8 8

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8 7.gif

8_7

8 9.gif

8_9

8 8.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

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[edit 8 8 Quick Notes]
[edit 8 8 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283
Gauss code -1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code 4 8 12 2 16 14 6 10
Conway Notation [2312]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 9, width is 4,

Braid index is 4

8 8 ML.gif 8 8 AP.gif
[{10, 2}, {1, 8}, {4, 9}, {8, 10}, {3, 5}, {2, 4}, {6, 3}, {5, 7}, {9, 6}, {7, 1}]

[edit Notes on presentations of 8 8]

Knot 8_8.
A graph, knot 8_8.
Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["8 8"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283
In[5]:= GaussCode[K]
Out[5]= -1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6
In[6]:= DTCode[K]
Out[6]= 4 8 12 2 16 14 6 10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [2312]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(4,\{1,1,1,2,-1,-3,2,-3,-3\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 4, 9, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
8 8 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 2}, {1, 8}, {4, 9}, {8, 10}, {3, 5}, {2, 4}, {6, 3}, {5, 7}, {9, 6}, {7, 1}]
In[14]:= Draw[ap]
8 8 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,5\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-4][-6]
Hyperbolic Volume 7.80134
A-Polynomial See Data:8 8/A-polynomial

[edit Notes for 8 8's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 0 }[/math]
Topological 4 genus [math]\displaystyle{ 0 }[/math]
Concordance genus [math]\displaystyle{ 0 }[/math]
Rasmussen s-Invariant 0

[edit Notes for 8 8's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 2 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 25, 0 }
Jones polynomial [math]\displaystyle{ -q^5+2 q^4-3 q^3+4 q^2-4 q+5-3 q^{-1} +2 q^{-2} - q^{-3} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^4 a^{-2} +z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +2 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+2 a z^5+z^5 a^{-1} +z^5 a^{-5} +2 a^2 z^4-9 z^4 a^{-2} -6 z^4 a^{-4} -z^4+a^3 z^3-3 z^3 a^{-1} -5 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+5 z^2 a^{-2} +4 z^2 a^{-4} -z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{10}-q^4+2 q^2+1+2 q^{-2} + q^{-4} + q^{-8} - q^{-10} - q^{-16} }[/math]
The G2 invariant [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-5 q^{32}+q^{30}+3 q^{28}-6 q^{26}+8 q^{24}-9 q^{22}+7 q^{20}-4 q^{18}-3 q^{16}+7 q^{14}-8 q^{12}+9 q^{10}-q^8-3 q^6+6 q^4-3 q^2+1+6 q^{-2} -10 q^{-4} +12 q^{-6} -5 q^{-8} +10 q^{-12} -14 q^{-14} +17 q^{-16} -11 q^{-18} +3 q^{-20} +4 q^{-22} -9 q^{-24} +14 q^{-26} -10 q^{-28} +6 q^{-30} + q^{-32} -5 q^{-34} +7 q^{-36} -5 q^{-38} +4 q^{-42} -8 q^{-44} +7 q^{-46} -3 q^{-48} -4 q^{-50} +10 q^{-52} -13 q^{-54} +10 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} -10 q^{-64} +9 q^{-66} -5 q^{-68} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_8.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^7+q^5-q^3+2 q+ q^{-1} + q^{-5} - q^{-7} + q^{-9} - q^{-11} }[/math]
2 [math]\displaystyle{ q^{20}-q^{18}-q^{16}+2 q^{14}-3 q^{12}-q^{10}+5 q^8-3 q^6-2 q^4+5 q^2- q^{-2} + q^{-4} +3 q^{-6} -2 q^{-10} +3 q^{-12} -5 q^{-16} +2 q^{-18} +2 q^{-20} -4 q^{-22} + q^{-24} +3 q^{-26} -2 q^{-28} - q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{39}+q^{37}+q^{35}-q^{31}+q^{29}+q^{27}-4 q^{25}-q^{23}+5 q^{21}+2 q^{19}-9 q^{17}-4 q^{15}+10 q^{13}+7 q^{11}-11 q^9-7 q^7+7 q^5+11 q^3-3 q-7 q^{-1} + q^{-3} +6 q^{-5} +5 q^{-7} -3 q^{-9} -6 q^{-11} + q^{-13} +9 q^{-15} - q^{-17} -9 q^{-19} -2 q^{-21} +10 q^{-23} +3 q^{-25} -10 q^{-27} -6 q^{-29} +9 q^{-31} +7 q^{-33} -6 q^{-35} -10 q^{-37} +3 q^{-39} +10 q^{-41} -8 q^{-45} -3 q^{-47} +6 q^{-49} +5 q^{-51} -3 q^{-53} -4 q^{-55} +2 q^{-59} + q^{-61} - q^{-63} }[/math]
4 [math]\displaystyle{ q^{64}-q^{62}-q^{60}-q^{56}+3 q^{54}-q^{52}+q^{50}+2 q^{48}-4 q^{46}+q^{44}-4 q^{42}+5 q^{40}+10 q^{38}-6 q^{36}-8 q^{34}-14 q^{32}+9 q^{30}+26 q^{28}+q^{26}-17 q^{24}-34 q^{22}+3 q^{20}+40 q^{18}+18 q^{16}-11 q^{14}-43 q^{12}-12 q^{10}+30 q^8+27 q^6+10 q^4-27 q^2-21+6 q^{-2} +18 q^{-4} +18 q^{-6} -3 q^{-8} -16 q^{-10} -15 q^{-12} +2 q^{-14} +19 q^{-16} +14 q^{-18} -10 q^{-20} -25 q^{-22} -3 q^{-24} +20 q^{-26} +24 q^{-28} -8 q^{-30} -33 q^{-32} -8 q^{-34} +19 q^{-36} +31 q^{-38} -3 q^{-40} -35 q^{-42} -17 q^{-44} +9 q^{-46} +37 q^{-48} +12 q^{-50} -26 q^{-52} -24 q^{-54} -9 q^{-56} +29 q^{-58} +24 q^{-60} -4 q^{-62} -19 q^{-64} -26 q^{-66} +8 q^{-68} +21 q^{-70} +13 q^{-72} - q^{-74} -22 q^{-76} -8 q^{-78} +4 q^{-80} +12 q^{-82} +11 q^{-84} -7 q^{-86} -7 q^{-88} -5 q^{-90} + q^{-92} +6 q^{-94} + q^{-96} -2 q^{-100} - q^{-102} + q^{-104} }[/math]
5 [math]\displaystyle{ -q^{95}+q^{93}+q^{91}+q^{87}-q^{85}-3 q^{83}-q^{81}+q^{79}+4 q^{75}+3 q^{73}-2 q^{71}-6 q^{69}-6 q^{67}+9 q^{63}+15 q^{61}+7 q^{59}-16 q^{57}-27 q^{55}-13 q^{53}+18 q^{51}+43 q^{49}+34 q^{47}-19 q^{45}-68 q^{43}-55 q^{41}+10 q^{39}+83 q^{37}+87 q^{35}+7 q^{33}-94 q^{31}-118 q^{29}-36 q^{27}+85 q^{25}+139 q^{23}+63 q^{21}-64 q^{19}-131 q^{17}-93 q^{15}+31 q^{13}+120 q^{11}+99 q^9+3 q^7-78 q^5-93 q^3-33 q+47 q^{-1} +77 q^{-3} +48 q^{-5} -9 q^{-7} -50 q^{-9} -53 q^{-11} -20 q^{-13} +30 q^{-15} +53 q^{-17} +35 q^{-19} -10 q^{-21} -54 q^{-23} -52 q^{-25} +3 q^{-27} +59 q^{-29} +57 q^{-31} +2 q^{-33} -62 q^{-35} -71 q^{-37} -3 q^{-39} +74 q^{-41} +77 q^{-43} +9 q^{-45} -78 q^{-47} -92 q^{-49} -16 q^{-51} +80 q^{-53} +103 q^{-55} +31 q^{-57} -73 q^{-59} -114 q^{-61} -50 q^{-63} +57 q^{-65} +111 q^{-67} +73 q^{-69} -30 q^{-71} -106 q^{-73} -89 q^{-75} +82 q^{-79} +96 q^{-81} +36 q^{-83} -52 q^{-85} -89 q^{-87} -58 q^{-89} +15 q^{-91} +68 q^{-93} +67 q^{-95} +21 q^{-97} -38 q^{-99} -63 q^{-101} -41 q^{-103} +7 q^{-105} +42 q^{-107} +45 q^{-109} +19 q^{-111} -19 q^{-113} -37 q^{-115} -28 q^{-117} - q^{-119} +20 q^{-121} +25 q^{-123} +15 q^{-125} -6 q^{-127} -17 q^{-129} -14 q^{-131} -3 q^{-133} +5 q^{-135} +9 q^{-137} +7 q^{-139} - q^{-141} -4 q^{-143} -3 q^{-145} - q^{-147} +2 q^{-151} + q^{-153} - q^{-155} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{10}-q^4+2 q^2+1+2 q^{-2} + q^{-4} + q^{-8} - q^{-10} - q^{-16} }[/math]
1,1 [math]\displaystyle{ q^{28}-2 q^{26}+4 q^{24}-6 q^{22}+9 q^{20}-12 q^{18}+14 q^{16}-20 q^{14}+19 q^{12}-24 q^{10}+22 q^8-22 q^6+18 q^4-4 q^2+20 q^{-2} -27 q^{-4} +44 q^{-6} -48 q^{-8} +56 q^{-10} -52 q^{-12} +48 q^{-14} -40 q^{-16} +24 q^{-18} -15 q^{-20} -4 q^{-22} +14 q^{-24} -24 q^{-26} +31 q^{-28} -32 q^{-30} +30 q^{-32} -24 q^{-34} +17 q^{-36} -12 q^{-38} +6 q^{-40} -2 q^{-42} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{26}-q^{22}+q^{18}-q^{16}-3 q^{14}+2 q^{10}-3 q^8-3 q^6+2 q^4+2 q^2+2+2 q^{-2} +5 q^{-4} +2 q^{-6} +2 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} -4 q^{-20} -3 q^{-22} + q^{-26} - q^{-28} - q^{-30} +2 q^{-32} + q^{-34} - q^{-36} - q^{-38} + q^{-42} }[/math]
3,0 [math]\displaystyle{ -q^{48}+q^{44}+q^{42}-2 q^{38}+2 q^{36}+3 q^{34}+q^{32}-4 q^{30}-5 q^{28}+4 q^{26}+6 q^{24}-q^{22}-10 q^{20}-8 q^{18}+8 q^{16}+9 q^{14}-3 q^{12}-13 q^{10}-8 q^8+8 q^6+6 q^4-4+4 q^{-2} +10 q^{-4} +8 q^{-6} +3 q^{-8} +3 q^{-10} +7 q^{-12} +2 q^{-14} -3 q^{-16} -2 q^{-18} +3 q^{-20} +2 q^{-22} -8 q^{-24} -9 q^{-26} +6 q^{-30} + q^{-32} -10 q^{-34} -7 q^{-36} +4 q^{-38} +8 q^{-40} +2 q^{-42} -8 q^{-44} -5 q^{-46} +3 q^{-48} +7 q^{-50} +4 q^{-52} -5 q^{-54} -4 q^{-56} +5 q^{-60} +4 q^{-62} -2 q^{-64} -3 q^{-66} -3 q^{-68} + q^{-70} +2 q^{-72} + q^{-74} - q^{-78} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{22}-q^{20}+2 q^{16}-3 q^{14}-3 q^{12}+2 q^{10}-3 q^8-3 q^6+5 q^4+q^2+2+4 q^{-2} +3 q^{-4} + q^{-6} +3 q^{-10} +2 q^{-12} -3 q^{-14} + q^{-16} + q^{-18} -5 q^{-20} - q^{-22} + q^{-24} -3 q^{-26} + q^{-28} + q^{-30} - q^{-32} + q^{-34} }[/math]
1,0,0 [math]\displaystyle{ -q^{13}-q^9-q^5+2 q^3+q+2 q^{-1} +2 q^{-3} + q^{-5} + q^{-7} + q^{-11} - q^{-13} - q^{-17} - q^{-21} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{28}+q^{22}+q^{20}-2 q^{18}-4 q^{16}-2 q^{14}-q^{12}-4 q^{10}-4 q^8+3 q^6+2 q^4+q^2+4+7 q^{-2} +4 q^{-4} +3 q^{-6} +5 q^{-8} +4 q^{-10} +2 q^{-14} +4 q^{-16} -2 q^{-18} -3 q^{-20} + q^{-22} -2 q^{-24} -6 q^{-26} -4 q^{-28} - q^{-32} -2 q^{-34} + q^{-36} +2 q^{-38} + q^{-44} }[/math]
1,0,0,0 [math]\displaystyle{ -q^{16}-q^{12}-q^{10}-q^6+2 q^4+q^2+2+2 q^{-2} +2 q^{-4} + q^{-6} + q^{-8} + q^{-10} + q^{-14} - q^{-16} - q^{-20} - q^{-22} - q^{-26} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{22}+q^{20}-2 q^{18}+2 q^{16}-3 q^{14}+3 q^{12}-4 q^{10}+3 q^8-q^6+q^4+3 q^2-2+6 q^{-2} -5 q^{-4} +7 q^{-6} -6 q^{-8} +5 q^{-10} -4 q^{-12} +3 q^{-14} - q^{-16} - q^{-18} +3 q^{-20} -3 q^{-22} +3 q^{-24} -3 q^{-26} +3 q^{-28} -3 q^{-30} + q^{-32} - q^{-34} }[/math]
1,0 [math]\displaystyle{ q^{36}-q^{32}-q^{30}+q^{28}+2 q^{26}-3 q^{22}-3 q^{20}+3 q^{16}-4 q^{12}-2 q^{10}+2 q^8+4 q^6-q^4-q^2+2+5 q^{-2} + q^{-4} - q^{-6} +3 q^{-10} +2 q^{-12} - q^{-14} - q^{-16} +2 q^{-18} +2 q^{-20} - q^{-22} -3 q^{-24} +3 q^{-28} + q^{-30} -4 q^{-32} -4 q^{-34} + q^{-36} +3 q^{-38} -3 q^{-42} -2 q^{-44} +2 q^{-46} +2 q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{30}-q^{28}+q^{26}-q^{24}+2 q^{22}-3 q^{20}-4 q^{16}+2 q^{14}-4 q^{12}-2 q^8+2 q^6+3 q^4+q^2+5- q^{-2} +7 q^{-4} -2 q^{-6} +6 q^{-8} -4 q^{-10} +6 q^{-12} -2 q^{-14} +5 q^{-16} -2 q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -4 q^{-28} + q^{-30} -4 q^{-32} + q^{-34} -3 q^{-36} +3 q^{-38} -2 q^{-40} +2 q^{-42} - q^{-44} + q^{-46} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{52}-q^{50}+2 q^{48}-2 q^{46}-3 q^{40}+4 q^{38}-5 q^{36}+4 q^{34}-5 q^{32}+q^{30}+3 q^{28}-6 q^{26}+8 q^{24}-9 q^{22}+7 q^{20}-4 q^{18}-3 q^{16}+7 q^{14}-8 q^{12}+9 q^{10}-q^8-3 q^6+6 q^4-3 q^2+1+6 q^{-2} -10 q^{-4} +12 q^{-6} -5 q^{-8} +10 q^{-12} -14 q^{-14} +17 q^{-16} -11 q^{-18} +3 q^{-20} +4 q^{-22} -9 q^{-24} +14 q^{-26} -10 q^{-28} +6 q^{-30} + q^{-32} -5 q^{-34} +7 q^{-36} -5 q^{-38} +4 q^{-42} -8 q^{-44} +7 q^{-46} -3 q^{-48} -4 q^{-50} +10 q^{-52} -13 q^{-54} +10 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} -10 q^{-64} +9 q^{-66} -5 q^{-68} +2 q^{-72} -4 q^{-74} +3 q^{-76} - q^{-78} + q^{-80} }[/math]

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["8 8"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 2 z^4+2 z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 25, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^5+2 q^4-3 q^3+4 q^2-4 q+5-3 q^{-1} +2 q^{-2} - q^{-3} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^4 a^{-2} +z^4-a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +2 z^2-a^2+ a^{-2} - a^{-4} +2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+2 a z^5+z^5 a^{-1} +z^5 a^{-5} +2 a^2 z^4-9 z^4 a^{-2} -6 z^4 a^{-4} -z^4+a^3 z^3-3 z^3 a^{-1} -5 z^3 a^{-3} -3 z^3 a^{-5} -2 a^2 z^2+5 z^2 a^{-2} +4 z^2 a^{-4} -z^2-a^3 z-a z+z a^{-1} +3 z a^{-3} +2 z a^{-5} +a^2- a^{-2} - a^{-4} +2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_129, K11n39, K11n45, K11n50, K11n132,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {10_129,}

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["8 8"];
In[4]:= {A = Alexander[K][t], J = Jones[K][q]}
KnotTheory::loading: Loading precomputed data in PD4Knots`.
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[4]= { [math]\displaystyle{ 2 t^2-6 t+9-6 t^{-1} +2 t^{-2} }[/math], [math]\displaystyle{ -q^5+2 q^4-3 q^3+4 q^2-4 q+5-3 q^{-1} +2 q^{-2} - q^{-3} }[/math] }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {10_129, K11n39, K11n45, K11n50, K11n132,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {10_129,}

Vassiliev invariants

V2 and V3: (2, 1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ -\frac{4}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{272}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ \frac{6271}{15} }[/math] [math]\displaystyle{ \frac{1516}{15} }[/math] [math]\displaystyle{ \frac{484}{45} }[/math] [math]\displaystyle{ \frac{113}{9} }[/math] [math]\displaystyle{ -\frac{449}{15} }[/math]

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 8 8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-1012345χ
11        1-1
9       1 1
7      21 -1
5     21  1
3    22   0
1   32    1
-1  13     2
-3 12      -1
-5 1       1
-71        -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{15}-2 q^{14}-q^{13}+6 q^{12}-4 q^{11}-6 q^{10}+12 q^9-4 q^8-13 q^7+17 q^6-q^5-18 q^4+19 q^3+2 q^2-20 q+17+3 q^{-1} -15 q^{-2} +10 q^{-3} +2 q^{-4} -7 q^{-5} +4 q^{-6} -2 q^{-8} + q^{-9} }[/math]
3 [math]\displaystyle{ -q^{30}+2 q^{29}+q^{28}-2 q^{27}-5 q^{26}+3 q^{25}+9 q^{24}-q^{23}-14 q^{22}-2 q^{21}+17 q^{20}+9 q^{19}-21 q^{18}-15 q^{17}+21 q^{16}+22 q^{15}-19 q^{14}-30 q^{13}+17 q^{12}+35 q^{11}-12 q^{10}-42 q^9+10 q^8+43 q^7-2 q^6-50 q^5+3 q^4+46 q^3+6 q^2-49 q-2+38 q^{-1} +10 q^{-2} -35 q^{-3} -6 q^{-4} +24 q^{-5} +6 q^{-6} -17 q^{-7} -3 q^{-8} +10 q^{-9} + q^{-10} -6 q^{-11} +4 q^{-13} -2 q^{-14} - q^{-15} +2 q^{-17} - q^{-18} }[/math]
4 [math]\displaystyle{ q^{50}-2 q^{49}-q^{48}+2 q^{47}+q^{46}+6 q^{45}-7 q^{44}-7 q^{43}+q^{41}+24 q^{40}-6 q^{39}-15 q^{38}-12 q^{37}-13 q^{36}+45 q^{35}+8 q^{34}-7 q^{33}-25 q^{32}-47 q^{31}+52 q^{30}+23 q^{29}+21 q^{28}-20 q^{27}-85 q^{26}+37 q^{25}+21 q^{24}+59 q^{23}+5 q^{22}-113 q^{21}+11 q^{20}+3 q^{19}+91 q^{18}+39 q^{17}-125 q^{16}-16 q^{15}-22 q^{14}+116 q^{13}+71 q^{12}-129 q^{11}-39 q^{10}-44 q^9+131 q^8+95 q^7-124 q^6-56 q^5-61 q^4+130 q^3+108 q^2-103 q-56-73 q^{-1} +103 q^{-2} +102 q^{-3} -66 q^{-4} -39 q^{-5} -70 q^{-6} +61 q^{-7} +71 q^{-8} -34 q^{-9} -10 q^{-10} -48 q^{-11} +24 q^{-12} +34 q^{-13} -17 q^{-14} +8 q^{-15} -23 q^{-16} +7 q^{-17} +11 q^{-18} -11 q^{-19} +10 q^{-20} -7 q^{-21} +2 q^{-22} +2 q^{-23} -6 q^{-24} +5 q^{-25} - q^{-26} + q^{-27} -2 q^{-29} + q^{-30} }[/math]
5 [math]\displaystyle{ -q^{75}+2 q^{74}+q^{73}-2 q^{72}-q^{71}-2 q^{70}-2 q^{69}+5 q^{68}+9 q^{67}-5 q^{65}-10 q^{64}-13 q^{63}+2 q^{62}+20 q^{61}+21 q^{60}+5 q^{59}-15 q^{58}-34 q^{57}-25 q^{56}+11 q^{55}+39 q^{54}+43 q^{53}+11 q^{52}-37 q^{51}-60 q^{50}-37 q^{49}+17 q^{48}+68 q^{47}+70 q^{46}+9 q^{45}-59 q^{44}-90 q^{43}-56 q^{42}+37 q^{41}+107 q^{40}+97 q^{39}+q^{38}-104 q^{37}-138 q^{36}-52 q^{35}+90 q^{34}+173 q^{33}+104 q^{32}-66 q^{31}-192 q^{30}-159 q^{29}+26 q^{28}+214 q^{27}+208 q^{26}+6 q^{25}-215 q^{24}-255 q^{23}-50 q^{22}+228 q^{21}+295 q^{20}+74 q^{19}-218 q^{18}-332 q^{17}-118 q^{16}+237 q^{15}+359 q^{14}+129 q^{13}-216 q^{12}-388 q^{11}-173 q^{10}+235 q^9+403 q^8+174 q^7-198 q^6-411 q^5-223 q^4+202 q^3+406 q^2+215 q-141-382 q^{-1} -253 q^{-2} +122 q^{-3} +346 q^{-4} +230 q^{-5} -60 q^{-6} -286 q^{-7} -232 q^{-8} +33 q^{-9} +222 q^{-10} +192 q^{-11} +7 q^{-12} -159 q^{-13} -156 q^{-14} -21 q^{-15} +101 q^{-16} +110 q^{-17} +31 q^{-18} -58 q^{-19} -76 q^{-20} -25 q^{-21} +28 q^{-22} +45 q^{-23} +18 q^{-24} -9 q^{-25} -23 q^{-26} -16 q^{-27} +3 q^{-28} +14 q^{-29} +4 q^{-30} +2 q^{-31} -8 q^{-33} -3 q^{-34} +5 q^{-35} -2 q^{-36} +2 q^{-37} +4 q^{-38} -3 q^{-39} -2 q^{-40} + q^{-41} - q^{-42} +2 q^{-44} - q^{-45} }[/math]
6 [math]\displaystyle{ q^{105}-2 q^{104}-q^{103}+2 q^{102}+q^{101}+2 q^{100}-2 q^{99}+4 q^{98}-7 q^{97}-9 q^{96}+3 q^{95}+4 q^{94}+11 q^{93}+2 q^{92}+18 q^{91}-14 q^{90}-27 q^{89}-13 q^{88}-7 q^{87}+16 q^{86}+9 q^{85}+65 q^{84}+6 q^{83}-30 q^{82}-37 q^{81}-47 q^{80}-21 q^{79}-24 q^{78}+111 q^{77}+60 q^{76}+30 q^{75}-10 q^{74}-58 q^{73}-90 q^{72}-139 q^{71}+78 q^{70}+64 q^{69}+116 q^{68}+100 q^{67}+49 q^{66}-82 q^{65}-267 q^{64}-54 q^{63}-70 q^{62}+100 q^{61}+200 q^{60}+269 q^{59}+82 q^{58}-267 q^{57}-171 q^{56}-313 q^{55}-89 q^{54}+154 q^{53}+472 q^{52}+354 q^{51}-85 q^{50}-146 q^{49}-531 q^{48}-389 q^{47}-69 q^{46}+540 q^{45}+606 q^{44}+210 q^{43}+35 q^{42}-628 q^{41}-682 q^{40}-390 q^{39}+472 q^{38}+759 q^{37}+508 q^{36}+292 q^{35}-615 q^{34}-899 q^{33}-705 q^{32}+340 q^{31}+827 q^{30}+750 q^{29}+534 q^{28}-557 q^{27}-1050 q^{26}-957 q^{25}+220 q^{24}+859 q^{23}+934 q^{22}+715 q^{21}-508 q^{20}-1162 q^{19}-1146 q^{18}+133 q^{17}+887 q^{16}+1075 q^{15}+844 q^{14}-464 q^{13}-1239 q^{12}-1287 q^{11}+41 q^{10}+880 q^9+1171 q^8+956 q^7-367 q^6-1237 q^5-1374 q^4-105 q^3+768 q^2+1170 q+1047-174 q^{-1} -1085 q^{-2} -1342 q^{-3} -283 q^{-4} +519 q^{-5} +1000 q^{-6} +1029 q^{-7} +62 q^{-8} -766 q^{-9} -1118 q^{-10} -382 q^{-11} +213 q^{-12} +667 q^{-13} +839 q^{-14} +210 q^{-15} -397 q^{-16} -743 q^{-17} -328 q^{-18} +309 q^{-20} +534 q^{-21} +212 q^{-22} -137 q^{-23} -382 q^{-24} -181 q^{-25} -65 q^{-26} +79 q^{-27} +264 q^{-28} +128 q^{-29} -25 q^{-30} -157 q^{-31} -59 q^{-32} -46 q^{-33} -11 q^{-34} +111 q^{-35} +54 q^{-36} + q^{-37} -56 q^{-38} -7 q^{-39} -19 q^{-40} -25 q^{-41} +43 q^{-42} +18 q^{-43} +4 q^{-44} -19 q^{-45} +5 q^{-46} -7 q^{-47} -17 q^{-48} +16 q^{-49} +4 q^{-50} +4 q^{-51} -6 q^{-52} +4 q^{-53} -2 q^{-54} -8 q^{-55} +5 q^{-56} +2 q^{-58} - q^{-59} + q^{-60} -2 q^{-62} + q^{-63} }[/math]
7 [math]\displaystyle{ -q^{140}+2 q^{139}+q^{138}-2 q^{137}-q^{136}-2 q^{135}+2 q^{134}-2 q^{132}+7 q^{131}+6 q^{130}-2 q^{129}-4 q^{128}-13 q^{127}-4 q^{126}-9 q^{124}+18 q^{123}+23 q^{122}+16 q^{121}+9 q^{120}-27 q^{119}-24 q^{118}-21 q^{117}-43 q^{116}+4 q^{115}+33 q^{114}+49 q^{113}+77 q^{112}+7 q^{111}-12 q^{110}-32 q^{109}-109 q^{108}-69 q^{107}-40 q^{106}+15 q^{105}+130 q^{104}+97 q^{103}+97 q^{102}+76 q^{101}-89 q^{100}-116 q^{99}-177 q^{98}-180 q^{97}+12 q^{96}+60 q^{95}+195 q^{94}+296 q^{93}+139 q^{92}+67 q^{91}-148 q^{90}-373 q^{89}-293 q^{88}-273 q^{87}-3 q^{86}+366 q^{85}+411 q^{84}+514 q^{83}+262 q^{82}-228 q^{81}-460 q^{80}-750 q^{79}-577 q^{78}-18 q^{77}+361 q^{76}+896 q^{75}+931 q^{74}+387 q^{73}-134 q^{72}-950 q^{71}-1232 q^{70}-794 q^{69}-228 q^{68}+840 q^{67}+1448 q^{66}+1226 q^{65}+685 q^{64}-599 q^{63}-1559 q^{62}-1605 q^{61}-1183 q^{60}+250 q^{59}+1527 q^{58}+1903 q^{57}+1700 q^{56}+190 q^{55}-1405 q^{54}-2132 q^{53}-2160 q^{52}-646 q^{51}+1183 q^{50}+2250 q^{49}+2583 q^{48}+1126 q^{47}-933 q^{46}-2318 q^{45}-2934 q^{44}-1548 q^{43}+654 q^{42}+2317 q^{41}+3230 q^{40}+1952 q^{39}-396 q^{38}-2323 q^{37}-3464 q^{36}-2271 q^{35}+160 q^{34}+2287 q^{33}+3675 q^{32}+2578 q^{31}+14 q^{30}-2306 q^{29}-3832 q^{28}-2785 q^{27}-181 q^{26}+2282 q^{25}+4005 q^{24}+3018 q^{23}+281 q^{22}-2337 q^{21}-4118 q^{20}-3156 q^{19}-411 q^{18}+2302 q^{17}+4257 q^{16}+3370 q^{15}+515 q^{14}-2338 q^{13}-4324 q^{12}-3478 q^{11}-681 q^{10}+2199 q^9+4373 q^8+3687 q^7+877 q^6-2110 q^5-4318 q^4-3738 q^3-1121 q^2+1787 q+4157+3852 q^{-1} +1387 q^{-2} -1500 q^{-3} -3863 q^{-4} -3738 q^{-5} -1611 q^{-6} +1019 q^{-7} +3423 q^{-8} +3585 q^{-9} +1790 q^{-10} -617 q^{-11} -2879 q^{-12} -3209 q^{-13} -1837 q^{-14} +175 q^{-15} +2267 q^{-16} +2766 q^{-17} +1751 q^{-18} +142 q^{-19} -1658 q^{-20} -2213 q^{-21} -1558 q^{-22} -363 q^{-23} +1133 q^{-24} +1662 q^{-25} +1269 q^{-26} +446 q^{-27} -695 q^{-28} -1157 q^{-29} -964 q^{-30} -445 q^{-31} +396 q^{-32} +763 q^{-33} +664 q^{-34} +369 q^{-35} -210 q^{-36} -452 q^{-37} -424 q^{-38} -286 q^{-39} +103 q^{-40} +261 q^{-41} +256 q^{-42} +195 q^{-43} -61 q^{-44} -142 q^{-45} -133 q^{-46} -121 q^{-47} +29 q^{-48} +67 q^{-49} +74 q^{-50} +90 q^{-51} -30 q^{-52} -47 q^{-53} -32 q^{-54} -40 q^{-55} +18 q^{-56} +8 q^{-57} +17 q^{-58} +44 q^{-59} -14 q^{-60} -21 q^{-61} -8 q^{-62} -13 q^{-63} +11 q^{-64} -3 q^{-65} +2 q^{-66} +22 q^{-67} -5 q^{-68} -8 q^{-69} -2 q^{-70} -6 q^{-71} +5 q^{-72} -3 q^{-73} +8 q^{-75} - q^{-76} -2 q^{-77} -2 q^{-79} + q^{-80} - q^{-81} +2 q^{-83} - q^{-84} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Rolfsen Knot Page master template (intermediate).

See/edit the Rolfsen_Splice_Base (expert).

Back to the top.

8 7.gif

8_7

8 9.gif

8_9

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