9 18
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![]() (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 18's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3 |
| Gauss code | -1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 2 10 8 6 |
| Conway Notation | [3222] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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![]() [{11, 5}, {1, 9}, {8, 10}, {9, 11}, {10, 6}, {5, 7}, {4, 8}, {6, 3}, {2, 4}, {3, 1}, {7, 2}] |
[edit Notes on presentations of 9 18]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 18"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,12,4,13 X5,14,6,15 X9,18,10,1 X17,6,18,7 X7,16,8,17 X15,8,16,9 X13,10,14,11 X11,2,12,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 9, -2, 1, -3, 5, -6, 7, -4, 8, -9, 2, -8, 3, -7, 6, -5, 4 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 14 16 18 2 10 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[3222] |
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}(4,\{-1,-1,-1,-2,1,-2,-2,-2,-3,2,-3\})} |
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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{ 4, 11, 4 } |
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[{11, 5}, {1, 9}, {8, 10}, {9, 11}, {10, 6}, {5, 7}, {4, 8}, {6, 3}, {2, 4}, {3, 1}, {7, 2}] |
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
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | |
| 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^{92}-2 q^{90}+6 q^{88}-12 q^{86}+23 q^{84}-40 q^{82}+60 q^{80}-84 q^{78}+110 q^{76}-132 q^{74}+142 q^{72}-136 q^{70}+114 q^{68}-76 q^{66}+18 q^{64}+52 q^{62}-119 q^{60}+184 q^{58}-236 q^{56}+268 q^{54}-280 q^{52}+260 q^{50}-224 q^{48}+162 q^{46}-97 q^{44}+24 q^{42}+38 q^{40}-92 q^{38}+123 q^{36}-134 q^{34}+136 q^{32}-120 q^{30}+103 q^{28}-72 q^{26}+56 q^{24}-34 q^{22}+23 q^{20}-10 q^{18}+6 q^{16}-2 q^{14}+q^{12}} |
| 2,0 |
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^{74}-q^{72}+q^{70}+2 q^{68}-4 q^{66}+2 q^{64}+4 q^{62}-8 q^{60}+3 q^{58}+5 q^{56}-10 q^{54}+q^{52}+5 q^{50}-6 q^{48}-2 q^{46}+2 q^{44}-3 q^{40}-2 q^{38}+7 q^{36}-2 q^{34}-5 q^{32}+10 q^{30}-6 q^{26}+9 q^{24}+q^{22}-3 q^{20}+4 q^{18}+q^{16}-q^{14}+q^{12}} |
| 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^{45}-q^{41}-2 q^{37}+q^{35}-q^{33}+q^{31}+q^{27}+2 q^{21}+2 q^{17}+2 q^{13}-q^{11}+q^9} |
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^{96}-q^{92}+2 q^{90}+3 q^{88}-2 q^{86}-q^{84}+5 q^{82}+3 q^{80}-6 q^{78}-4 q^{76}+4 q^{74}-3 q^{72}-10 q^{70}+4 q^{66}-5 q^{64}-3 q^{62}+4 q^{60}-q^{58}-6 q^{56}+q^{54}+4 q^{52}-5 q^{50}-2 q^{48}+9 q^{46}+2 q^{44}-5 q^{42}+4 q^{40}+8 q^{38}-2 q^{34}+4 q^{32}+5 q^{30}-q^{28}+3 q^{24}+q^{22}-q^{20}+q^{18}} |
| 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^{56}-q^{52}-q^{50}-2 q^{46}+q^{44}-q^{42}+q^{38}+q^{34}+q^{30}+2 q^{26}+2 q^{22}+q^{20}+2 q^{16}-q^{14}+q^{12}} |
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^{74}+q^{72}-3 q^{70}+4 q^{68}-6 q^{66}+8 q^{64}-8 q^{62}+8 q^{60}-7 q^{58}+5 q^{56}-2 q^{54}-3 q^{52}+7 q^{50}-12 q^{48}+14 q^{46}-16 q^{44}+16 q^{42}-15 q^{40}+12 q^{38}-7 q^{36}+4 q^{34}+q^{32}-4 q^{30}+8 q^{28}-8 q^{26}+9 q^{24}-7 q^{22}+7 q^{20}-4 q^{18}+3 q^{16}-q^{14}+q^{12}} |
| 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^{120}-q^{116}-q^{114}+2 q^{112}+3 q^{110}-q^{108}-5 q^{106}-2 q^{104}+6 q^{102}+6 q^{100}-4 q^{98}-9 q^{96}-q^{94}+9 q^{92}+5 q^{90}-7 q^{88}-8 q^{86}+2 q^{84}+7 q^{82}-7 q^{78}-2 q^{76}+5 q^{74}+2 q^{72}-6 q^{70}-4 q^{68}+4 q^{66}+5 q^{64}-3 q^{62}-6 q^{60}+2 q^{58}+7 q^{56}+q^{54}-8 q^{52}-3 q^{50}+8 q^{48}+8 q^{46}-4 q^{44}-8 q^{42}+9 q^{38}+5 q^{36}-3 q^{34}-5 q^{32}+q^{30}+4 q^{28}+2 q^{26}-q^{24}-q^{22}+q^{18}} |
D4 Invariants.
| Weight | Invariant |
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| 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^{102}-q^{100}+2 q^{98}-2 q^{96}+4 q^{94}-5 q^{92}+5 q^{90}-6 q^{88}+7 q^{86}-7 q^{84}+5 q^{82}-5 q^{80}+4 q^{78}-3 q^{76}-3 q^{74}+2 q^{72}-5 q^{70}+7 q^{68}-12 q^{66}+10 q^{64}-12 q^{62}+13 q^{60}-13 q^{58}+9 q^{56}-10 q^{54}+9 q^{52}-4 q^{50}+3 q^{48}-q^{46}+7 q^{42}-4 q^{40}+6 q^{38}-6 q^{36}+9 q^{34}-4 q^{32}+6 q^{30}-4 q^{28}+5 q^{26}-q^{24}+2 q^{22}-q^{20}+q^{18}} |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 |
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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 18"];
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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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In[5]:=
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Conway[K][z]
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Out[5]=
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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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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 41, -4 } |
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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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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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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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a246,}
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}} ): {}
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 18"];
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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 4 t^2-10 t+13-10 t^{-1} +4 t^{-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^{-2} -2 q^{-3} +5 q^{-4} -6 q^{-5} +7 q^{-6} -7 q^{-7} +6 q^{-8} -4 q^{-9} +2 q^{-10} - q^{-11} } } |
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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{K11a246,} |
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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{} |
Vassiliev invariants
| V2 and V3: | (6, -15) |
| 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=} -4 is the signature of 9 18. 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
| 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^{-4} -2 q^{-5} + q^{-6} +6 q^{-7} -10 q^{-8} + q^{-9} +20 q^{-10} -25 q^{-11} -3 q^{-12} +39 q^{-13} -37 q^{-14} -10 q^{-15} +52 q^{-16} -39 q^{-17} -16 q^{-18} +51 q^{-19} -30 q^{-20} -19 q^{-21} +38 q^{-22} -15 q^{-23} -15 q^{-24} +20 q^{-25} -4 q^{-26} -8 q^{-27} +6 q^{-28} -2 q^{-30} + q^{-31} } |
| 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^{-7} + q^{-8} +2 q^{-9} +2 q^{-10} -8 q^{-11} +13 q^{-13} +5 q^{-14} -27 q^{-15} -5 q^{-16} +37 q^{-17} +22 q^{-18} -65 q^{-19} -29 q^{-20} +78 q^{-21} +57 q^{-22} -104 q^{-23} -76 q^{-24} +114 q^{-25} +107 q^{-26} -128 q^{-27} -126 q^{-28} +127 q^{-29} +145 q^{-30} -124 q^{-31} -155 q^{-32} +111 q^{-33} +160 q^{-34} -95 q^{-35} -157 q^{-36} +74 q^{-37} +148 q^{-38} -50 q^{-39} -134 q^{-40} +26 q^{-41} +116 q^{-42} -7 q^{-43} -93 q^{-44} -7 q^{-45} +68 q^{-46} +18 q^{-47} -48 q^{-48} -17 q^{-49} +26 q^{-50} +17 q^{-51} -15 q^{-52} -10 q^{-53} +5 q^{-54} +7 q^{-55} -3 q^{-56} -2 q^{-57} +2 q^{-59} - q^{-60} } |
| 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^{-8} -2 q^{-9} + q^{-10} +2 q^{-11} -2 q^{-12} +4 q^{-13} -9 q^{-14} +3 q^{-15} +11 q^{-16} -7 q^{-17} +9 q^{-18} -31 q^{-19} +9 q^{-20} +39 q^{-21} -12 q^{-22} +10 q^{-23} -89 q^{-24} +15 q^{-25} +106 q^{-26} +10 q^{-27} +14 q^{-28} -221 q^{-29} -15 q^{-30} +218 q^{-31} +103 q^{-32} +53 q^{-33} -423 q^{-34} -123 q^{-35} +321 q^{-36} +266 q^{-37} +168 q^{-38} -626 q^{-39} -300 q^{-40} +355 q^{-41} +433 q^{-42} +333 q^{-43} -748 q^{-44} -465 q^{-45} +310 q^{-46} +527 q^{-47} +486 q^{-48} -762 q^{-49} -557 q^{-50} +218 q^{-51} +529 q^{-52} +582 q^{-53} -679 q^{-54} -567 q^{-55} +97 q^{-56} +456 q^{-57} +621 q^{-58} -520 q^{-59} -512 q^{-60} -36 q^{-61} +323 q^{-62} +599 q^{-63} -308 q^{-64} -397 q^{-65} -153 q^{-66} +155 q^{-67} +506 q^{-68} -105 q^{-69} -239 q^{-70} -195 q^{-71} +4 q^{-72} +343 q^{-73} +19 q^{-74} -83 q^{-75} -151 q^{-76} -73 q^{-77} +171 q^{-78} +42 q^{-79} +7 q^{-80} -70 q^{-81} -67 q^{-82} +58 q^{-83} +17 q^{-84} +23 q^{-85} -17 q^{-86} -31 q^{-87} +15 q^{-88} +10 q^{-90} - q^{-91} -9 q^{-92} +4 q^{-93} - q^{-94} +2 q^{-95} -2 q^{-97} + q^{-98} } |
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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