kS\NtX
#f!_ sTgXl~
\!
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
\!
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
\!
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
8hNfNkS\JNgVK
BQ' u~Q'
@U'
8hNfNkS\JNgVK
BQ'%hj""&HI""J>t`NdT/pK u~Q'
@U'
8hNfNkS\JNgVK
BQ'%hj""&HI""J>t`NdT/pK u~Q'
@U'
8hNfNkS\JNgVK
BQ'%hj""&HI""J>t`NdT/pK
u~Q'M/?$"ez"j\s"fO"
shoulder knot
"!!!!H_3
@U'
8hNfNkS\JNgVK
BQ'%hj""&HI""J>t`NdT/pK
u~Q'M/?$"ez"j\s"fO"
shoulder knot
"!!!!H_3
@U'
8hNfNkS\JNgVK
BQ'%hj""&HI""J>t`NdT/pK
u~Q'M/?$"ez"j\s"fO"
shoulder knot
"!!!!H_3
@U'kSNlV"NCHJ%N.5K"
!!!
cut the Gordian knot, nodo d’amore
kS\NcA>0ND$?kS\
half hitch
kS\NcA>0ND$?kS\
half hitch square kont,
reef knot
(
^kS)
kS\NcA>0ND$?kS\
half hitch square kont,
reef knot
(
^kS)
running knot
kS\NcABQ"*hSu~Q
sheep shank
kS\NcABQ"*hSu~Q
sheep shank
"oSkS
kS\NcABQ"*hSu~Q
sheep shank
"oSkS Z#kS
kS\NgV?A
\m_"s&j s/
kS\NgV?A
\m_"s&j s/
H_3
kS\NgV?A
\m_"s&j s/
H_3
(pure) braid
tXG7&kS\
ksG$k
<@rjK}CF%5J1lP[I1J$#
jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~
tXG7&kS\
ksG$k
)
&<r;/7F$/
<@rjK}CF%5J1lP[I1J$#
jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~
tXG7&kS\
ksG$k
)
&<r;/7F$/
)
ksG$J$
<@rjK}CF%5J1lP[I1J$#
jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~
tXG7&kS\
ksG$k
)
&<r;/7F$/
)
ksG$J$
<@rjK}CF%5J1lP[I1J$# jA !tXG7&kS\a3!5uVK"+,+HH
roiJ$h&Kda~^l?_~
tXG7&kS\
ksG$k
)
&<r;/7F$/
)
ksG$J$
<@rjK}CF%5J1lP[I1J$#
jA !tXG7&kS\a3!5uVK"+,+HH roiJ$h&Kda~^l?_~
J1JkS\Nc
Trefoil
J0UkS\K=)
!J8NzkS\K
=)
J1JkS\Nc
Trefoil
J0UkS\K=)
Figure eight
!J8NzkS\K=)
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$&K
3kS\O?H18)
sDNkS\,18"HO)
jA !sDNkS\,V18WJ1?"$=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)
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$&K
3kS\O?H18)
3kSOJK+)
3kSNQA
3kSOJK+)
3kSNQA
3kSOJK+)
3kSNQA
3kSOJK+)
3kSNQA
3kSOJK+)
3kSNQA
3kSOJK+)
3kSNQA
kS\N&f6,rA*}+i
,&9
(1777 – 1855)
10^k/% m_\Nm_tNQ,=(r?(?#JE'$XK
S*&5P<kN!' js/G ()1
kS\N&f6,rA*}+i
,&9
(1777 – 1855)
10^k/%
100
^k/%N/ii& 7e<^sm_\Nm_tNQ,=(r?(?#JE'$XK S*&5P<kN!'
js/G ()1
kS\N&f6,rA*}+i
,&9
(1777 – 1855)
10^k/%
100
^k/%N/ii& 7e<^sm_\Nm_tNQ,=(r?(?#JE'$XK S*&5P<kN!'
Hopf
js/G ()1kS\N&f6,rA*}+i
,&9NoRNj9F#s0 1kt#s*J&#j"`&H`=sK
26R@ '6Ra(<FkN2X
1kt#s*
P%G%F$H kS\Nlw=
kS\N&f6,rA*}+i
,&9NoRNj9F#s0
1kt#s*J&#j"`&H`=sK
(1824 – 1907)
26R@(1867)
'6Ra(<FkN2X1kt#s*
P%G%F$H kS\Nlw=
kS\N&f6,rA*}+i
,&9NoRNj9F#s0
1kt#s*J&#j"`&H`=sK
(1824 – 1907)
26R@(1867)
'6Ra(<FkN2X1kt#s* P%G%F$H
P%G%F$H
(1831 – 1901)
kS\Nlw=Intermezzo AkQs III $
4ZGm_gC?OsI+UJj{Krnj^7?J8
^K#4ZrZC?j7J$G""3*KQA7?@1 G"&^Nh&KsDNXC+r0;kG7g&+#
kS\N&f6,rA3F3FT
"l/5s@<
(1888 – 1971)
'"l/5s@<?`0!JkS\TQLK
(1920’s)
i$G^$9?< '2DNkS\^0,18kS\r=o9
kS\N&f6,rA3F3FT
"l/5s@<
(1888 – 1971)
'"l/5s@<?`0!JkS\TQLK
(1920’s)
i$G^$9?<(1893 – 1971)
'2DNkS\^0,18kS\r=o9
kS\N&f6,rA3F3FT
"l/5s@<
(1888 – 1971)
'"l/5s@<?`0!JkS\TQLK
(1920’s)
i$G^$9?<(1893 – 1971)
'2DNkS\^0,18kS\r=o9
() \A*K!NIjQAr-Bs\9H\j"&
(Reidemeister moves (1932))
kS\N&f6,rAVl$/9k<
t)<s&8g<s:
(1952 – )
nQGD@ 8g<s:?`0JkS\TQLKU#<k:^u^
kS\N&f6,rAVl$/9k<
t)<s&8g<s:
(1952 – )
nQGD@8g<s:?`0JkS\TQLK U#<k:^u^
kS\N&f6,rAVl$/9k<
t)<s&8g<s:
(1952 – )
nQGD@A H_32 ; 8g<s:?`0JkS\TQLK U#<k:^u^
kS\N&f6,rAVl$/9k<
t)<s&8g<s:
(1952 – )
nQGD@A H_32 ; 8g<s:?`0JkS\TQLK U#<k:^u^!
(1990)
kS\TQLA=N&O\*+)
dj ?BN&NkS\H6jN&NkS\Oc&N+)
?BN&NkS\ dj !3NsDOL?7F\vKc&N+)
kS\TQLA=N&O\*+)
dj ?BN&NkS\H6jN&NkS\Oc&N+)
6jN&NkS\ dj !3NsDOL?7F\vKc&N+)
kS\TQLA=N&O\*+)
dj ?BN&NkS\H6jN&NkS\Oc&N+)
?BN&NkS\ 6jN&NkS\ dj !3NsDOL?7F\vKc&N+)
kS\TQLA=N&O\*+)
dj ?BN&NkS\H6jN&NkS\Oc&N+)
?BN&NkS\ 6jN&NkS\ dj !3NsDOL?7F\vKc&N+)
kS\,18+I&+N=j
|,'2DNkS\^0N=o9kS\,18
, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$
183HrZ@9kKO"B]KQA7F+;lP h$#
8NzkS\a=N@|
kS\,18+I&+N=j
|,'2DNkS\^0N=o9kS\,18
, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$
183HrZ@9kKO"B]KQA7F+;lP h$#
8NzkS\a=N@|
kS\,18+I&+N=j
|,'2DNkS\^0N=o9kS\,18
, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$
183HrZ@9kKO"B]KQA7F+;lP h$#
8NzkS\a=N@|
kS\,18+I&+N=j
|,'2DNkS\^0N=o9kS\,18
, i$G^$9?<&`<Vr?s+9lP\j"& 3N"k4j:`OJ$
183HrZ@9kKO"B]KQA7F+;lPh
$#JJ1HOBiJ$K 8NzkS\a=N@| )
!
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
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#
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#
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#
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#
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
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
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
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
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
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
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V H NT<9@1~lX(k3HOG-J$W 3H,,+k
kS\TQLJ8g<s:?`0KKaj^7g&#
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V H NT<9@1~lX(k3HOG-J$W 3H,,+k
kS\TQLJ8g<s:?`0KKaj^7g&#
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V H NT<9@1~lX(k3HOG-J$W 3H,,+k
kS\TQLJ8g<s:?`0KKaj^7g&#
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V H NT<9@1~lX(k3HOG-J$W 3H,,+k
kS\TQLJ8g<s:?`0KKaj^7g&#
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V H NT<9@1~lX(k3HOG-J$W 3H,,+k
kS\TQLJ8g<s:?`0KKaj^7g&#
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V H NT<9@1~lX(k3HOG-J$W 3H,,+k
kS\TQLJ8g<s:?`0KKaj^7g&#
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V
14
H15
NT<9@1~lX(k3HOG-J$W 3H,,+kkS\TQLJ8g<s:?`0KKaj^7g&#
15Q:kNTQL
T '
15
Q:kNuV4FJ16!
DN5hjJkK"'T<9r9i$I5;F\j"&"
S = f+1; 1g"f'V9Ndf "kU!
1MX8 O"
V1DNT<9r#K0+9W HV1DNT<9rDK0+9W
N2DN`nr-Bs+jV;P@ilk#
f ,TQL () f NM,eN2DN`nGQoiJ$ TQLrQ$kH"
V
14
H15
NT<9@1~lX(k3HOG-J$W 3H,,+kkS\TQLJ8g<s:?`0KKaj^7g&#
H_32A Jones ?`0XNM16
n \N3+iJkH_34N Bn O
GQJi;Krjak3HKhj"V2WKJk# et*JHj7$,G-k*
3NH_32O! !NANbNH=NU5G8. 5l"!NX80JH_3X80Kr~?9#
H_32A Jones ?`0XNM16
n \N3+iJkH_34N Bn O
GQJi;Krjak3HKhj"V2WKJk# et*JHj7$,G-k*
3NH_32O! !NANbNH=NU5G8. 5l"!NX80JH_3X80Kr~?9#
H_32A Jones ?`0XNM16
n \N3+iJkH_34N Bn O
GQJi;Krjak3HKhj"V2WKJk# et*JHj7$,G-k*
3NH_32O! !NANbNH=NU5G8. 5l"!NX80JH_3X80Kr~?9#
H_3+ikS\"m_\X
H_3rD8kH"kS\"m_\KJk
H_3+ikS\"m_\X
H_3rD8kH"kS\"m_\KJk
H_3+ikS\"m_\X
H_3rD8kH"kS\"m_\KJk
8g<s:?`0NC'E1
T = f kS\JHm_\Kg"'V18W" S = ft (t
1
2 ) NJiNQbv7?K0t8t?`0 g kS\^?Om_\ L N8g<s:?`0 VL(t) O"
!N2DN-AGC'U1ilk#
+@JkS\r{G=9H VÆ(t) = 1 91$sX80
t 1
V
L
+
(t) tV
L
(t) = (t 1
2
t 1
2
)V
L
0
(t)
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