)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.37/38

kSN@U

kS\}@O3l+iI3XT/N+)
IN,n,kS\}@K7?J
C-r?(kN+)|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.37/38

ms@4Z=j{NO:7+?

3GOJ/F4Z"H$&H3m,_=G9#0lF$kuV+iM(?},>F,G^9,'|\tXq&

2008/Y)(mg,Jq&T1Viq

 p.38/38