8 10

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8 9.gif

8_9

8 11.gif

8_11

8 10.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

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[edit 8 10 Quick Notes]
[edit 8 10 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X3849 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X7283
Gauss code -1, 8, -2, 1, -4, 5, -8, 2, -3, 7, -6, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code 4 8 12 2 14 16 6 10
Conway Notation [3,21,2]


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

Length is 8, width is 3,

Braid index is 3

8 10 ML.gif 8 10 AP.gif
[{9, 4}, {3, 7}, {6, 8}, {7, 9}, {8, 11}, {5, 10}, {4, 6}, {2, 5}, {1, 3}, {11, 2}, {10, 1}]

[edit Notes on presentations of 8 10]

Knot 8_10.
A graph, knot 8_10.
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 10"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3849 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,1,12,16 X15,11,16,10 X7283
In[5]:= GaussCode[K]
Out[5]= -1, 8, -2, 1, -4, 5, -8, 2, -3, 7, -6, 4, -5, 3, -7, 6
In[6]:= DTCode[K]
Out[6]= 4 8 12 2 14 16 6 10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [3,21,2]
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,-2,1,1,-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.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.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 10 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{9, 4}, {3, 7}, {6, 8}, {7, 9}, {8, 11}, {5, 10}, {4, 6}, {2, 5}, {1, 3}, {11, 2}, {10, 1}]
In[14]:= Draw[ap]
8 10 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
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 \{4,6\}}
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-8]
Hyperbolic Volume 8.65115
A-Polynomial See Data:8 10/A-polynomial

[edit Notes for 8 10's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus
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 1}
Rasmussen s-Invariant 2

[edit Notes for 8 10'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+6 t-7+6 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+3 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 { 27, 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^5-4 q^4+5 q^3-4 q^2+5 q-3+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+9 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+6 a^{-2} -3 a^{-4} -2}
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} +5 z^6 a^{-2} +3 z^6 a^{-4} +2 z^6+a z^5+z^5 a^{-1} +3 z^5 a^{-3} +3 z^5 a^{-5} -13 z^4 a^{-2} -5 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4-3 a z^3-8 z^3 a^{-1} -9 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +5 z^2+2 a z+5 z a^{-1} +6 z a^{-3} +2 z a^{-5} -z a^{-7} -6 a^{-2} -3 a^{-4} -2}
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-q^2+2 q^{-2} + q^{-4} +4 q^{-6} + q^{-8} + q^{-10} - q^{-12} -2 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}+3 q^{28}-4 q^{26}+2 q^{24}-q^{22}-5 q^{20}+9 q^{18}-12 q^{16}+10 q^{14}-6 q^{12}-5 q^{10}+12 q^8-16 q^6+14 q^4-9 q^2-2+9 q^{-2} -13 q^{-4} +9 q^{-6} - q^{-8} -4 q^{-10} +12 q^{-12} -7 q^{-14} +3 q^{-16} +7 q^{-18} -10 q^{-20} +19 q^{-22} -14 q^{-24} +9 q^{-26} +7 q^{-28} -12 q^{-30} +23 q^{-32} -19 q^{-34} +14 q^{-36} - q^{-38} -7 q^{-40} +13 q^{-42} -17 q^{-44} +11 q^{-46} - q^{-48} -7 q^{-50} +8 q^{-52} -8 q^{-54} -2 q^{-56} +8 q^{-58} -14 q^{-60} +9 q^{-62} -6 q^{-64} -3 q^{-66} +9 q^{-68} -14 q^{-70} +14 q^{-72} -8 q^{-74} +2 q^{-76} +2 q^{-78} -7 q^{-80} +6 q^{-82} -6 q^{-84} +5 q^{-86} -2 q^{-88} + q^{-92} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }
Further Quantum Invariants
Further quantum knot invariants for 8_10.

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-q+2 q^{-1} + q^{-3} + q^{-5} + q^{-7} -2 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}+3 q^{10}-5 q^6+3 q^4+3 q^2-5+2 q^{-2} +5 q^{-4} -2 q^{-6} +4 q^{-10} + q^{-12} -2 q^{-14} - q^{-16} +4 q^{-18} -4 q^{-20} -4 q^{-22} +6 q^{-24} -2 q^{-26} -4 q^{-28} +3 q^{-30} - 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}-4 q^{25}-2 q^{23}+6 q^{21}+5 q^{19}-6 q^{17}-10 q^{15}+3 q^{13}+13 q^{11}+q^9-15 q^7-6 q^5+12 q^3+12 q-11 q^{-1} -11 q^{-3} +8 q^{-5} +16 q^{-7} -3 q^{-9} -13 q^{-11} +2 q^{-13} +14 q^{-15} + q^{-17} -10 q^{-19} -3 q^{-21} +6 q^{-23} +8 q^{-25} -4 q^{-27} -11 q^{-29} -4 q^{-31} +14 q^{-33} +6 q^{-35} -13 q^{-37} -13 q^{-39} +13 q^{-41} +13 q^{-43} -10 q^{-45} -13 q^{-47} +4 q^{-49} +11 q^{-51} - q^{-53} -6 q^{-55} + q^{-57} +3 q^{-59} - q^{-63} + 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}+6 q^{46}-6 q^{42}-6 q^{40}-4 q^{38}+15 q^{36}+12 q^{34}-2 q^{32}-16 q^{30}-25 q^{28}+9 q^{26}+26 q^{24}+24 q^{22}-4 q^{20}-45 q^{18}-22 q^{16}+11 q^{14}+45 q^{12}+32 q^{10}-33 q^8-46 q^6-26 q^4+37 q^2+60+ q^{-2} -40 q^{-4} -49 q^{-6} +13 q^{-8} +60 q^{-10} +25 q^{-12} -26 q^{-14} -52 q^{-16} - q^{-18} +46 q^{-20} +28 q^{-22} -16 q^{-24} -42 q^{-26} -9 q^{-28} +27 q^{-30} +29 q^{-32} -4 q^{-34} -30 q^{-36} -22 q^{-38} +5 q^{-40} +32 q^{-42} +21 q^{-44} -7 q^{-46} -40 q^{-48} -31 q^{-50} +28 q^{-52} +46 q^{-54} +27 q^{-56} -41 q^{-58} -66 q^{-60} +3 q^{-62} +48 q^{-64} +55 q^{-66} -17 q^{-68} -64 q^{-70} -19 q^{-72} +21 q^{-74} +50 q^{-76} +10 q^{-78} -32 q^{-80} -19 q^{-82} -4 q^{-84} +22 q^{-86} +10 q^{-88} -7 q^{-90} -3 q^{-92} -6 q^{-94} +4 q^{-96} +2 q^{-98} -2 q^{-100} + q^{-102} -2 q^{-104} + q^{-106} - 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}-4 q^{73}+8 q^{69}+8 q^{67}+2 q^{65}-7 q^{63}-16 q^{61}-14 q^{59}+4 q^{57}+26 q^{55}+29 q^{53}+10 q^{51}-23 q^{49}-49 q^{47}-39 q^{45}+7 q^{43}+59 q^{41}+71 q^{39}+31 q^{37}-43 q^{35}-97 q^{33}-82 q^{31}+2 q^{29}+98 q^{27}+127 q^{25}+61 q^{23}-66 q^{21}-153 q^{19}-129 q^{17}+2 q^{15}+149 q^{13}+181 q^{11}+66 q^9-107 q^7-208 q^5-143 q^3+53 q+209 q^{-1} +185 q^{-3} +16 q^{-5} -176 q^{-7} -215 q^{-9} -60 q^{-11} +146 q^{-13} +214 q^{-15} +94 q^{-17} -100 q^{-19} -198 q^{-21} -105 q^{-23} +78 q^{-25} +170 q^{-27} +104 q^{-29} -55 q^{-31} -147 q^{-33} -93 q^{-35} +40 q^{-37} +124 q^{-39} +82 q^{-41} -32 q^{-43} -108 q^{-45} -81 q^{-47} +14 q^{-49} +92 q^{-51} +84 q^{-53} +12 q^{-55} -69 q^{-57} -102 q^{-59} -55 q^{-61} +46 q^{-63} +112 q^{-65} +105 q^{-67} +9 q^{-69} -122 q^{-71} -164 q^{-73} -60 q^{-75} +104 q^{-77} +204 q^{-79} +139 q^{-81} -71 q^{-83} -236 q^{-85} -193 q^{-87} +11 q^{-89} +218 q^{-91} +241 q^{-93} +50 q^{-95} -176 q^{-97} -246 q^{-99} -100 q^{-101} +113 q^{-103} +213 q^{-105} +131 q^{-107} -46 q^{-109} -160 q^{-111} -128 q^{-113} + q^{-115} +98 q^{-117} +99 q^{-119} +27 q^{-121} -47 q^{-123} -64 q^{-125} -31 q^{-127} +17 q^{-129} +32 q^{-131} +20 q^{-133} - q^{-135} -13 q^{-137} -12 q^{-139} -4 q^{-141} +6 q^{-143} +5 q^{-145} - q^{-151} -2 q^{-153} + q^{-155} +2 q^{-157} - q^{-159} + 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}+4 q^{106}-2 q^{104}-10 q^{102}-7 q^{100}-2 q^{98}+8 q^{96}+10 q^{94}+21 q^{92}+8 q^{90}-16 q^{88}-30 q^{86}-32 q^{84}-12 q^{82}+8 q^{80}+61 q^{78}+66 q^{76}+31 q^{74}-25 q^{72}-81 q^{70}-100 q^{68}-88 q^{66}+27 q^{64}+121 q^{62}+165 q^{60}+121 q^{58}+8 q^{56}-137 q^{54}-260 q^{52}-194 q^{50}-43 q^{48}+171 q^{46}+308 q^{44}+312 q^{42}+121 q^{40}-207 q^{38}-392 q^{36}-416 q^{34}-182 q^{32}+171 q^{30}+505 q^{28}+550 q^{26}+238 q^{24}-187 q^{22}-585 q^{20}-648 q^{18}-349 q^{16}+245 q^{14}+694 q^{12}+721 q^{10}+362 q^8-285 q^6-773 q^4-829 q^2-300+394 q^{-2} +838 q^{-4} +803 q^{-6} +225 q^{-8} -492 q^{-10} -923 q^{-12} -684 q^{-14} -47 q^{-16} +604 q^{-18} +871 q^{-20} +539 q^{-22} -128 q^{-24} -712 q^{-26} -727 q^{-28} -295 q^{-30} +311 q^{-32} +677 q^{-34} +545 q^{-36} +55 q^{-38} -455 q^{-40} -555 q^{-42} -298 q^{-44} +159 q^{-46} +453 q^{-48} +393 q^{-50} +60 q^{-52} -302 q^{-54} -376 q^{-56} -203 q^{-58} +123 q^{-60} +320 q^{-62} +280 q^{-64} +44 q^{-66} -222 q^{-68} -309 q^{-70} -203 q^{-72} +56 q^{-74} +255 q^{-76} +318 q^{-78} +173 q^{-80} -83 q^{-82} -316 q^{-84} -375 q^{-86} -189 q^{-88} +105 q^{-90} +417 q^{-92} +486 q^{-94} +266 q^{-96} -200 q^{-98} -580 q^{-100} -612 q^{-102} -288 q^{-104} +321 q^{-106} +768 q^{-108} +775 q^{-110} +203 q^{-112} -517 q^{-114} -928 q^{-116} -808 q^{-118} -103 q^{-120} +689 q^{-122} +1086 q^{-124} +708 q^{-126} -81 q^{-128} -797 q^{-130} -1043 q^{-132} -579 q^{-134} +221 q^{-136} +876 q^{-138} +865 q^{-140} +366 q^{-142} -297 q^{-144} -761 q^{-146} -678 q^{-148} -199 q^{-150} +362 q^{-152} +563 q^{-154} +434 q^{-156} +85 q^{-158} -283 q^{-160} -399 q^{-162} -258 q^{-164} +21 q^{-166} +177 q^{-168} +223 q^{-170} +139 q^{-172} -26 q^{-174} -120 q^{-176} -119 q^{-178} -38 q^{-180} +10 q^{-182} +53 q^{-184} +63 q^{-186} +17 q^{-188} -18 q^{-190} -28 q^{-192} -12 q^{-194} -10 q^{-196} +5 q^{-198} +17 q^{-200} +6 q^{-202} -2 q^{-204} -5 q^{-206} + q^{-208} -4 q^{-210} - q^{-212} +5 q^{-214} - q^{-218} -2 q^{-220} + q^{-222} - 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-q^2+2 q^{-2} + q^{-4} +4 q^{-6} + q^{-8} + q^{-10} - q^{-12} -2 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}+6 q^{16}-12 q^{14}+19 q^{12}-28 q^{10}+36 q^8-44 q^6+39 q^4-40 q^2+26-10 q^{-2} -8 q^{-4} +34 q^{-6} -40 q^{-8} +70 q^{-10} -67 q^{-12} +80 q^{-14} -70 q^{-16} +60 q^{-18} -49 q^{-20} +24 q^{-22} -12 q^{-24} -12 q^{-26} +23 q^{-28} -34 q^{-30} +34 q^{-32} -34 q^{-34} +32 q^{-36} -26 q^{-38} +18 q^{-40} -14 q^{-42} +11 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^{10}-q^8-3 q^6-q^4+q^2-2-2 q^{-2} +3 q^{-4} + q^{-6} + q^{-8} +3 q^{-10} +6 q^{-12} +4 q^{-14} +5 q^{-16} +6 q^{-18} +2 q^{-20} -4 q^{-22} -3 q^{-24} -3 q^{-26} -7 q^{-28} -4 q^{-30} - q^{-36} +2 q^{-40} + q^{-42} + 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}-3 q^{26}-q^{24}+3 q^{22}+6 q^{20}-7 q^{16}-6 q^{14}+3 q^{12}+8 q^{10}-12 q^6-9 q^4+3 q^2+10- q^{-2} -9 q^{-4} -3 q^{-6} +8 q^{-8} +10 q^{-10} - q^{-12} -2 q^{-14} +3 q^{-16} +10 q^{-18} +4 q^{-20} +4 q^{-24} +11 q^{-26} +5 q^{-28} + q^{-30} +9 q^{-34} +4 q^{-36} -11 q^{-38} -18 q^{-40} -9 q^{-42} +3 q^{-44} -4 q^{-46} -13 q^{-48} -14 q^{-50} +4 q^{-52} +10 q^{-54} +4 q^{-56} -5 q^{-58} -4 q^{-60} +7 q^{-62} +10 q^{-64} +4 q^{-66} - q^{-68} -2 q^{-70} +2 q^{-72} + q^{-74} - q^{-76} - q^{-78} - q^{-80} - 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}+q^{10}+q^8-4 q^6-2 q^2-7+ q^{-2} +3 q^{-4} +9 q^{-8} +10 q^{-10} +6 q^{-12} +3 q^{-14} + q^{-16} -2 q^{-18} -6 q^{-20} -6 q^{-22} + q^{-24} -4 q^{-26} -3 q^{-28} +5 q^{-30} -2 q^{-32} -2 q^{-34} +3 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^7-2 q^3- q^{-1} +2 q^{-3} +2 q^{-5} +4 q^{-7} +4 q^{-9} +2 q^{-11} + q^{-13} -2 q^{-15} - q^{-17} -3 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}+5 q^{20}-5 q^{18}+2 q^{16}+6 q^{14}-16 q^{12}+24 q^{10}-21 q^8+7 q^6+7 q^4-39 q^2+35-47 q^{-2} +14 q^{-4} +4 q^{-6} -21 q^{-8} +47 q^{-10} -7 q^{-12} +34 q^{-14} +21 q^{-16} +16 q^{-18} +19 q^{-22} -26 q^{-24} -13 q^{-26} +7 q^{-28} -54 q^{-30} +29 q^{-32} -29 q^{-34} -2 q^{-36} +27 q^{-38} -32 q^{-40} +31 q^{-42} -7 q^{-44} -10 q^{-46} +21 q^{-48} -19 q^{-50} +5 q^{-52} +5 q^{-54} -9 q^{-56} +9 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^{12}+2 q^{10}-2 q^6-q^4-6 q^2-10-7 q^{-2} -4 q^{-4} -3 q^{-6} + q^{-8} +14 q^{-10} +17 q^{-12} +15 q^{-14} +16 q^{-16} +17 q^{-18} +2 q^{-20} -3 q^{-22} -4 q^{-24} -11 q^{-26} -14 q^{-28} -8 q^{-30} -4 q^{-32} -6 q^{-34} -2 q^{-36} +3 q^{-38} +2 q^{-40} -2 q^{-42} +2 q^{-44} +3 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^8-2 q^4-q^2-1- q^{-2} +2 q^{-4} +2 q^{-6} +5 q^{-8} +4 q^{-10} +5 q^{-12} +2 q^{-14} + q^{-16} -2 q^{-18} -2 q^{-20} -2 q^{-22} -3 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}-3 q^{10}+3 q^8-4 q^6+4 q^4-4 q^2+3- q^{-2} + q^{-4} +4 q^{-6} -3 q^{-8} +8 q^{-10} -6 q^{-12} +9 q^{-14} -7 q^{-16} +6 q^{-18} -4 q^{-20} +2 q^{-22} - q^{-24} -2 q^{-26} +3 q^{-28} -5 q^{-30} +4 q^{-32} -4 q^{-34} +3 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}+2 q^{16}+2 q^{14}-2 q^{12}-4 q^{10}-q^8+3 q^6+q^4-5 q^2-5+ q^{-2} +4 q^{-4} +2 q^{-6} -3 q^{-8} + q^{-10} +5 q^{-12} +6 q^{-14} + q^{-16} +2 q^{-18} +4 q^{-20} +5 q^{-22} -3 q^{-26} - q^{-28} +2 q^{-30} - q^{-32} -5 q^{-34} -4 q^{-36} + q^{-38} +2 q^{-40} -3 q^{-42} -5 q^{-44} +5 q^{-48} + q^{-50} -3 q^{-52} -3 q^{-54} + q^{-56} +3 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}+2 q^{14}-2 q^{12}+3 q^{10}-4 q^8+q^6-6 q^4-6-2 q^{-2} -2 q^{-4} + q^{-6} +5 q^{-8} +3 q^{-10} +13 q^{-12} +5 q^{-14} +14 q^{-16} +9 q^{-20} -5 q^{-22} +3 q^{-24} -9 q^{-26} -3 q^{-28} -7 q^{-30} -2 q^{-32} -2 q^{-34} -3 q^{-36} + q^{-38} -3 q^{-40} +5 q^{-42} -3 q^{-44} +2 q^{-46} -3 q^{-48} +3 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}+3 q^{28}-4 q^{26}+2 q^{24}-q^{22}-5 q^{20}+9 q^{18}-12 q^{16}+10 q^{14}-6 q^{12}-5 q^{10}+12 q^8-16 q^6+14 q^4-9 q^2-2+9 q^{-2} -13 q^{-4} +9 q^{-6} - q^{-8} -4 q^{-10} +12 q^{-12} -7 q^{-14} +3 q^{-16} +7 q^{-18} -10 q^{-20} +19 q^{-22} -14 q^{-24} +9 q^{-26} +7 q^{-28} -12 q^{-30} +23 q^{-32} -19 q^{-34} +14 q^{-36} - q^{-38} -7 q^{-40} +13 q^{-42} -17 q^{-44} +11 q^{-46} - q^{-48} -7 q^{-50} +8 q^{-52} -8 q^{-54} -2 q^{-56} +8 q^{-58} -14 q^{-60} +9 q^{-62} -6 q^{-64} -3 q^{-66} +9 q^{-68} -14 q^{-70} +14 q^{-72} -8 q^{-74} +2 q^{-76} +2 q^{-78} -7 q^{-80} +6 q^{-82} -6 q^{-84} +5 q^{-86} -2 q^{-88} + q^{-92} -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 10"];
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+6 t-7+6 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+3 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]= { 27, 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^5-4 q^4+5 q^3-4 q^2+5 q-3+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+9 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+6 a^{-2} -3 a^{-4} -2}
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} +5 z^6 a^{-2} +3 z^6 a^{-4} +2 z^6+a z^5+z^5 a^{-1} +3 z^5 a^{-3} +3 z^5 a^{-5} -13 z^4 a^{-2} -5 z^4 a^{-4} +2 z^4 a^{-6} -6 z^4-3 a z^3-8 z^3 a^{-1} -9 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +5 z^2+2 a z+5 z a^{-1} +6 z a^{-3} +2 z a^{-5} -z a^{-7} -6 a^{-2} -3 a^{-4} -2}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_143, K11n106,}

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 10"];
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+6 t-7+6 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^5-4 q^4+5 q^3-4 q^2+5 q-3+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]= {10_143, K11n106,}
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: (3, 3)
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 12} 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 24} 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 72} 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 110} 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 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 288} 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 432} 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 64} 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 56} 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 288} 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 288} 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 1320} 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 120} 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{19711}{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 \frac{618}{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{9062}{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{161}{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 \frac{511}{10}}

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 10. 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      31 -2
7     21  1
5    23   1
3   32    1
1  13     2
-1 12      -1
-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}^{2}\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^{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}^{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=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}^{3}\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^{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}^{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=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}+q^{15}+4 q^{14}-9 q^{13}+3 q^{12}+12 q^{11}-19 q^{10}+3 q^9+20 q^8-24 q^7+2 q^6+23 q^5-21 q^4-2 q^3+21 q^2-14 q-5+14 q^{-1} -6 q^{-2} -5 q^{-3} +6 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^{31}-q^{30}+5 q^{28}-3 q^{27}-8 q^{26}+5 q^{25}+17 q^{24}-10 q^{23}-25 q^{22}+8 q^{21}+40 q^{20}-10 q^{19}-51 q^{18}+8 q^{17}+59 q^{16}-2 q^{15}-69 q^{14}+q^{13}+66 q^{12}+10 q^{11}-71 q^{10}-8 q^9+59 q^8+21 q^7-58 q^6-20 q^5+44 q^4+31 q^3-39 q^2-28 q+25+31 q^{-1} -16 q^{-2} -28 q^{-3} +7 q^{-4} +22 q^{-5} -16 q^{-7} -3 q^{-8} +9 q^{-9} +4 q^{-10} -5 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^{52}+q^{51}-3 q^{50}+4 q^{49}-5 q^{48}+5 q^{47}+3 q^{46}-13 q^{45}+7 q^{44}-9 q^{43}+22 q^{42}+15 q^{41}-39 q^{40}-8 q^{39}-22 q^{38}+64 q^{37}+55 q^{36}-68 q^{35}-48 q^{34}-67 q^{33}+111 q^{32}+127 q^{31}-75 q^{30}-93 q^{29}-136 q^{28}+136 q^{27}+195 q^{26}-56 q^{25}-111 q^{24}-195 q^{23}+127 q^{22}+228 q^{21}-28 q^{20}-100 q^{19}-222 q^{18}+100 q^{17}+220 q^{16}-2 q^{15}-67 q^{14}-224 q^{13}+64 q^{12}+187 q^{11}+24 q^{10}-23 q^9-206 q^8+17 q^7+136 q^6+50 q^5+28 q^4-171 q^3-30 q^2+74 q+59+69 q^{-1} -112 q^{-2} -53 q^{-3} +11 q^{-4} +39 q^{-5} +82 q^{-6} -47 q^{-7} -40 q^{-8} -23 q^{-9} +6 q^{-10} +59 q^{-11} -6 q^{-12} -12 q^{-13} -21 q^{-14} -11 q^{-15} +25 q^{-16} +3 q^{-17} +2 q^{-18} -7 q^{-19} -8 q^{-20} +6 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^{78}-q^{77}+3 q^{76}-q^{75}-4 q^{74}+3 q^{73}-q^{71}+8 q^{70}-14 q^{68}-5 q^{67}-q^{66}+11 q^{65}+29 q^{64}+12 q^{63}-29 q^{62}-53 q^{61}-34 q^{60}+28 q^{59}+103 q^{58}+84 q^{57}-30 q^{56}-150 q^{55}-163 q^{54}-4 q^{53}+217 q^{52}+261 q^{51}+52 q^{50}-250 q^{49}-376 q^{48}-150 q^{47}+287 q^{46}+487 q^{45}+243 q^{44}-273 q^{43}-583 q^{42}-354 q^{41}+244 q^{40}+652 q^{39}+453 q^{38}-208 q^{37}-683 q^{36}-518 q^{35}+140 q^{34}+694 q^{33}+584 q^{32}-112 q^{31}-676 q^{30}-584 q^{29}+39 q^{28}+647 q^{27}+617 q^{26}-31 q^{25}-604 q^{24}-576 q^{23}-39 q^{22}+552 q^{21}+590 q^{20}+45 q^{19}-490 q^{18}-534 q^{17}-123 q^{16}+419 q^{15}+536 q^{14}+137 q^{13}-331 q^{12}-468 q^{11}-215 q^{10}+236 q^9+443 q^8+235 q^7-137 q^6-348 q^5-283 q^4+30 q^3+288 q^2+274 q+55-179 q^{-1} -259 q^{-2} -126 q^{-3} +92 q^{-4} +209 q^{-5} +156 q^{-6} -6 q^{-7} -144 q^{-8} -158 q^{-9} -55 q^{-10} +78 q^{-11} +132 q^{-12} +81 q^{-13} -17 q^{-14} -92 q^{-15} -84 q^{-16} -18 q^{-17} +48 q^{-18} +66 q^{-19} +37 q^{-20} -18 q^{-21} -44 q^{-22} -30 q^{-23} -4 q^{-24} +20 q^{-25} +27 q^{-26} +8 q^{-27} -11 q^{-28} -11 q^{-29} -7 q^{-30} -2 q^{-31} +9 q^{-32} +6 q^{-33} -2 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^{109}+q^{108}-3 q^{107}+q^{106}+q^{105}+6 q^{104}-8 q^{103}-2 q^{102}+6 q^{101}-9 q^{100}+4 q^{99}+9 q^{98}+17 q^{97}-20 q^{96}-17 q^{95}+4 q^{94}-25 q^{93}+14 q^{92}+44 q^{91}+63 q^{90}-30 q^{89}-60 q^{88}-44 q^{87}-106 q^{86}+13 q^{85}+138 q^{84}+228 q^{83}+54 q^{82}-106 q^{81}-200 q^{80}-385 q^{79}-128 q^{78}+254 q^{77}+596 q^{76}+403 q^{75}+23 q^{74}-401 q^{73}-946 q^{72}-607 q^{71}+171 q^{70}+1060 q^{69}+1066 q^{68}+522 q^{67}-390 q^{66}-1601 q^{65}-1407 q^{64}-293 q^{63}+1306 q^{62}+1782 q^{61}+1311 q^{60}-12 q^{59}-1998 q^{58}-2199 q^{57}-998 q^{56}+1186 q^{55}+2193 q^{54}+2031 q^{53}+560 q^{52}-2005 q^{51}-2646 q^{50}-1607 q^{49}+862 q^{48}+2225 q^{47}+2411 q^{46}+1026 q^{45}-1785 q^{44}-2715 q^{43}-1919 q^{42}+568 q^{41}+2039 q^{40}+2470 q^{39}+1259 q^{38}-1529 q^{37}-2570 q^{36}-1982 q^{35}+369 q^{34}+1780 q^{33}+2364 q^{32}+1346 q^{31}-1263 q^{30}-2334 q^{29}-1942 q^{28}+172 q^{27}+1454 q^{26}+2191 q^{25}+1420 q^{24}-901 q^{23}-2001 q^{22}-1882 q^{21}-122 q^{20}+997 q^{19}+1934 q^{18}+1520 q^{17}-391 q^{16}-1511 q^{15}-1750 q^{14}-488 q^{13}+391 q^{12}+1502 q^{11}+1535 q^{10}+193 q^9-844 q^8-1418 q^7-755 q^6-260 q^5+865 q^4+1296 q^3+624 q^2-127 q-840-720 q^{-1} -704 q^{-2} +171 q^{-3} +767 q^{-4} +680 q^{-5} +361 q^{-6} -193 q^{-7} -361 q^{-8} -731 q^{-9} -278 q^{-10} +173 q^{-11} +381 q^{-12} +424 q^{-13} +205 q^{-14} +64 q^{-15} -419 q^{-16} -323 q^{-17} -161 q^{-18} +28 q^{-19} +190 q^{-20} +229 q^{-21} +249 q^{-22} -91 q^{-23} -132 q^{-24} -165 q^{-25} -109 q^{-26} -24 q^{-27} +78 q^{-28} +183 q^{-29} +32 q^{-30} +13 q^{-31} -52 q^{-32} -65 q^{-33} -68 q^{-34} -16 q^{-35} +68 q^{-36} +20 q^{-37} +32 q^{-38} +4 q^{-39} -9 q^{-40} -33 q^{-41} -21 q^{-42} +15 q^{-43} +12 q^{-45} +6 q^{-46} +5 q^{-47} -9 q^{-48} -8 q^{-49} +4 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^{145}-q^{144}+3 q^{143}-q^{142}-q^{141}-3 q^{140}-q^{139}+10 q^{138}-3 q^{137}-5 q^{136}+5 q^{135}-5 q^{134}-2 q^{133}-8 q^{132}+33 q^{130}+3 q^{129}-13 q^{128}-4 q^{127}-29 q^{126}-13 q^{125}-21 q^{124}+13 q^{123}+96 q^{122}+52 q^{121}+8 q^{120}-34 q^{119}-131 q^{118}-116 q^{117}-95 q^{116}+25 q^{115}+261 q^{114}+276 q^{113}+212 q^{112}+2 q^{111}-378 q^{110}-509 q^{109}-510 q^{108}-162 q^{107}+512 q^{106}+890 q^{105}+980 q^{104}+500 q^{103}-550 q^{102}-1333 q^{101}-1700 q^{100}-1131 q^{99}+415 q^{98}+1774 q^{97}+2613 q^{96}+2089 q^{95}+51 q^{94}-2106 q^{93}-3671 q^{92}-3343 q^{91}-825 q^{90}+2168 q^{89}+4631 q^{88}+4818 q^{87}+2013 q^{86}-1899 q^{85}-5485 q^{84}-6312 q^{83}-3356 q^{82}+1267 q^{81}+5938 q^{80}+7656 q^{79}+4877 q^{78}-363 q^{77}-6095 q^{76}-8724 q^{75}-6237 q^{74}-678 q^{73}+5881 q^{72}+9406 q^{71}+7380 q^{70}+1744 q^{69}-5432 q^{68}-9753 q^{67}-8236 q^{66}-2654 q^{65}+4922 q^{64}+9773 q^{63}+8701 q^{62}+3378 q^{61}-4308 q^{60}-9623 q^{59}-8990 q^{58}-3863 q^{57}+3885 q^{56}+9349 q^{55}+8943 q^{54}+4156 q^{53}-3406 q^{52}-9030 q^{51}-8916 q^{50}-4321 q^{49}+3157 q^{48}+8713 q^{47}+8661 q^{46}+4390 q^{45}-2770 q^{44}-8353 q^{43}-8548 q^{42}-4474 q^{41}+2528 q^{40}+7984 q^{39}+8242 q^{38}+4565 q^{37}-2034 q^{36}-7511 q^{35}-8097 q^{34}-4735 q^{33}+1607 q^{32}+6945 q^{31}+7724 q^{30}+4942 q^{29}-870 q^{28}-6211 q^{27}-7459 q^{26}-5180 q^{25}+177 q^{24}+5333 q^{23}+6882 q^{22}+5385 q^{21}+766 q^{20}-4265 q^{19}-6306 q^{18}-5507 q^{17}-1574 q^{16}+3085 q^{15}+5380 q^{14}+5419 q^{13}+2444 q^{12}-1778 q^{11}-4386 q^{10}-5126 q^9-3029 q^8+566 q^7+3117 q^6+4495 q^5+3415 q^4+586 q^3-1862 q^2-3669 q-3388-1398 q^{-1} +620 q^{-2} +2585 q^{-3} +3042 q^{-4} +1904 q^{-5} +400 q^{-6} -1494 q^{-7} -2398 q^{-8} -1972 q^{-9} -1098 q^{-10} +483 q^{-11} +1570 q^{-12} +1706 q^{-13} +1431 q^{-14} +283 q^{-15} -763 q^{-16} -1208 q^{-17} -1380 q^{-18} -724 q^{-19} +88 q^{-20} +618 q^{-21} +1094 q^{-22} +845 q^{-23} +334 q^{-24} -108 q^{-25} -678 q^{-26} -725 q^{-27} -504 q^{-28} -223 q^{-29} +290 q^{-30} +467 q^{-31} +459 q^{-32} +383 q^{-33} -4 q^{-34} -227 q^{-35} -323 q^{-36} -357 q^{-37} -118 q^{-38} +22 q^{-39} +144 q^{-40} +274 q^{-41} +166 q^{-42} +61 q^{-43} -44 q^{-44} -157 q^{-45} -106 q^{-46} -85 q^{-47} -43 q^{-48} +71 q^{-49} +76 q^{-50} +74 q^{-51} +38 q^{-52} -30 q^{-53} -20 q^{-54} -34 q^{-55} -45 q^{-56} -4 q^{-57} +10 q^{-58} +26 q^{-59} +24 q^{-60} -5 q^{-61} +3 q^{-62} -2 q^{-63} -14 q^{-64} -6 q^{-65} -4 q^{-66} +6 q^{-67} +8 q^{-68} -2 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.

8 9.gif

8_9

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8_11

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