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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.1/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.2/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.2/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.2/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.3/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.3/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.3/38
shoulder knot
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.3/38
shoulder knot
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.3/38
shoulder knot
cut the Gordian knot, nodo d’amore
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.3/38
half hitch
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.4/38
half hitch
square kont,
reef knot
()
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.4/38
half hitch
square kont,
reef knot
()
running knot
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.4/38
sheep shank
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.5/38
sheep shank
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.5/38
sheep shank
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.5/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.6/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.6/38
(pure) braid
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.6/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.7/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.7/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.7/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.7/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.7/38
Trefoil
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.8/38
Trefoil
Figure eight
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.8/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.9/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.9/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.9/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.10/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.10/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.10/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.10/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.10/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.10/38
(1777 – 1855)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.11/38
(1777 – 1855)
100
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.11/38
(1777 – 1855)
100
¦
Hopf
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.11/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.12/38
(1824 – 1907)
(1867)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.12/38
(1824 – 1907)
(1867)
(1831 – 1901)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.12/38
IntermezzoIII
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.13/38
(1888 – 1971)
(1920’s)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.14/38
(1888 – 1971)
(1920’s)
(1893 – 1971)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.14/38
(1888 – 1971)
(1920’s)
(1893 – 1971)
(Reidemeister moves (1932))
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.14/38
(1952 – )
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.15/38
(1952 – )
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.15/38
(1952 – )
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.15/38
(1952 – )
! (1990)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.15/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.16/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.16/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.16/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.16/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.17/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.17/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.17/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.17/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.17/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.18/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.18/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.18/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.18/38
Ì
ËÌË
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.19/38
Ì
ËÌË
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.19/38
Ì
ËÌË
Ë Ì
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.19/38
Ë
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Ì
ËÌË
Ì
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.19/38
Ë
Ì Ì
Ì
ËÌË
Ì
℄Ò ℄ Ò ℄ ℄ ℄,
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.19/38
Ë
Ì Ì
Ì
ËÌË
Ì
℄Ò ℄ Ò ℄ ℄ ℄,
℄
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.19/38
Ì15
Ë
16!
¢
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Ì15
Ë
16!
¢
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Ì15
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16!
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Ì15
Ë
16!
¢
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Ì15
Ë
16!
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Ì15
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16!
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Ì15
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16!
¢
14 15
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Ì15
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16!
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14 15
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.20/38
Jones
ÒÒ
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.21/38
Jones
ÒÒ
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.21/38
Jones
ÒÒ
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.21/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.22/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.22/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.22/38
ËØ(Ø Ì
)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.23/38
ËØ(Ø Ì
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.23/38
ËØ(Ø Ì
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ÎÆØ
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.23/38
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ÄÎÄØ
ÎÆØ
Ø ÎÄ Ø ØÎÄ Ø Ø Ø ÎÄ Ø
Ä Ä Ä “”
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.23/38
ÎÆØ
Ø ÎÄ Ø ØÎÄ Ø Ø Ø ÎÄ Ø
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
Ø ÎÄ Ø ØÎÄ Ø Ø Ø ÎÄ Ø
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
Ø ÎÄ Ø ØÎÄ Ø Ø Ø ÎÄ Ø
Ø Ø ÎÆÆØ
Ø
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
Ø ÎÄ Ø ØÎÄ Ø Ø Ø ÎÄ Ø
Î Ø Ø ÒØ Ø Ø Ø Ø ¡ Ó
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
ÎÆØ
Ø ÎÄ Ø ØÎÄ Ø Ø Ø ÎÄ Ø
Î trefoilØ
Ò
Ø ¡ Ø Ø Ø Ø Ó
Ø Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.24/38
K K
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.25/38
ÎÃ Ø Ø Ø Ø ÎÃ Ø Ø Ø Ø
K K
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.25/38
ÎÃ Ø Ø Ø Ø ÎÃ Ø Ø Ø Ø
K K
TL FL
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.25/38
ÎÃ Ø Ø Ø Ø ÎÃ Ø Ø Ø Ø
K K
ÎÌÄØ ÎÄØØ
Ø Ø Ø TL
Ø Ø
Ø Ø Ø
FL
Ø Ø
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.25/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.26/38
¡¡¡
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.26/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.26/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.26/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.26/38
( () )
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.26/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.27/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.27/38
¡¡¡
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.27/38
¡¡¡
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.27/38
¡¡¡
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.27/38
¡¡¡
¡¡¡
(cut and paste)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.27/38
Ë Ã
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.28/38
Ë Ã
Ã
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.28/38
Ë Ã
Ã
Ë Ã
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.28/38
Ë
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.29/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.29/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.29/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.30/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.30/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.30/38
ideal knotsetc
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.30/38
ideal knotsetc
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.30/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.31/38
(Frisch-Wasserman-Delbruck)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.31/38
(Frisch-Wasserman-Delbruck)
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.31/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.32/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.32/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.32/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.33/38
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à à ÄÌÛ
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à (writhe) ÏÖ
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.33/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.34/38
ÝÀТÖ
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ÐÜÖÝ Ü
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.34/38
x
y
ÝÀТÖ
ÐÜÖÝ Ü Ö
Ý Ü ¡ Ü¢Ý
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.34/38
x
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.35/38
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Ý Ü
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.35/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.35/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.36/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.36/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.36/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.36/38
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Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.36/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.37/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.37/38
Figures by Kusner and Sullivan. Taken from http://www.gang.umass.edu/˜kusner/knot pix/ 2008 – p.38/38
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