8 1
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 8 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7 |
| Gauss code | -1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5 |
| Dowker-Thistlethwaite code | 4 10 16 14 12 2 8 6 |
| Conway Notation | [62] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 10, width is 5, Braid index is 5 |
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 [{10, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}, {8, 10}, {1, 9}] |
[edit Notes on presentations of 8 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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["8 1"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 16 14 12 2 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[62] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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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}(5,\{-1,-1,-2,1,-2,-3,2,4,-3,4\})} |
In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 5, 10, 5 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{10, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}, {8, 10}, {1, 9}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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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 -3 t+7-3 t^{-1} } |
| 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 1-3 z^2} |
| 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 | { 13, 0 } |
| 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^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} } |
| 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 a^6-z^2 a^4-a^4-z^2 a^2-z^2+ a^{-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 a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-4 a^3 z^5+a z^5-5 a^6 z^4-8 a^4 z^4-2 a^2 z^4+z^4+7 a^5 z^3+5 a^3 z^3-a z^3+z^3 a^{-1} +6 a^6 z^2+7 a^4 z^2+z^2 a^{-2} -3 a^5 z-3 a^3 z-a^6-a^4- a^{-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^{20}+q^{18}-q^{12}-q^{10}+ q^{-2} + q^{-6} + q^{-8} } |
| 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^{94}+q^{90}-q^{88}+2 q^{80}-2 q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2 q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^8+ q^{-2} - q^{-6} + q^{-10} + q^{-14} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} } |
Further Quantum Invariants
Further quantum knot invariants for 8_1.
A1 Invariants.
| Weight | Invariant |
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| 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^{13}-q^7+ q^{-1} + q^{-5} } |
| 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^{38}-q^{34}-q^{28}+q^{24}+q^{12}+q^{10}-1+ q^{-8} + q^{-14} } |
| 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^{75}-q^{71}-q^{69}+q^{65}-q^{61}+q^{57}+q^{55}-q^{51}+q^{37}+q^{35}-q^{31}-q^{25}-q^{23}-q^{17}+q^{13}+2 q^{11}+q^9+q^7-q^5+q+ q^{-1} - q^{-5} + q^{-9} - q^{-13} - q^{-15} + q^{-19} + q^{-27} } |
| 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^{124}-q^{120}-q^{118}-q^{116}+q^{114}+q^{112}+q^{110}-2 q^{106}+q^{102}+q^{100}+q^{98}-q^{96}-q^{94}-q^{92}+q^{88}+q^{74}+q^{72}-q^{68}-2 q^{66}+q^{62}-q^{58}-2 q^{56}+2 q^{52}+q^{50}-q^{46}+2 q^{42}+q^{40}+q^{32}-q^{28}-q^{26}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}+q^{10}+2 q^8+q^6-q^4-2 q^2+2 q^{-2} +4 q^{-4} + q^{-6} -2 q^{-8} - q^{-10} + q^{-12} +3 q^{-14} -2 q^{-18} - q^{-20} +2 q^{-24} - q^{-28} - q^{-30} - q^{-32} + q^{-34} + q^{-44} } |
| 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^{185}-q^{181}-q^{179}-q^{177}+q^{173}+2 q^{171}+q^{169}-q^{165}-2 q^{163}-q^{161}+q^{159}+2 q^{157}+q^{155}-q^{151}-2 q^{149}-q^{147}+q^{143}+q^{141}+q^{139}-q^{135}+q^{123}+q^{121}-q^{117}-2 q^{115}-2 q^{113}+2 q^{109}+2 q^{107}-2 q^{103}-2 q^{101}-q^{99}+2 q^{97}+4 q^{95}+2 q^{93}-q^{91}-2 q^{89}-2 q^{87}+2 q^{83}+2 q^{81}-2 q^{77}-2 q^{75}+q^{71}+2 q^{69}+q^{67}-q^{65}-2 q^{63}-q^{61}+q^{57}+q^{55}-q^{53}-2 q^{51}-q^{49}+q^{45}-q^{41}+q^{37}+3 q^{35}+2 q^{33}-q^{25}+q^{23}+q^{21}+q^{19}+q^{17}-3 q^{13}-3 q^{11}-q^9+q^7+3 q^5+2 q^3-3 q^{-1} -3 q^{-3} +3 q^{-7} +3 q^{-9} -2 q^{-13} -2 q^{-15} +3 q^{-19} +2 q^{-21} -2 q^{-25} +2 q^{-29} +2 q^{-31} + q^{-33} - q^{-35} -2 q^{-37} - q^{-39} + q^{-41} + q^{-43} - q^{-49} - q^{-51} + q^{-65} } |
| 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^{258}-q^{254}-q^{252}-q^{250}+2 q^{244}+2 q^{242}+q^{240}-q^{236}-2 q^{234}-3 q^{232}+q^{228}+2 q^{226}+2 q^{224}+q^{222}-3 q^{218}-2 q^{216}-q^{214}+q^{210}+2 q^{208}+2 q^{206}-q^{200}-q^{198}-q^{196}+q^{192}+q^{184}+q^{182}-q^{178}-2 q^{176}-2 q^{174}-2 q^{172}+q^{170}+3 q^{168}+3 q^{166}+2 q^{164}-q^{162}-3 q^{160}-4 q^{158}-q^{156}+2 q^{154}+4 q^{152}+5 q^{150}+2 q^{148}-2 q^{146}-5 q^{144}-4 q^{142}-2 q^{140}+q^{138}+4 q^{136}+3 q^{134}+q^{132}-2 q^{130}-3 q^{128}-3 q^{126}-q^{124}+3 q^{122}+3 q^{120}+3 q^{118}+q^{116}-q^{114}-3 q^{112}-4 q^{110}-q^{108}+q^{106}+2 q^{104}+2 q^{102}+q^{100}-2 q^{98}-4 q^{96}-2 q^{94}+q^{92}+3 q^{90}+3 q^{88}+2 q^{86}-q^{84}-4 q^{82}-q^{80}+2 q^{78}+3 q^{76}+3 q^{74}+2 q^{72}-q^{70}-3 q^{68}-q^{66}+q^{64}+q^{62}-q^{58}-2 q^{56}-2 q^{54}+q^{50}-q^{46}-2 q^{44}-2 q^{42}-q^{40}+q^{38}+2 q^{36}+3 q^{34}+2 q^{32}+q^{30}-q^{26}-3 q^{24}-3 q^{22}+3 q^{18}+4 q^{16}+4 q^{14}+3 q^{12}-2 q^{10}-4 q^8-4 q^6-q^4+2 q^2+5+6 q^{-2} + q^{-4} -2 q^{-6} -5 q^{-8} -4 q^{-10} -2 q^{-12} +2 q^{-14} +4 q^{-16} + q^{-18} -2 q^{-22} - q^{-24} - q^{-26} + q^{-30} - q^{-32} + q^{-34} + q^{-36} +2 q^{-38} - q^{-42} - q^{-44} -2 q^{-46} + q^{-48} +2 q^{-50} +3 q^{-52} + q^{-54} - q^{-58} -2 q^{-60} + q^{-66} - q^{-74} - q^{-78} + q^{-90} } |
A2 Invariants.
| Weight | Invariant |
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| 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^{20}+q^{18}-q^{12}-q^{10}+ q^{-2} + q^{-6} + q^{-8} } |
| 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^{52}+2 q^{48}-2 q^{46}+2 q^{44}-4 q^{42}+2 q^{40}-2 q^{38}-2 q^{32}+2 q^{30}-3 q^{28}+4 q^{26}-2 q^{24}+4 q^{22}-2 q^{20}+4 q^{18}+2 q^{14}+2 q^{12}-2 q^6-2 q^4-2 q^2-2+ q^{-4} +2 q^{-8} +2 q^{-12} +2 q^{-16} + q^{-20} } |
| 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^{52}+q^{50}+q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{18}+2 q^{16}+q^{14}+q^{12}+q^{10}-q^4-2 q^2-2- q^{-2} + q^{-4} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} } |
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^{40}+q^{36}-q^{32}-q^{30}-q^{28}+q^{24}+2 q^{22}+q^{20}+2 q^{18}-q^{14}-q^{12}-q^{10}-q^8-q^6+ q^{-4} + q^{-8} +2 q^{-10} + q^{-12} + q^{-16} } |
| 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^{27}+q^{25}+q^{23}-q^{17}-q^{15}-q^{13}+ q^{-3} + q^{-7} + q^{-9} + q^{-11} } |
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^{40}+q^{36}+q^{32}-q^{30}+q^{28}-q^{24}-q^{20}-2 q^{16}+q^{14}-q^{12}+q^{10}-q^8+q^6+ q^{-4} + q^{-8} + q^{-12} + q^{-16} } |
| 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^{66}+q^{58}-q^{54}-q^{52}-q^{46}-q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+q^{30}+q^{28}-q^{18}-q^{16}-q^{10}-q^8+ q^{-6} + q^{-14} + q^{-16} + q^{-18} + q^{-26} } |
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^{94}+q^{90}-q^{88}+2 q^{80}-2 q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2 q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^8+ q^{-2} - q^{-6} + q^{-10} + q^{-14} + q^{-20} + q^{-24} + q^{-28} + q^{-34} + q^{-38} } |
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["8 1"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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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 t+7-3 t^{-1} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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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 1-3 z^2} |
In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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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]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 13, 0 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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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^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} } |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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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 a^6-z^2 a^4-a^4-z^2 a^2-z^2+ a^{-2} } |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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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 a^5 z^7+a^3 z^7+a^6 z^6+2 a^4 z^6+a^2 z^6-5 a^5 z^5-4 a^3 z^5+a z^5-5 a^6 z^4-8 a^4 z^4-2 a^2 z^4+z^4+7 a^5 z^3+5 a^3 z^3-a z^3+z^3 a^{-1} +6 a^6 z^2+7 a^4 z^2+z^2 a^{-2} -3 a^5 z-3 a^3 z-a^6-a^4- a^{-2} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
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}} ): {K11n70,}
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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["8 1"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ 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 t+7-3 t^{-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^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} } } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{K11n70,} |
Vassiliev invariants
| V2 and V3: | (-3, 3) |
| V2,1 through V6,9: |
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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=} 0 is the signature of 8 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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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^6-q^5+2 q^3-2 q^2+2-3 q^{-1} + q^{-2} +2 q^{-3} -3 q^{-4} + q^{-5} +3 q^{-6} -3 q^{-7} +3 q^{-9} -3 q^{-10} +3 q^{-12} -2 q^{-13} - q^{-14} +2 q^{-15} - q^{-16} - q^{-17} + q^{-18} } |
| 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^{12}-q^{11}+2 q^8-2 q^7-q^6+3 q^4-q^3-2 q^2-q+4-2 q^{-2} -2 q^{-3} +3 q^{-4} +2 q^{-5} -2 q^{-6} - q^{-7} +2 q^{-8} + q^{-9} -3 q^{-10} +2 q^{-12} -3 q^{-14} + q^{-15} +2 q^{-16} - q^{-17} -2 q^{-18} +2 q^{-19} +2 q^{-20} -2 q^{-21} -2 q^{-22} +2 q^{-23} +2 q^{-24} -2 q^{-25} -2 q^{-26} + q^{-27} +3 q^{-28} - q^{-29} -2 q^{-30} +2 q^{-32} - q^{-34} - q^{-35} + q^{-36} } |
| 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^{20}-q^{19}+2 q^{15}-3 q^{14}+q^{11}+4 q^{10}-5 q^9-q^8-q^7+3 q^6+7 q^5-7 q^4-3 q^3-2 q^2+6 q+10-9 q^{-1} -5 q^{-2} -4 q^{-3} +7 q^{-4} +12 q^{-5} -8 q^{-6} -6 q^{-7} -6 q^{-8} +7 q^{-9} +12 q^{-10} -8 q^{-11} -5 q^{-12} -5 q^{-13} +6 q^{-14} +11 q^{-15} -8 q^{-16} -4 q^{-17} -4 q^{-18} +5 q^{-19} +11 q^{-20} -8 q^{-21} -3 q^{-22} -3 q^{-23} +3 q^{-24} +10 q^{-25} -7 q^{-26} -2 q^{-27} -2 q^{-28} + q^{-29} +8 q^{-30} -6 q^{-31} - q^{-32} - q^{-33} +6 q^{-35} -5 q^{-36} +5 q^{-40} -5 q^{-41} +5 q^{-45} -4 q^{-46} - q^{-47} - q^{-48} +5 q^{-50} -2 q^{-51} - q^{-52} - q^{-53} - q^{-54} +3 q^{-55} - q^{-58} - q^{-59} + q^{-60} } |
| 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^{30}-q^{29}+q^{24}-2 q^{23}+q^{21}+q^{19}+q^{18}-4 q^{17}-q^{16}+2 q^{15}+2 q^{14}+2 q^{13}+q^{12}-6 q^{11}-3 q^{10}+4 q^9+4 q^8+3 q^7-2 q^6-8 q^5-3 q^4+6 q^3+7 q^2+3 q-5-11 q^{-1} -3 q^{-2} +9 q^{-3} +9 q^{-4} +4 q^{-5} -7 q^{-6} -13 q^{-7} -5 q^{-8} +9 q^{-9} +12 q^{-10} +5 q^{-11} -7 q^{-12} -13 q^{-13} -5 q^{-14} +7 q^{-15} +13 q^{-16} +5 q^{-17} -7 q^{-18} -11 q^{-19} -4 q^{-20} +5 q^{-21} +12 q^{-22} +4 q^{-23} -6 q^{-24} -10 q^{-25} -5 q^{-26} +4 q^{-27} +11 q^{-28} +5 q^{-29} -4 q^{-30} -10 q^{-31} -6 q^{-32} +3 q^{-33} +10 q^{-34} +6 q^{-35} -2 q^{-36} -9 q^{-37} -7 q^{-38} +2 q^{-39} +8 q^{-40} +6 q^{-41} -7 q^{-43} -7 q^{-44} +6 q^{-46} +6 q^{-47} + q^{-48} -4 q^{-49} -5 q^{-50} -2 q^{-51} +3 q^{-52} +5 q^{-53} + q^{-54} -2 q^{-55} -3 q^{-56} -2 q^{-57} + q^{-58} +3 q^{-59} + q^{-60} - q^{-61} -2 q^{-62} - q^{-63} + q^{-64} +2 q^{-65} + q^{-66} - q^{-67} -2 q^{-68} - q^{-69} +3 q^{-71} +2 q^{-72} - q^{-73} -2 q^{-74} -2 q^{-75} - q^{-76} +2 q^{-77} +3 q^{-78} - q^{-80} - q^{-81} -2 q^{-82} +2 q^{-84} + q^{-85} - q^{-88} - q^{-89} + q^{-90} } |
| 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^{42}-q^{41}-q^{36}+2 q^{35}-2 q^{34}+q^{33}+q^{30}-2 q^{29}+2 q^{28}-4 q^{27}+2 q^{26}+q^{25}+q^{24}+3 q^{23}-3 q^{22}+q^{21}-7 q^{20}+3 q^{19}+q^{18}+2 q^{17}+5 q^{16}-4 q^{15}+q^{14}-9 q^{13}+5 q^{12}+q^{10}+5 q^9-5 q^8+3 q^7-8 q^6+8 q^5-2 q^4-3 q^3+3 q^2-6 q+6-5 q^{-1} +13 q^{-2} -3 q^{-3} -6 q^{-4} -9 q^{-6} +6 q^{-7} -3 q^{-8} +18 q^{-9} -2 q^{-10} -6 q^{-11} - q^{-12} -12 q^{-13} +3 q^{-14} -3 q^{-15} +20 q^{-16} - q^{-17} -5 q^{-18} -11 q^{-20} +2 q^{-21} -4 q^{-22} +18 q^{-23} -2 q^{-24} -5 q^{-25} + q^{-26} -10 q^{-27} +3 q^{-28} -5 q^{-29} +16 q^{-30} -2 q^{-31} -4 q^{-32} +2 q^{-33} -9 q^{-34} +3 q^{-35} -7 q^{-36} +14 q^{-37} - q^{-39} +3 q^{-40} -9 q^{-41} +2 q^{-42} -10 q^{-43} +11 q^{-44} +3 q^{-45} +2 q^{-46} +4 q^{-47} -9 q^{-48} -13 q^{-50} +9 q^{-51} +5 q^{-52} +5 q^{-53} +5 q^{-54} -9 q^{-55} - q^{-56} -15 q^{-57} +6 q^{-58} +6 q^{-59} +7 q^{-60} +7 q^{-61} -7 q^{-62} - q^{-63} -15 q^{-64} +2 q^{-65} +4 q^{-66} +7 q^{-67} +8 q^{-68} -4 q^{-69} + q^{-70} -14 q^{-71} - q^{-72} + q^{-73} +5 q^{-74} +7 q^{-75} - q^{-76} +5 q^{-77} -11 q^{-78} -2 q^{-79} - q^{-80} +2 q^{-81} +4 q^{-82} +7 q^{-84} -8 q^{-85} - q^{-86} - q^{-87} + q^{-89} +7 q^{-91} -7 q^{-92} +7 q^{-98} -6 q^{-99} - q^{-100} - q^{-101} + q^{-104} +7 q^{-105} -4 q^{-106} - q^{-107} -2 q^{-108} - q^{-109} - q^{-110} +6 q^{-112} - q^{-113} - q^{-115} - q^{-116} -2 q^{-117} - q^{-118} +3 q^{-119} + q^{-121} - q^{-124} - q^{-125} + q^{-126} } |
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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