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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.2/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.2/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.2/38

8hNfNkS\JNgVK

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.3/38

8hNfNkS\JNgVK

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.3/38

8hNfNkS\JNgVK

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.3/38

8hNfNkS\JNgVK

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.3/38

8hNfNkS\JNgVK

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.3/38

8hNfNkS\JNgVK

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.3/38

kS\NcA>0ND$?kS\

half hitch

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.4/38

kS\NcA>0ND$?kS\

half hitchsquare kont,reef knot()^kS

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.4/38

kS\NcA>0ND$?kS\

half hitchsquare kont,reef knot()

running knot

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.4/38

kS\NcABQ"*hSu~Q

sheep shank

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.5/38

kS\NcABQ"*hSu~Q"oSkS
sheep shank

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.5/38

kS\NcABQ"*hSu~Q

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.5/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.6/38

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

H_3

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.6/38

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

H_3

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot(pure) braid

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.6/38

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ksG$k

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.7/38

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.7/38

tXG7&kS\))

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.7/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.7/38

tXG7&kS\))

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.7/38

J1JkS\Nc
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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.8/38

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.8/38

sDNkS\,18"HO)

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.9/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.9/38

3kSOJK+)

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.10/38

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.10/38

3kSOJK+)

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.10/38

3kSOJK+)

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.10/38

3kSOJK+)

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.10/38

3kSOJK+)

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.10/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.11/38

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

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7e<^s

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kS\N&f6,rA*}+i,&9 (1777 – 1855)

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.11/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.12/38

kS\N&f6,rA*}+i,&9NoRNj9F#s0
1kt#s*J&#j"`&H`=sK(1824 – 1907)
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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.12/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.13/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.14/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.14/38

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))

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.14/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.15/38

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

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.15/38

kS\TQLA=N&O\*+)

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.16/38

kS\TQLA=N&O\*+)

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.16/38

kS\TQLA=N&O\*+)

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6jN&NkS\

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.16/38

kS\TQLA=N&O\*+)

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pix/!|\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"&Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.17/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.17/38

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183HrZ@9kKO"B]KQA7F+;lPh$#Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.17/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.17/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.17/38

kS\,18+I&+N=j

RH(kS\a=N@|)!



Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

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kS\,18+I&+N=j

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kS\,18+I&+N=j

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kS\,18+I&+N=j

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pix/!|\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)#Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.19/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.19/38

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pix/!|\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"
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pix/!|\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"
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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
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f ,TQL () f O [i℄ e18MrHk!|\tXq& /Y)(mg,Jq&T1Viq
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pix/

2008

– p.19/38

15Q:kNTQL

T '15 Q:kNuV4FJ16!DN5hjJkK"'T<9r9i$I5;F\j"&"S = f+1; 1g"f 'V9Ndf  "kU!
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pix/!|\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!
         Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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!
 
 
 

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pix/!|\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!
 
 

 
     Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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#

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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$

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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$
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V14 H 15 NT<9@1~lX(k3HOG-J$W
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pix/!|\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
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N2DN`nr-Bs+jV;P@ilk#
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V14 H 15 NT<9@1~lX(k3HOG-J$W
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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.20/38

H_32AJones ?`0XNM16

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

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.21/38

H_32AJones ?`0XNM16

n \N3+iJkH_34N Bn OGQJi;Krjak3HKhj"V2WKJk#
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.21/38

H_32AJones ?`0XNM16

n \N3+iJkH_34N Bn OGQJi;Krjak3HKhj"V2WKJk#
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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.21/38

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.22/38

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.22/38

H_3+ikS\"m_\X

H_3rD8kH"kS\"m_\KJk
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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#

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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'
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VÆ(t) = 1 u t VL+ (t) tVL (t) = (t 2

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/

t

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1
2 VL0 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
2 VL0 t

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

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1
2 VL0 t



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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

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1
2 VL0 t



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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

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1
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

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1
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

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1
2 VL0 t



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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

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1
2 VL0 t



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

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1
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pix/!|\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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pix/

o

t
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1
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2008

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

t



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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/

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

RH(kS\H=N@|K

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pix/!|\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
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.25/38

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

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VK (t) = t 4 + t 3 + t 16=VK+ (t) = t4 + t3 + t
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.25/38

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

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VK (t) = t 4 + t 3 + t 16=VK+ (t) = t4 + t3 + t
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.25/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.26/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.26/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.27/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.28/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.30/38

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

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.30/38

>,nHNX"nQGDJ8g<s:K}WOXJ,[Xt"ds&P/9?<X80JIKkS\HGtJ9< ;*aJe#gXKKFigures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.30/38

>,nHNX"nQGDJ8g<s:K}WOXJ,[Xt"ds&P/9?<X80JIKkS\HGtJ9< ;*aJe#gXKKv?X*kS\}@'kS\NAN#(5rO+j"l$JAraakJkS\N(Mk.<"&Av?X"
ideal knots"?Qr@t"etcK

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.30/38

>,nHNX"nQGDJ8g<s:K}WOXJ,[Xt"ds&P/9?<X80JIKkS\HGtJ9< ;*aJe#gXKKv?X*kS\}@'kS\NAN#(5rO+j"l$JAraakJkS\N(Mk.<"&Av?X"
ideal knots"?Qr@t"etcKb,RJX"DNAFigures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.30/38

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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.31/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.31/38

DNA

G*-7j\K@Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.32/38

DNA

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.32/38

DNA

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.32/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.33/38

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K1 H K2 Nm_tr Lk"M8ltr T w"
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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.33/38

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.34/38

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j dl  r
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.34/38

m_tN,&9NQ,x0

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pix/

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,&9Q,x0+iI89kbNZ Z(y x)  (dx  dy)1
Lk(C1; C2) =4 C1 C2
jy xj3Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.35/38

,&9Q,x0+iI89kbNZ Z(y x)  (dx  dy)1
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9KeN0GdPMrD1?bN,?Qr@t'
ZZ1j(y x)  (dx  dy)jAC (C1) =4 C1C1n
jy xj3C1 N#(5rO+kL#
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.35/38

DNAkS\

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.36/38

DNAkS\

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.36/38

DNAkS\

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.36/38

DNAkS\

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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.36/38

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kS\}@O3l+iI3XT/N+)

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.37/38

kSN@U

kS\}@O3l+iI3XT/N+)
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pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.37/38

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

Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot

pix/!|\tXq&2008 /Y)(mg,Jq&T1Viq – p.38/38