8 7

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8 6.gif

8_6

8 8.gif

8_8

8 7.gif
(KnotPlot image)

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[edit 8 7 Quick Notes]
[edit 8 7 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X7,15,8,14 X13,7,14,6 X15,9,16,8 X9,2,10,3
Gauss code -1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3
Dowker-Thistlethwaite code 4 10 12 14 2 16 6 8
Conway Notation [4112]


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

Length is 8, width is 3,

Braid index is 3

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

[edit Notes on presentations of 8 7]

Knot 8_7.
A graph, knot 8_7.
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 7"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X7,15,8,14 X13,7,14,6 X15,9,16,8 X9,2,10,3
In[5]:= GaussCode[K]
Out[5]= -1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3
In[6]:= DTCode[K]
Out[6]= 4 10 12 14 2 16 6 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [4112]
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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(3,\{1,1,1,1,-2,1,-2,-2\})}
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]= { 3, 8, 3 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.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 7 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{10, 6}, {1, 8}, {7, 9}, {8, 10}, {9, 5}, {6, 4}, {5, 3}, {4, 2}, {3, 1}, {2, 7}]
In[14]:= Draw[ap]
8 7 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{3,6\}}
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-8]
Hyperbolic Volume 7.0222
A-Polynomial See Data:8 7/A-polynomial

[edit Notes for 8 7's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}
Rasmussen s-Invariant 2

[edit Notes for 8 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} }
Conway polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+3 z^4+2 z^2+1}
2nd Alexander ideal (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature { 23, 2 }
Jones polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+2 q^5-3 q^4+4 q^3-4 q^2+4 q-2+2 q^{-1} - q^{-2} }
HOMFLY-PT polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-2} +5 z^4 a^{-2} -z^4 a^{-4} -z^4+8 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+4 a^{-2} -2 a^{-4} -1}
Kauffman polynomial (db, data sources) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+a z^5-z^5 a^{-1} +2 z^5 a^{-5} -12 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4 a^{-6} -7 z^4-3 a z^3-3 z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -1}
The A2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+1+2 q^{-2} +2 q^{-6} + q^{-10} - q^{-14} - q^{-18} }
The G2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-q^{30}+2 q^{28}-3 q^{26}+q^{24}-q^{22}-3 q^{20}+7 q^{18}-8 q^{16}+5 q^{14}-3 q^{12}-2 q^{10}+6 q^8-10 q^6+7 q^4-3 q^2+6 q^{-2} -6 q^{-4} +4 q^{-6} +3 q^{-8} - q^{-10} +4 q^{-12} -4 q^{-14} +2 q^{-16} +4 q^{-18} -5 q^{-20} +10 q^{-22} -7 q^{-24} +5 q^{-26} +3 q^{-28} -7 q^{-30} +10 q^{-32} -11 q^{-34} +8 q^{-36} -2 q^{-38} -3 q^{-40} +8 q^{-42} -8 q^{-44} +6 q^{-46} -3 q^{-50} +3 q^{-52} -3 q^{-54} - q^{-56} +4 q^{-58} -5 q^{-60} +5 q^{-62} -3 q^{-64} -2 q^{-66} +4 q^{-68} -7 q^{-70} +5 q^{-72} -5 q^{-74} +2 q^{-76} - q^{-78} -3 q^{-80} +4 q^{-82} -4 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-90} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }
Further Quantum Invariants
Further quantum knot invariants for 8_7.

A1 Invariants.

Weight Invariant
1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^5+q^3+2 q^{-1} + q^{-7} - q^{-9} + q^{-11} - q^{-13} }
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{16}-q^{14}-2 q^{12}+2 q^{10}-3 q^6+2 q^4+3 q^2-2+2 q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-10} -2 q^{-14} - q^{-16} +3 q^{-18} -2 q^{-20} -2 q^{-22} +4 q^{-24} - q^{-26} -2 q^{-28} +2 q^{-30} - q^{-32} - q^{-34} + q^{-36} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{33}+q^{31}+2 q^{29}-3 q^{25}-2 q^{23}+4 q^{21}+3 q^{19}-4 q^{17}-6 q^{15}+q^{13}+6 q^{11}+2 q^9-7 q^7-2 q^5+6 q^3+7 q-3 q^{-1} -5 q^{-3} +3 q^{-5} +7 q^{-7} - q^{-9} -6 q^{-11} +2 q^{-13} +5 q^{-15} -6 q^{-19} -2 q^{-21} +3 q^{-23} +3 q^{-25} - q^{-27} -5 q^{-29} -2 q^{-31} +7 q^{-33} +4 q^{-35} -6 q^{-37} -6 q^{-39} +5 q^{-41} +6 q^{-43} -3 q^{-45} -5 q^{-47} +2 q^{-49} +3 q^{-51} - q^{-53} -2 q^{-55} + q^{-57} - q^{-61} + q^{-65} + q^{-67} - q^{-69} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{56}-q^{54}-2 q^{52}+q^{48}+5 q^{46}-4 q^{42}-4 q^{40}-3 q^{38}+10 q^{36}+7 q^{34}-2 q^{32}-8 q^{30}-13 q^{28}+4 q^{26}+10 q^{24}+9 q^{22}-18 q^{18}-8 q^{16}+2 q^{14}+15 q^{12}+14 q^{10}-8 q^8-13 q^6-12 q^4+10 q^2+23+5 q^{-2} -9 q^{-4} -20 q^{-6} +21 q^{-10} +10 q^{-12} -5 q^{-14} -19 q^{-16} -4 q^{-18} +15 q^{-20} +8 q^{-22} -4 q^{-24} -15 q^{-26} -4 q^{-28} +10 q^{-30} +10 q^{-32} -2 q^{-34} -10 q^{-36} -6 q^{-38} + q^{-40} +11 q^{-42} +7 q^{-44} + q^{-46} -13 q^{-48} -15 q^{-50} +8 q^{-52} +15 q^{-54} +14 q^{-56} -10 q^{-58} -25 q^{-60} -3 q^{-62} +12 q^{-64} +23 q^{-66} + q^{-68} -21 q^{-70} -9 q^{-72} +16 q^{-76} +7 q^{-78} -8 q^{-80} -4 q^{-82} -6 q^{-84} +5 q^{-86} +4 q^{-88} - q^{-90} +2 q^{-92} -5 q^{-94} + q^{-98} +3 q^{-102} - q^{-104} - q^{-108} - q^{-110} + q^{-112} }
5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{85}+q^{83}+2 q^{81}-q^{77}-3 q^{75}-3 q^{73}+6 q^{69}+6 q^{67}+q^{65}-5 q^{63}-10 q^{61}-8 q^{59}+3 q^{57}+16 q^{55}+15 q^{53}+3 q^{51}-11 q^{49}-22 q^{47}-15 q^{45}+4 q^{43}+22 q^{41}+23 q^{39}+9 q^{37}-12 q^{35}-30 q^{33}-24 q^{31}-3 q^{29}+22 q^{27}+33 q^{25}+21 q^{23}-9 q^{21}-34 q^{19}-38 q^{17}-14 q^{15}+28 q^{13}+51 q^{11}+31 q^9-8 q^7-48 q^5-50 q^3-5 q+48 q^{-1} +59 q^{-3} +23 q^{-5} -33 q^{-7} -63 q^{-9} -34 q^{-11} +25 q^{-13} +60 q^{-15} +39 q^{-17} -15 q^{-19} -52 q^{-21} -40 q^{-23} +7 q^{-25} +45 q^{-27} +34 q^{-29} -7 q^{-31} -36 q^{-33} -30 q^{-35} +4 q^{-37} +33 q^{-39} +24 q^{-41} -7 q^{-43} -26 q^{-45} -20 q^{-47} +2 q^{-49} +23 q^{-51} +20 q^{-53} +5 q^{-55} -15 q^{-57} -23 q^{-59} -15 q^{-61} +3 q^{-63} +25 q^{-65} +32 q^{-67} +14 q^{-69} -24 q^{-71} -47 q^{-73} -31 q^{-75} +12 q^{-77} +54 q^{-79} +54 q^{-81} +2 q^{-83} -57 q^{-85} -69 q^{-87} -21 q^{-89} +46 q^{-91} +77 q^{-93} +39 q^{-95} -30 q^{-97} -71 q^{-99} -49 q^{-101} +13 q^{-103} +56 q^{-105} +49 q^{-107} +3 q^{-109} -37 q^{-111} -43 q^{-113} -12 q^{-115} +22 q^{-117} +29 q^{-119} +14 q^{-121} -7 q^{-123} -20 q^{-125} -14 q^{-127} +2 q^{-129} +11 q^{-131} +10 q^{-133} +3 q^{-135} -5 q^{-137} -8 q^{-139} -4 q^{-141} +3 q^{-143} +5 q^{-145} +2 q^{-147} + q^{-149} -2 q^{-151} -3 q^{-153} - q^{-155} + q^{-157} + q^{-161} + q^{-163} - q^{-165} }
6 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{120}-q^{118}-2 q^{116}+q^{112}+3 q^{110}+q^{108}+3 q^{106}-2 q^{104}-8 q^{102}-5 q^{100}-q^{98}+6 q^{96}+7 q^{94}+14 q^{92}+3 q^{90}-12 q^{88}-18 q^{86}-17 q^{84}-3 q^{82}+6 q^{80}+33 q^{78}+30 q^{76}+8 q^{74}-15 q^{72}-34 q^{70}-34 q^{68}-29 q^{66}+17 q^{64}+42 q^{62}+47 q^{60}+28 q^{58}-q^{56}-31 q^{54}-67 q^{52}-42 q^{50}-12 q^{48}+30 q^{46}+56 q^{44}+66 q^{42}+42 q^{40}-27 q^{38}-62 q^{36}-86 q^{34}-62 q^{32}-8 q^{30}+75 q^{28}+117 q^{26}+81 q^{24}+18 q^{22}-81 q^{20}-134 q^{18}-123 q^{16}-12 q^{14}+104 q^{12}+154 q^{10}+136 q^8+15 q^6-116 q^4-190 q^2-123+12 q^{-2} +137 q^{-4} +195 q^{-6} +116 q^{-8} -37 q^{-10} -177 q^{-12} -176 q^{-14} -74 q^{-16} +70 q^{-18} +179 q^{-20} +159 q^{-22} +32 q^{-24} -123 q^{-26} -164 q^{-28} -104 q^{-30} +17 q^{-32} +128 q^{-34} +140 q^{-36} +53 q^{-38} -75 q^{-40} -117 q^{-42} -85 q^{-44} +2 q^{-46} +83 q^{-48} +94 q^{-50} +35 q^{-52} -51 q^{-54} -74 q^{-56} -49 q^{-58} +13 q^{-60} +56 q^{-62} +58 q^{-64} +18 q^{-66} -36 q^{-68} -54 q^{-70} -42 q^{-72} +5 q^{-74} +37 q^{-76} +54 q^{-78} +40 q^{-80} +2 q^{-82} -46 q^{-84} -73 q^{-86} -52 q^{-88} -13 q^{-90} +59 q^{-92} +102 q^{-94} +94 q^{-96} +4 q^{-98} -96 q^{-100} -138 q^{-102} -118 q^{-104} +5 q^{-106} +136 q^{-108} +202 q^{-110} +116 q^{-112} -44 q^{-114} -177 q^{-116} -226 q^{-118} -111 q^{-120} +80 q^{-122} +236 q^{-124} +213 q^{-126} +65 q^{-128} -113 q^{-130} -237 q^{-132} -188 q^{-134} -24 q^{-136} +157 q^{-138} +200 q^{-140} +125 q^{-142} -8 q^{-144} -144 q^{-146} -162 q^{-148} -76 q^{-150} +53 q^{-152} +108 q^{-154} +97 q^{-156} +40 q^{-158} -50 q^{-160} -84 q^{-162} -59 q^{-164} +4 q^{-166} +35 q^{-168} +45 q^{-170} +37 q^{-172} -10 q^{-174} -33 q^{-176} -29 q^{-178} -4 q^{-180} +7 q^{-182} +16 q^{-184} +23 q^{-186} -12 q^{-190} -14 q^{-192} -4 q^{-194} - q^{-196} +5 q^{-198} +13 q^{-200} +2 q^{-202} -3 q^{-204} -6 q^{-206} -2 q^{-208} -3 q^{-210} +5 q^{-214} + q^{-216} + q^{-218} - q^{-220} - q^{-224} - q^{-226} + q^{-228} }

A2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+1+2 q^{-2} +2 q^{-6} + q^{-10} - q^{-14} - q^{-18} }
1,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-2 q^{18}+4 q^{16}-8 q^{14}+13 q^{12}-18 q^{10}+18 q^8-24 q^6+20 q^4-16 q^2+10+4 q^{-2} -4 q^{-4} +22 q^{-6} -22 q^{-8} +34 q^{-10} -34 q^{-12} +34 q^{-14} -34 q^{-16} +26 q^{-18} -22 q^{-20} +12 q^{-22} -4 q^{-24} -4 q^{-26} +7 q^{-28} -12 q^{-30} +14 q^{-32} -14 q^{-34} +13 q^{-36} -12 q^{-38} +12 q^{-40} -10 q^{-42} +7 q^{-44} -6 q^{-46} +4 q^{-48} -2 q^{-50} + q^{-52} }
2,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-q^{14}-q^{12}-q^8-3 q^6+2 q^2+2+ q^{-2} +4 q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} +2 q^{-12} - q^{-16} + q^{-18} - q^{-20} -3 q^{-22} + q^{-26} - q^{-28} - q^{-30} + q^{-32} - q^{-36} - q^{-38} + q^{-46} }
3,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{36}+q^{32}+2 q^{30}+q^{28}-2 q^{26}-q^{24}+q^{22}+4 q^{20}-6 q^{16}-7 q^{14}-q^{12}+3 q^{10}-6 q^6-4 q^4+5 q^2+10+6 q^{-2} -2 q^{-4} + q^{-6} +5 q^{-8} +8 q^{-10} + q^{-12} +2 q^{-16} +4 q^{-18} + q^{-20} -4 q^{-22} - q^{-24} -3 q^{-28} -7 q^{-30} -5 q^{-32} +3 q^{-34} +3 q^{-36} -3 q^{-38} -6 q^{-40} + q^{-42} +8 q^{-44} +5 q^{-46} -3 q^{-48} -6 q^{-50} + q^{-52} +6 q^{-54} +2 q^{-56} -4 q^{-58} -5 q^{-60} +3 q^{-64} - q^{-68} -2 q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} - q^{-84} }

A3 Invariants.

Weight Invariant
0,1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}-q^{12}-3 q^6+q^4-2+2 q^{-2} +3 q^{-4} - q^{-6} +4 q^{-8} +5 q^{-10} +2 q^{-12} +2 q^{-14} + q^{-16} -3 q^{-20} -3 q^{-22} + q^{-24} -3 q^{-26} -2 q^{-28} +3 q^{-30} - q^{-32} -2 q^{-34} +2 q^{-36} - q^{-40} + q^{-42} }
1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7-q^3+q+2 q^{-3} + q^{-5} +2 q^{-7} +2 q^{-9} + q^{-11} + q^{-13} - q^{-15} -2 q^{-19} - q^{-23} }
1,0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-2 q^{22}+3 q^{20}-3 q^{18}+q^{16}+5 q^{14}-10 q^{12}+11 q^{10}-12 q^8+q^6+q^4-16 q^2+16-18 q^{-2} +15 q^{-4} +6 q^{-6} +22 q^{-10} +2 q^{-12} +11 q^{-14} +4 q^{-16} +4 q^{-18} -8 q^{-20} +7 q^{-22} -16 q^{-24} -18 q^{-30} +16 q^{-32} -15 q^{-34} +2 q^{-36} +9 q^{-38} -13 q^{-40} +10 q^{-42} - q^{-44} -6 q^{-46} +11 q^{-48} -7 q^{-50} +2 q^{-52} +4 q^{-54} -8 q^{-56} +6 q^{-58} -3 q^{-60} - q^{-62} +3 q^{-64} -2 q^{-66} + q^{-68} }

A4 Invariants.

Weight Invariant
0,1,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{16}-q^8-2 q^6-q^4-2 q^2-3+ q^{-4} + q^{-6} + q^{-8} +7 q^{-10} +7 q^{-12} +5 q^{-14} +6 q^{-16} +8 q^{-18} + q^{-20} - q^{-22} -4 q^{-26} -6 q^{-28} -3 q^{-30} -2 q^{-32} -4 q^{-34} -2 q^{-36} + q^{-38} -2 q^{-42} + q^{-44} +2 q^{-46} + q^{-52} }
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8-q^4+2 q^{-4} + q^{-6} +3 q^{-8} +2 q^{-10} +3 q^{-12} + q^{-14} + q^{-16} - q^{-18} - q^{-20} - q^{-22} -2 q^{-24} - q^{-28} }

B2 Invariants.

Weight Invariant
0,1 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{14}+q^{12}-2 q^{10}+2 q^8-3 q^6+3 q^4-2 q^2+2+ q^{-4} +3 q^{-6} -2 q^{-8} +5 q^{-10} -4 q^{-12} +6 q^{-14} -5 q^{-16} +4 q^{-18} -3 q^{-20} + q^{-22} - q^{-24} - q^{-26} +2 q^{-28} -3 q^{-30} +3 q^{-32} -2 q^{-34} +2 q^{-36} -2 q^{-38} + q^{-40} - q^{-42} }
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-q^{20}-q^{18}+q^{16}+q^{14}-2 q^{12}-3 q^{10}+3 q^6+q^4-2 q^2-2+2 q^{-2} +3 q^{-4} +2 q^{-6} -2 q^{-8} +2 q^{-12} +3 q^{-14} +2 q^{-20} +3 q^{-22} -2 q^{-26} +2 q^{-30} -3 q^{-34} -2 q^{-36} + q^{-38} + q^{-40} -2 q^{-42} -3 q^{-44} +3 q^{-48} + q^{-50} -2 q^{-52} -2 q^{-54} +2 q^{-58} + q^{-60} - q^{-62} - q^{-64} + q^{-68} }

D4 Invariants.

Weight Invariant
1,0,0,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-q^{16}+q^{14}-2 q^{12}+2 q^{10}-3 q^8+q^6-3 q^4+q^2-2+ q^{-6} +3 q^{-8} + q^{-10} +7 q^{-12} +2 q^{-14} +8 q^{-16} - q^{-18} +6 q^{-20} -3 q^{-22} +3 q^{-24} -5 q^{-26} - q^{-28} -4 q^{-30} - q^{-32} - q^{-34} -2 q^{-36} -2 q^{-40} +3 q^{-42} -2 q^{-44} + q^{-46} -2 q^{-48} +2 q^{-50} - q^{-52} + q^{-54} - q^{-56} + q^{-58} }

G2 Invariants.

Weight Invariant
1,0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{32}-q^{30}+2 q^{28}-3 q^{26}+q^{24}-q^{22}-3 q^{20}+7 q^{18}-8 q^{16}+5 q^{14}-3 q^{12}-2 q^{10}+6 q^8-10 q^6+7 q^4-3 q^2+6 q^{-2} -6 q^{-4} +4 q^{-6} +3 q^{-8} - q^{-10} +4 q^{-12} -4 q^{-14} +2 q^{-16} +4 q^{-18} -5 q^{-20} +10 q^{-22} -7 q^{-24} +5 q^{-26} +3 q^{-28} -7 q^{-30} +10 q^{-32} -11 q^{-34} +8 q^{-36} -2 q^{-38} -3 q^{-40} +8 q^{-42} -8 q^{-44} +6 q^{-46} -3 q^{-50} +3 q^{-52} -3 q^{-54} - q^{-56} +4 q^{-58} -5 q^{-60} +5 q^{-62} -3 q^{-64} -2 q^{-66} +4 q^{-68} -7 q^{-70} +5 q^{-72} -5 q^{-74} +2 q^{-76} - q^{-78} -3 q^{-80} +4 q^{-82} -4 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-90} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }

.

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 7"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} }
In[5]:= Conway[K][z]
Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6+3 z^4+2 z^2+1}
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]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 23, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+2 q^5-3 q^4+4 q^3-4 q^2+4 q-2+2 q^{-1} - q^{-2} }
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 a^{-2} +5 z^4 a^{-2} -z^4 a^{-4} -z^4+8 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+4 a^{-2} -2 a^{-4} -1}
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+a z^5-z^5 a^{-1} +2 z^5 a^{-5} -12 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4 a^{-6} -7 z^4-3 a z^3-3 z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n24,}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

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 7"];
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]= { Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^6+2 q^5-3 q^4+4 q^3-4 q^2+4 q-2+2 q^{-1} - q^{-2} } }
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]= {K11n24,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (2, 2)
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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{124}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{28}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{64}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{256}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{992}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{224}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{7951}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1052}{5}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{6596}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{65}{9}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{929}{15}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 8 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -3-2-1012345χ
13        1-1
11       1 1
9      21 -1
7     21  1
5    22   0
3   22    0
1  13     2
-1 11      0
-3 1       1
-51        -1
Integral Khovanov Homology

(db, data source)

  
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{17}-2 q^{16}+4 q^{14}-6 q^{13}+q^{12}+9 q^{11}-12 q^{10}+q^9+14 q^8-16 q^7+16 q^5-14 q^4-2 q^3+14 q^2-9 q-3+10 q^{-1} -4 q^{-2} -4 q^{-3} +5 q^{-4} - q^{-5} -2 q^{-6} + q^{-7} }
3 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{33}+2 q^{32}-q^{30}-2 q^{29}+3 q^{28}+q^{27}-4 q^{26}-q^{25}+7 q^{24}-11 q^{22}+q^{21}+16 q^{20}-q^{19}-22 q^{18}+q^{17}+26 q^{16}+2 q^{15}-31 q^{14}-2 q^{13}+30 q^{12}+6 q^{11}-31 q^{10}-7 q^9+26 q^8+12 q^7-26 q^6-10 q^5+18 q^4+17 q^3-18 q^2-14 q+10+19 q^{-1} -8 q^{-2} -15 q^{-3} +2 q^{-4} +14 q^{-5} + q^{-6} -11 q^{-7} -3 q^{-8} +7 q^{-9} +3 q^{-10} -4 q^{-11} -2 q^{-12} + q^{-13} +2 q^{-14} - q^{-15} }
4 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{54}-2 q^{53}+q^{51}-q^{50}+5 q^{49}-5 q^{48}+q^{47}-6 q^{45}+12 q^{44}-8 q^{43}+6 q^{42}+q^{41}-17 q^{40}+14 q^{39}-12 q^{38}+21 q^{37}+10 q^{36}-33 q^{35}+5 q^{34}-24 q^{33}+43 q^{32}+32 q^{31}-44 q^{30}-10 q^{29}-46 q^{28}+58 q^{27}+56 q^{26}-43 q^{25}-17 q^{24}-69 q^{23}+60 q^{22}+70 q^{21}-37 q^{20}-13 q^{19}-79 q^{18}+53 q^{17}+66 q^{16}-29 q^{15}-q^{14}-79 q^{13}+39 q^{12}+55 q^{11}-18 q^{10}+11 q^9-72 q^8+20 q^7+40 q^6-4 q^5+26 q^4-61 q^3-q^2+20 q+7+40 q^{-1} -43 q^{-2} -14 q^{-3} -2 q^{-4} +6 q^{-5} +45 q^{-6} -21 q^{-7} -13 q^{-8} -15 q^{-9} -4 q^{-10} +35 q^{-11} -3 q^{-12} -4 q^{-13} -14 q^{-14} -10 q^{-15} +18 q^{-16} +2 q^{-17} +2 q^{-18} -5 q^{-19} -7 q^{-20} +5 q^{-21} + q^{-22} +2 q^{-23} - q^{-24} -2 q^{-25} + q^{-26} }
5 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{80}+2 q^{79}-q^{77}+q^{76}-2 q^{75}-3 q^{74}+3 q^{73}+3 q^{72}+4 q^{70}-4 q^{69}-10 q^{68}-q^{67}+6 q^{66}+8 q^{65}+11 q^{64}-3 q^{63}-19 q^{62}-17 q^{61}+21 q^{59}+32 q^{58}+12 q^{57}-26 q^{56}-51 q^{55}-31 q^{54}+27 q^{53}+72 q^{52}+58 q^{51}-19 q^{50}-94 q^{49}-93 q^{48}+5 q^{47}+113 q^{46}+127 q^{45}+19 q^{44}-125 q^{43}-160 q^{42}-43 q^{41}+125 q^{40}+186 q^{39}+71 q^{38}-125 q^{37}-202 q^{36}-86 q^{35}+109 q^{34}+209 q^{33}+109 q^{32}-107 q^{31}-209 q^{30}-108 q^{29}+91 q^{28}+201 q^{27}+117 q^{26}-87 q^{25}-194 q^{24}-105 q^{23}+70 q^{22}+179 q^{21}+111 q^{20}-68 q^{19}-163 q^{18}-96 q^{17}+41 q^{16}+145 q^{15}+105 q^{14}-39 q^{13}-122 q^{12}-85 q^{11}+3 q^{10}+98 q^9+93 q^8-2 q^7-68 q^6-64 q^5-32 q^4+39 q^3+64 q^2+28 q-12-28 q^{-1} -43 q^{-2} -14 q^{-3} +19 q^{-4} +30 q^{-5} +28 q^{-6} +11 q^{-7} -23 q^{-8} -37 q^{-9} -23 q^{-10} +6 q^{-11} +32 q^{-12} +36 q^{-13} +7 q^{-14} -25 q^{-15} -34 q^{-16} -19 q^{-17} +11 q^{-18} +30 q^{-19} +25 q^{-20} -4 q^{-21} -20 q^{-22} -20 q^{-23} -7 q^{-24} +11 q^{-25} +18 q^{-26} +7 q^{-27} -6 q^{-28} -8 q^{-29} -6 q^{-30} -2 q^{-31} +7 q^{-32} +5 q^{-33} - q^{-34} -2 q^{-35} - q^{-36} -2 q^{-37} + q^{-38} +2 q^{-39} - q^{-40} }
6 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{111}-2 q^{110}+q^{108}-q^{107}+2 q^{106}+5 q^{104}-7 q^{103}-3 q^{102}+2 q^{101}-5 q^{100}+5 q^{99}+5 q^{98}+16 q^{97}-15 q^{96}-9 q^{95}-q^{94}-15 q^{93}+7 q^{92}+17 q^{91}+39 q^{90}-22 q^{89}-18 q^{88}-12 q^{87}-40 q^{86}+3 q^{85}+40 q^{84}+86 q^{83}-14 q^{82}-28 q^{81}-43 q^{80}-103 q^{79}-22 q^{78}+74 q^{77}+176 q^{76}+43 q^{75}-17 q^{74}-98 q^{73}-232 q^{72}-108 q^{71}+92 q^{70}+312 q^{69}+176 q^{68}+58 q^{67}-141 q^{66}-413 q^{65}-272 q^{64}+43 q^{63}+436 q^{62}+354 q^{61}+206 q^{60}-118 q^{59}-569 q^{58}-463 q^{57}-72 q^{56}+485 q^{55}+487 q^{54}+366 q^{53}-32 q^{52}-635 q^{51}-594 q^{50}-195 q^{49}+465 q^{48}+529 q^{47}+466 q^{46}+58 q^{45}-627 q^{44}-637 q^{43}-267 q^{42}+426 q^{41}+508 q^{40}+493 q^{39}+106 q^{38}-589 q^{37}-623 q^{36}-284 q^{35}+394 q^{34}+461 q^{33}+481 q^{32}+124 q^{31}-535 q^{30}-583 q^{29}-286 q^{28}+351 q^{27}+399 q^{26}+456 q^{25}+147 q^{24}-449 q^{23}-524 q^{22}-297 q^{21}+270 q^{20}+312 q^{19}+424 q^{18}+189 q^{17}-321 q^{16}-437 q^{15}-309 q^{14}+159 q^{13}+191 q^{12}+364 q^{11}+230 q^{10}-166 q^9-310 q^8-289 q^7+50 q^6+47 q^5+262 q^4+229 q^3-26 q^2-157 q-210-8 q^{-1} -78 q^{-2} +127 q^{-3} +162 q^{-4} +48 q^{-5} -26 q^{-6} -89 q^{-7} +10 q^{-8} -128 q^{-9} +11 q^{-10} +51 q^{-11} +37 q^{-12} +27 q^{-13} +10 q^{-14} +73 q^{-15} -92 q^{-16} -31 q^{-17} -32 q^{-18} -17 q^{-19} +3 q^{-20} +34 q^{-21} +108 q^{-22} -23 q^{-23} -7 q^{-24} -42 q^{-25} -43 q^{-26} -39 q^{-27} +4 q^{-28} +83 q^{-29} +13 q^{-30} +23 q^{-31} -13 q^{-32} -24 q^{-33} -44 q^{-34} -21 q^{-35} +37 q^{-36} +8 q^{-37} +23 q^{-38} +6 q^{-39} - q^{-40} -22 q^{-41} -18 q^{-42} +10 q^{-43} - q^{-44} +9 q^{-45} +5 q^{-46} +5 q^{-47} -7 q^{-48} -7 q^{-49} +3 q^{-50} -2 q^{-51} +2 q^{-52} + q^{-53} +2 q^{-54} - q^{-55} -2 q^{-56} + q^{-57} }
7 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{147}+2 q^{146}-q^{144}+q^{143}-2 q^{142}-2 q^{140}-q^{139}+7 q^{138}+q^{137}-q^{136}+4 q^{135}-7 q^{134}-3 q^{133}-7 q^{132}-6 q^{131}+18 q^{130}+6 q^{129}+2 q^{128}+9 q^{127}-13 q^{126}-8 q^{125}-18 q^{124}-17 q^{123}+32 q^{122}+16 q^{121}+11 q^{120}+19 q^{119}-26 q^{118}-19 q^{117}-35 q^{116}-33 q^{115}+50 q^{114}+42 q^{113}+44 q^{112}+36 q^{111}-59 q^{110}-64 q^{109}-82 q^{108}-65 q^{107}+84 q^{106}+117 q^{105}+145 q^{104}+100 q^{103}-110 q^{102}-182 q^{101}-239 q^{100}-177 q^{99}+112 q^{98}+273 q^{97}+387 q^{96}+302 q^{95}-98 q^{94}-371 q^{93}-566 q^{92}-481 q^{91}+35 q^{90}+446 q^{89}+775 q^{88}+721 q^{87}+87 q^{86}-499 q^{85}-990 q^{84}-979 q^{83}-256 q^{82}+491 q^{81}+1163 q^{80}+1255 q^{79}+467 q^{78}-436 q^{77}-1292 q^{76}-1498 q^{75}-682 q^{74}+333 q^{73}+1361 q^{72}+1686 q^{71}+877 q^{70}-205 q^{69}-1364 q^{68}-1814 q^{67}-1047 q^{66}+81 q^{65}+1343 q^{64}+1884 q^{63}+1141 q^{62}+27 q^{61}-1276 q^{60}-1903 q^{59}-1218 q^{58}-113 q^{57}+1238 q^{56}+1897 q^{55}+1230 q^{54}+155 q^{53}-1174 q^{52}-1863 q^{51}-1238 q^{50}-193 q^{49}+1136 q^{48}+1838 q^{47}+1215 q^{46}+200 q^{45}-1082 q^{44}-1784 q^{43}-1203 q^{42}-233 q^{41}+1039 q^{40}+1741 q^{39}+1174 q^{38}+253 q^{37}-952 q^{36}-1669 q^{35}-1174 q^{34}-305 q^{33}+876 q^{32}+1589 q^{31}+1138 q^{30}+369 q^{29}-731 q^{28}-1485 q^{27}-1145 q^{26}-447 q^{25}+610 q^{24}+1354 q^{23}+1087 q^{22}+539 q^{21}-412 q^{20}-1199 q^{19}-1075 q^{18}-619 q^{17}+259 q^{16}+1017 q^{15}+966 q^{14}+687 q^{13}-38 q^{12}-808 q^{11}-897 q^{10}-728 q^9-101 q^8+599 q^7+719 q^6+716 q^5+273 q^4-369 q^3-582 q^2-668 q-345+191 q^{-1} +370 q^{-2} +554 q^{-3} +398 q^{-4} -24 q^{-5} -204 q^{-6} -425 q^{-7} -363 q^{-8} -60 q^{-9} +41 q^{-10} +264 q^{-11} +294 q^{-12} +103 q^{-13} +60 q^{-14} -126 q^{-15} -191 q^{-16} -75 q^{-17} -113 q^{-18} +15 q^{-19} +86 q^{-20} +25 q^{-21} +109 q^{-22} +35 q^{-23} - q^{-24} +48 q^{-25} -67 q^{-26} -60 q^{-27} -43 q^{-28} -89 q^{-29} +14 q^{-30} +23 q^{-31} +49 q^{-32} +128 q^{-33} +34 q^{-34} -34 q^{-36} -108 q^{-37} -52 q^{-38} -45 q^{-39} -4 q^{-40} +91 q^{-41} +66 q^{-42} +49 q^{-43} +20 q^{-44} -52 q^{-45} -37 q^{-46} -53 q^{-47} -45 q^{-48} +25 q^{-49} +29 q^{-50} +42 q^{-51} +33 q^{-52} -9 q^{-53} -5 q^{-54} -20 q^{-55} -33 q^{-56} -6 q^{-57} +2 q^{-58} +16 q^{-59} +19 q^{-60} -2 q^{-61} +4 q^{-62} - q^{-63} -11 q^{-64} -5 q^{-65} -4 q^{-66} +4 q^{-67} +7 q^{-68} - q^{-69} +2 q^{-71} -2 q^{-72} - q^{-73} -2 q^{-74} + q^{-75} +2 q^{-76} - q^{-77} }

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.

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8_6

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8_8

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