9 2
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X3,12,4,13 X5,18,6,1 X7,16,8,17 X9,14,10,15 X13,10,14,11 X15,8,16,9 X17,6,18,7 X11,2,12,3 |
| Gauss code | -1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3 |
| Dowker-Thistlethwaite code | 4 12 18 16 14 2 10 8 6 |
| Conway Notation | [72] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 |
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 [{11, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 10}, {9, 11}, {10, 1}] |
[edit Notes on presentations of 9 2]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 2"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,12,4,13 X5,18,6,1 X7,16,8,17 X9,14,10,15 X13,10,14,11 X15,8,16,9 X17,6,18,7 X11,2,12,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 9, -2, 1, -3, 8, -4, 7, -5, 6, -9, 2, -6, 5, -7, 4, -8, 3 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 18 16 14 2 10 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[72] |
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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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[math]\displaystyle{ \textrm{BR}(5,\{-1,-1,-1,-2,1,-2,-3,2,-3,-4,3,-4\}) }[/math] |
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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{ 5, 12, 5 } |
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Show[BraidPlot[br]]
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-Graphics- |
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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[{11, 8}, {7, 9}, {8, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 10}, {9, 11}, {10, 1}] |
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 | [math]\displaystyle{ 4 t-7+4 t^{-1} }[/math] |
| Conway polynomial | [math]\displaystyle{ 4 z^2+1 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 15, -2 } |
| Jones polynomial | [math]\displaystyle{ q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ -a^{10}+z^2 a^8+a^8+z^2 a^6+z^2 a^4+z^2 a^2+a^2 }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-4 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+11 z^4 a^{10}-7 z^2 a^{10}+a^{10}+2 z^7 a^9-10 z^5 a^9+13 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+8 z^4 a^8-6 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z^6 a^6-2 z^4 a^6+z^5 a^5-z^3 a^5+z^4 a^4+z^3 a^3+z^2 a^2-a^2 }[/math] |
| The A2 invariant | [math]\displaystyle{ -q^{32}-q^{30}+q^{24}+q^{22}+q^8+q^6+q^2 }[/math] |
| The G2 invariant | [math]\displaystyle{ q^{156}+q^{152}-q^{150}+q^{142}-2 q^{140}+q^{138}-q^{136}-q^{134}-2 q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2 q^{102}+q^{98}+q^{94}+q^{92}-2 q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 9_2.
A1 Invariants.
| Weight | Invariant |
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| 1 | [math]\displaystyle{ -q^{21}+q^{15}+q^5+q }[/math] |
| 2 | [math]\displaystyle{ q^{60}-q^{56}-q^{50}+q^{46}-q^{30}-q^{28}+q^{18}+q^{16}+q^{14}+q^8+q^2 }[/math] |
| 3 | [math]\displaystyle{ -q^{117}+q^{113}+q^{111}-q^{107}+q^{103}-q^{99}-q^{97}+q^{93}+q^{73}+q^{71}-q^{67}-q^{61}-q^{59}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{39}+q^{37}-q^{33}-q^{31}+q^{29}+2 q^{27}-q^{23}+q^{21}+2 q^{19}+q^{17}+q^{11}+q^3 }[/math] |
| 4 | [math]\displaystyle{ q^{192}-q^{188}-q^{186}-q^{184}+q^{182}+q^{180}+q^{178}-2 q^{174}+q^{170}+q^{168}+q^{166}-q^{164}-q^{162}-q^{160}+q^{156}-q^{134}-q^{132}+q^{128}+2 q^{126}-q^{122}+q^{118}+2 q^{116}-2 q^{112}-q^{110}+q^{106}-2 q^{102}-q^{100}+q^{96}+q^{94}+q^{90}+q^{88}+q^{86}-q^{82}-q^{80}+q^{76}-q^{72}-q^{70}-q^{68}-q^{66}+q^{62}-q^{60}-2 q^{58}-2 q^{56}+2 q^{52}+q^{50}-2 q^{46}+q^{44}+2 q^{42}+q^{40}-q^{38}-2 q^{36}+q^{34}+2 q^{32}+q^{30}-q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{14}+q^4 }[/math] |
| 5 | [math]\displaystyle{ -q^{285}+q^{281}+q^{279}+q^{277}-q^{273}-2 q^{271}-q^{269}+q^{265}+2 q^{263}+q^{261}-q^{259}-2 q^{257}-q^{255}+q^{251}+2 q^{249}+q^{247}-q^{243}-q^{241}-q^{239}+q^{235}+q^{213}+q^{211}-q^{207}-2 q^{205}-2 q^{203}+2 q^{199}+2 q^{197}-2 q^{193}-2 q^{191}-q^{189}+2 q^{187}+4 q^{185}+2 q^{183}-q^{181}-2 q^{179}-2 q^{177}+2 q^{173}+2 q^{171}-2 q^{167}-2 q^{165}-q^{163}+q^{159}+q^{157}-q^{155}-q^{153}-q^{151}+q^{147}+2 q^{145}+q^{143}-q^{139}-q^{137}+q^{135}+2 q^{133}+q^{131}-q^{127}+q^{123}-q^{119}-2 q^{117}-q^{115}+q^{113}+2 q^{111}+q^{109}-q^{105}-q^{103}-2 q^{101}+q^{97}+2 q^{95}+2 q^{93}+q^{91}-2 q^{89}-3 q^{87}-2 q^{85}+q^{81}+q^{79}-q^{77}-2 q^{75}-3 q^{73}-q^{71}+q^{69}+q^{67}-q^{63}+q^{57}-q^{53}+2 q^{49}+3 q^{47}+q^{45}-q^{43}-q^{41}-q^{39}+q^{37}+2 q^{35}+q^{33}+q^{23}+q^{21}+q^{19}+q^{17}+q^5 }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ -q^{32}-q^{30}+q^{24}+q^{22}+q^8+q^6+q^2 }[/math] |
| 1,1 | [math]\displaystyle{ q^{84}+2 q^{80}-2 q^{78}+2 q^{76}-4 q^{74}+2 q^{72}-4 q^{70}+4 q^{62}-q^{60}+4 q^{58}-4 q^{56}+2 q^{54}-4 q^{52}+2 q^{50}-2 q^{48}+2 q^{46}-2 q^{42}-2 q^{40}-2 q^{38}-2 q^{36}+2 q^{30}+2 q^{28}+2 q^{26}+2 q^{24}+q^{20}+2 q^{16}+2 q^{12}+2 q^8+q^4 }[/math] |
| 2,0 | [math]\displaystyle{ q^{82}+q^{80}+q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}+q^{64}+q^{62}+q^{60}-q^{44}-2 q^{42}-2 q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+2 q^{26}+q^{24}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^4 }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ q^{66}+q^{62}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{46}-q^{42}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{16}+q^{12}+2 q^{10}+q^8+q^4 }[/math] |
| 1,0,0 | [math]\displaystyle{ -q^{43}-q^{41}-q^{39}+q^{33}+q^{31}+q^{29}+q^{11}+q^9+q^7+q^3 }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ -q^{66}-q^{62}-q^{58}+q^{56}-q^{54}+q^{52}+q^{50}+q^{46}+q^{42}-2 q^{40}+q^{38}-q^{36}+q^{34}-q^{32}+q^{30}+q^{16}+q^{12}+q^8+q^4 }[/math] |
| 1,0 | [math]\displaystyle{ q^{108}+q^{100}-q^{96}-q^{94}-q^{88}-q^{86}+q^{82}-q^{76}-q^{68}-q^{66}+q^{56}+q^{54}+q^{48}+q^{46}+q^{26}+q^{18}+q^{16}+q^{14}+q^6 }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{156}+q^{152}-q^{150}+q^{142}-2 q^{140}+q^{138}-q^{136}-q^{134}-2 q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2 q^{102}+q^{98}+q^{94}+q^{92}-2 q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/math] |
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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["9 2"];
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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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[math]\displaystyle{ 4 t-7+4 t^{-1} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ 4 z^2+1 }[/math] |
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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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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{ 15, -2 } |
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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[math]\displaystyle{ q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }[/math] |
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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[math]\displaystyle{ -a^{10}+z^2 a^8+a^8+z^2 a^6+z^2 a^4+z^2 a^2+a^2 }[/math] |
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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[math]\displaystyle{ z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-4 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+11 z^4 a^{10}-7 z^2 a^{10}+a^{10}+2 z^7 a^9-10 z^5 a^9+13 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+8 z^4 a^8-6 z^2 a^8+a^8+z^7 a^7-3 z^5 a^7+z^3 a^7+z^6 a^6-2 z^4 a^6+z^5 a^5-z^3 a^5+z^4 a^4+z^3 a^3+z^2 a^2-a^2 }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {7_4,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {K11n13,}
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["9 2"];
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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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{ [math]\displaystyle{ 4 t-7+4 t^{-1} }[/math], [math]\displaystyle{ q^{-1} - q^{-2} +2 q^{-3} -2 q^{-4} +2 q^{-5} -2 q^{-6} +2 q^{-7} - q^{-8} + q^{-9} - q^{-10} }[/math] } |
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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{7_4,} |
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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{K11n13,} |
Vassiliev invariants
| V2 and V3: | (4, -10) |
| 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of 9 2. 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 [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ J_n }[/math] |
| 2 | [math]\displaystyle{ q^{-2} - q^{-3} +2 q^{-5} -2 q^{-6} +3 q^{-8} -2 q^{-9} +2 q^{-11} -2 q^{-12} +2 q^{-14} -3 q^{-15} +3 q^{-17} -3 q^{-18} +3 q^{-20} -3 q^{-21} +3 q^{-23} -2 q^{-24} - q^{-25} +2 q^{-26} - q^{-27} - q^{-28} + q^{-29} }[/math] |
| 3 | [math]\displaystyle{ q^{-3} - q^{-4} +2 q^{-7} -2 q^{-8} + q^{-10} +3 q^{-11} -3 q^{-12} -2 q^{-13} +2 q^{-14} +5 q^{-15} -4 q^{-16} -4 q^{-17} +2 q^{-18} +6 q^{-19} -3 q^{-20} -6 q^{-21} +2 q^{-22} +6 q^{-23} -2 q^{-24} -5 q^{-25} +2 q^{-26} +5 q^{-27} -3 q^{-28} -4 q^{-29} +2 q^{-30} +4 q^{-31} -3 q^{-32} -3 q^{-33} +2 q^{-34} +3 q^{-35} -2 q^{-36} -2 q^{-37} +2 q^{-38} +2 q^{-39} -2 q^{-40} -2 q^{-41} +2 q^{-42} +2 q^{-43} -2 q^{-44} -2 q^{-45} +2 q^{-46} +2 q^{-47} - q^{-48} -3 q^{-49} + q^{-50} +2 q^{-51} -2 q^{-53} + q^{-55} + q^{-56} - q^{-57} }[/math] |
| 4 | [math]\displaystyle{ q^{-4} - q^{-5} +2 q^{-9} -2 q^{-10} + q^{-11} +2 q^{-14} -4 q^{-15} +2 q^{-16} + q^{-17} + q^{-18} + q^{-19} -7 q^{-20} +3 q^{-21} +3 q^{-22} +2 q^{-23} -10 q^{-25} +5 q^{-26} +4 q^{-27} +3 q^{-28} -2 q^{-29} -12 q^{-30} +5 q^{-31} +5 q^{-32} +5 q^{-33} -3 q^{-34} -13 q^{-35} +5 q^{-36} +5 q^{-37} +5 q^{-38} -2 q^{-39} -12 q^{-40} +4 q^{-41} +4 q^{-42} +5 q^{-43} - q^{-44} -11 q^{-45} +4 q^{-46} +4 q^{-47} +4 q^{-48} -11 q^{-50} +3 q^{-51} +3 q^{-52} +3 q^{-53} +2 q^{-54} -10 q^{-55} +2 q^{-56} +2 q^{-57} +2 q^{-58} +4 q^{-59} -8 q^{-60} + q^{-61} + q^{-62} + q^{-63} +5 q^{-64} -6 q^{-65} +5 q^{-69} -5 q^{-70} +5 q^{-74} -5 q^{-75} +5 q^{-79} -4 q^{-80} - q^{-81} - q^{-82} +5 q^{-84} -2 q^{-85} - q^{-86} - q^{-87} - q^{-88} +3 q^{-89} - q^{-92} - q^{-93} + q^{-94} }[/math] |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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