10 18
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 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 X15,20,16,1 X9,17,10,16 X7,19,8,18 X17,9,18,8 X19,7,20,6 X13,10,14,11 X11,2,12,3 |
| Gauss code | -1, 10, -2, 1, -3, 8, -6, 7, -5, 9, -10, 2, -9, 3, -4, 5, -7, 6, -8, 4 |
| Dowker-Thistlethwaite code | 4 12 14 18 16 2 10 20 8 6 |
| Conway Notation | [41122] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 12, width is 5, Braid index is 5 |
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 [{12, 3}, {2, 10}, {9, 11}, {10, 12}, {11, 4}, {3, 5}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 9}, {1, 8}] |
[edit Notes on presentations of 10 18]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 18"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,12,4,13 X5,14,6,15 X15,20,16,1 X9,17,10,16 X7,19,8,18 X17,9,18,8 X19,7,20,6 X13,10,14,11 X11,2,12,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 10, -2, 1, -3, 8, -6, 7, -5, 9, -10, 2, -9, 3, -4, 5, -7, 6, -8, 4 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 14 18 16 2 10 20 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[41122] |
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,-1,-2,1,-2,3,-2,3,4,-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, 12, 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[{12, 3}, {2, 10}, {9, 11}, {10, 12}, {11, 4}, {3, 5}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 9}, {1, 8}] |
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 -4 t^2+14 t-19+14 t^{-1} -4 t^{-2} } |
| 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 -4 z^4-2 z^2+1} |
| 2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
| Determinant and Signature | { 55, -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^3-2 q^2+4 q-6+8 q^{-1} -9 q^{-2} +9 q^{-3} -7 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} } |
| 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^2 a^6-z^4 a^4+a^4-2 z^4 a^2-3 z^2 a^2-a^2-z^4-z^2+z^2 a^{-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^3 z^9+a z^9+3 a^4 z^8+5 a^2 z^8+2 z^8+4 a^5 z^7+3 a^3 z^7+a z^7+2 z^7 a^{-1} +4 a^6 z^6-5 a^4 z^6-15 a^2 z^6+z^6 a^{-2} -5 z^6+3 a^7 z^5-6 a^5 z^5-12 a^3 z^5-10 a z^5-7 z^5 a^{-1} +a^8 z^4-5 a^6 z^4+6 a^4 z^4+17 a^2 z^4-4 z^4 a^{-2} +z^4-4 a^7 z^3+5 a^5 z^3+14 a^3 z^3+11 a z^3+6 z^3 a^{-1} -a^8 z^2+a^6 z^2-3 a^4 z^2-8 a^2 z^2+4 z^2 a^{-2} +z^2-2 a^5 z-4 a^3 z-4 a z-2 z a^{-1} +a^4+a^2- 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^{22}-q^{20}-q^{18}+2 q^{16}-q^{14}+q^{12}+q^{10}-q^8+q^6-2 q^4+q^2- q^{-2} +2 q^{-4} + q^{-10} } |
| 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^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-q^{104}-4 q^{102}+12 q^{100}-16 q^{98}+20 q^{96}-18 q^{94}+6 q^{92}+6 q^{90}-21 q^{88}+32 q^{86}-37 q^{84}+33 q^{82}-20 q^{80}+24 q^{76}-41 q^{74}+49 q^{72}-40 q^{70}+20 q^{68}+4 q^{66}-28 q^{64}+39 q^{62}-29 q^{60}+10 q^{58}+17 q^{56}-33 q^{54}+30 q^{52}-8 q^{50}-24 q^{48}+52 q^{46}-63 q^{44}+51 q^{42}-17 q^{40}-25 q^{38}+62 q^{36}-79 q^{34}+72 q^{32}-43 q^{30}+35 q^{26}-58 q^{24}+60 q^{22}-41 q^{20}+9 q^{18}+20 q^{16}-38 q^{14}+34 q^{12}-13 q^{10}-16 q^8+38 q^6-44 q^4+28 q^2+1-33 q^{-2} +58 q^{-4} -57 q^{-6} +40 q^{-8} -10 q^{-10} -21 q^{-12} +42 q^{-14} -47 q^{-16} +39 q^{-18} -21 q^{-20} +2 q^{-22} +13 q^{-24} -20 q^{-26} +20 q^{-28} -14 q^{-30} +9 q^{-32} - q^{-34} -3 q^{-36} +4 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} } |
Further Quantum Invariants
Further quantum knot invariants for 10_18.
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^{15}-2 q^{13}+2 q^{11}-2 q^9+2 q^7-q^3+2 q-2 q^{-1} +2 q^{-3} - q^{-5} + q^{-7} } |
| 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^{42}-2 q^{40}-q^{38}+5 q^{36}-4 q^{34}-2 q^{32}+10 q^{30}-7 q^{28}-6 q^{26}+13 q^{24}-6 q^{22}-9 q^{20}+10 q^{18}+q^{16}-6 q^{14}+7 q^{10}-q^8-9 q^6+9 q^4+5 q^2-12+6 q^{-2} +8 q^{-4} -11 q^{-6} +8 q^{-10} -5 q^{-12} -2 q^{-14} +4 q^{-16} - q^{-18} - q^{-20} + q^{-22} } |
| 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^{81}-2 q^{79}-q^{77}+2 q^{75}+3 q^{73}-q^{71}-5 q^{69}+3 q^{67}+3 q^{65}-5 q^{63}-4 q^{61}+10 q^{59}+6 q^{57}-17 q^{55}-11 q^{53}+25 q^{51}+21 q^{49}-30 q^{47}-31 q^{45}+27 q^{43}+41 q^{41}-20 q^{39}-43 q^{37}+6 q^{35}+41 q^{33}+5 q^{31}-31 q^{29}-17 q^{27}+18 q^{25}+28 q^{23}-8 q^{21}-32 q^{19}-3 q^{17}+37 q^{15}+13 q^{13}-39 q^{11}-22 q^9+37 q^7+31 q^5-31 q^3-38 q+22 q^{-1} +44 q^{-3} -7 q^{-5} -43 q^{-7} -7 q^{-9} +37 q^{-11} +17 q^{-13} -25 q^{-15} -23 q^{-17} +14 q^{-19} +22 q^{-21} -5 q^{-23} -16 q^{-25} - q^{-27} +11 q^{-29} +2 q^{-31} -6 q^{-33} -2 q^{-35} +3 q^{-37} + q^{-39} - q^{-41} - q^{-43} + q^{-45} } |
| 4 | |
| 5 |
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^{22}-q^{20}-q^{18}+2 q^{16}-q^{14}+q^{12}+q^{10}-q^8+q^6-2 q^4+q^2- q^{-2} +2 q^{-4} + q^{-10} } |
| 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^{60}-4 q^{58}+10 q^{56}-20 q^{54}+32 q^{52}-48 q^{50}+70 q^{48}-90 q^{46}+108 q^{44}-126 q^{42}+144 q^{40}-156 q^{38}+156 q^{36}-148 q^{34}+128 q^{32}-88 q^{30}+28 q^{28}+48 q^{26}-128 q^{24}+212 q^{22}-288 q^{20}+340 q^{18}-376 q^{16}+378 q^{14}-351 q^{12}+302 q^{10}-226 q^8+144 q^6-44 q^4-44 q^2+118-172 q^{-2} +204 q^{-4} -210 q^{-6} +192 q^{-8} -164 q^{-10} +130 q^{-12} -96 q^{-14} +64 q^{-16} -40 q^{-18} +25 q^{-20} -12 q^{-22} +6 q^{-24} -2 q^{-26} + q^{-28} } |
| 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^{56}-q^{54}-2 q^{52}+q^{50}+3 q^{48}-5 q^{44}+2 q^{42}+7 q^{40}-3 q^{38}-7 q^{36}+3 q^{34}+8 q^{32}-4 q^{30}-8 q^{28}+4 q^{26}+4 q^{24}-5 q^{22}-q^{20}+3 q^{18}-2 q^{16}+4 q^{12}-q^{10}-4 q^8+4 q^6+9 q^4-3 q^2-5+6 q^{-2} +4 q^{-4} -6 q^{-6} -6 q^{-8} +2 q^{-10} +4 q^{-12} -2 q^{-14} -3 q^{-16} +2 q^{-18} +3 q^{-20} - q^{-24} + q^{-28} } |
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^{48}-2 q^{46}+4 q^{42}-5 q^{40}+8 q^{36}-8 q^{34}-4 q^{32}+11 q^{30}-6 q^{28}-5 q^{26}+11 q^{24}-q^{22}-3 q^{20}+3 q^{18}+3 q^{16}-2 q^{14}-6 q^{12}+5 q^{10}+q^8-11 q^6+5 q^4+6 q^2-9+5 q^{-2} +6 q^{-4} -6 q^{-6} +3 q^{-8} +2 q^{-10} -3 q^{-12} +2 q^{-14} + q^{-16} - q^{-18} + q^{-20} } |
| 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^{29}-q^{27}-q^{23}+2 q^{21}-q^{19}+2 q^{17}+q^{13}-q^{11}-2 q^5+q^3-q+ q^{-1} - q^{-3} +2 q^{-5} + q^{-9} + q^{-13} } |
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^{48}-2 q^{46}+4 q^{44}-6 q^{42}+7 q^{40}-10 q^{38}+12 q^{36}-12 q^{34}+12 q^{32}-9 q^{30}+6 q^{28}-q^{26}-5 q^{24}+11 q^{22}-17 q^{20}+21 q^{18}-23 q^{16}+24 q^{14}-22 q^{12}+19 q^{10}-13 q^8+7 q^6-q^4-4 q^2+7-11 q^{-2} +12 q^{-4} -12 q^{-6} +11 q^{-8} -8 q^{-10} +7 q^{-12} -4 q^{-14} +3 q^{-16} - q^{-18} + q^{-20} } |
| 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^{78}-2 q^{74}-2 q^{72}+2 q^{70}+5 q^{68}-6 q^{64}-5 q^{62}+5 q^{60}+10 q^{58}-11 q^{54}-8 q^{52}+7 q^{50}+12 q^{48}-q^{46}-12 q^{44}-4 q^{42}+10 q^{40}+8 q^{38}-6 q^{36}-8 q^{34}+4 q^{32}+9 q^{30}-q^{28}-8 q^{26}-q^{24}+7 q^{22}+2 q^{20}-7 q^{18}-4 q^{16}+7 q^{14}+6 q^{12}-7 q^{10}-11 q^8+3 q^6+13 q^4+4 q^2-11-10 q^{-2} +7 q^{-4} +13 q^{-6} + q^{-8} -10 q^{-10} -5 q^{-12} +6 q^{-14} +6 q^{-16} - q^{-18} -5 q^{-20} - q^{-22} +3 q^{-24} +2 q^{-26} - q^{-28} - q^{-30} + q^{-34} } |
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^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-q^{104}-4 q^{102}+12 q^{100}-16 q^{98}+20 q^{96}-18 q^{94}+6 q^{92}+6 q^{90}-21 q^{88}+32 q^{86}-37 q^{84}+33 q^{82}-20 q^{80}+24 q^{76}-41 q^{74}+49 q^{72}-40 q^{70}+20 q^{68}+4 q^{66}-28 q^{64}+39 q^{62}-29 q^{60}+10 q^{58}+17 q^{56}-33 q^{54}+30 q^{52}-8 q^{50}-24 q^{48}+52 q^{46}-63 q^{44}+51 q^{42}-17 q^{40}-25 q^{38}+62 q^{36}-79 q^{34}+72 q^{32}-43 q^{30}+35 q^{26}-58 q^{24}+60 q^{22}-41 q^{20}+9 q^{18}+20 q^{16}-38 q^{14}+34 q^{12}-13 q^{10}-16 q^8+38 q^6-44 q^4+28 q^2+1-33 q^{-2} +58 q^{-4} -57 q^{-6} +40 q^{-8} -10 q^{-10} -21 q^{-12} +42 q^{-14} -47 q^{-16} +39 q^{-18} -21 q^{-20} +2 q^{-22} +13 q^{-24} -20 q^{-26} +20 q^{-28} -14 q^{-30} +9 q^{-32} - q^{-34} -3 q^{-36} +4 q^{-38} -4 q^{-40} +3 q^{-42} - q^{-44} + q^{-46} } |
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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["10 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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Failed to parse (SVG (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+14 t-19+14 t^{-1} -4 t^{-2} } |
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 -4 z^4-2 z^2+1} |
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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{ 55, -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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Failed to parse (SVG (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^3-2 q^2+4 q-6+8 q^{-1} -9 q^{-2} +9 q^{-3} -7 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} } |
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 z^2 a^6-z^4 a^4+a^4-2 z^4 a^2-3 z^2 a^2-a^2-z^4-z^2+z^2 a^{-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^3 z^9+a z^9+3 a^4 z^8+5 a^2 z^8+2 z^8+4 a^5 z^7+3 a^3 z^7+a z^7+2 z^7 a^{-1} +4 a^6 z^6-5 a^4 z^6-15 a^2 z^6+z^6 a^{-2} -5 z^6+3 a^7 z^5-6 a^5 z^5-12 a^3 z^5-10 a z^5-7 z^5 a^{-1} +a^8 z^4-5 a^6 z^4+6 a^4 z^4+17 a^2 z^4-4 z^4 a^{-2} +z^4-4 a^7 z^3+5 a^5 z^3+14 a^3 z^3+11 a z^3+6 z^3 a^{-1} -a^8 z^2+a^6 z^2-3 a^4 z^2-8 a^2 z^2+4 z^2 a^{-2} +z^2-2 a^5 z-4 a^3 z-4 a z-2 z a^{-1} +a^4+a^2- a^{-2} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {10_24,}
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]:=
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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["10 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+14 t-19+14 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^3-2 q^2+4 q-6+8 q^{-1} -9 q^{-2} +9 q^{-3} -7 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} } } |
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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{10_24,} |
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: | (-2, 1) |
| 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=} -2 is the signature of 10 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
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^{10}-2 q^9+6 q^7-8 q^6-3 q^5+19 q^4-16 q^3-14 q^2+38 q-18-32 q^{-1} +55 q^{-2} -14 q^{-3} -50 q^{-4} +63 q^{-5} -6 q^{-6} -57 q^{-7} +57 q^{-8} + q^{-9} -48 q^{-10} +38 q^{-11} +4 q^{-12} -29 q^{-13} +19 q^{-14} +3 q^{-15} -12 q^{-16} +7 q^{-17} + q^{-18} -3 q^{-19} + q^{-20} } |
| 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^{21}-2 q^{20}+2 q^{18}+3 q^{17}-7 q^{16}-4 q^{15}+10 q^{14}+12 q^{13}-19 q^{12}-19 q^{11}+21 q^{10}+39 q^9-27 q^8-56 q^7+19 q^6+81 q^5-7 q^4-100 q^3-17 q^2+117 q+44-122 q^{-1} -77 q^{-2} +124 q^{-3} +106 q^{-4} -116 q^{-5} -136 q^{-6} +107 q^{-7} +158 q^{-8} -92 q^{-9} -176 q^{-10} +78 q^{-11} +182 q^{-12} -56 q^{-13} -186 q^{-14} +43 q^{-15} +168 q^{-16} -20 q^{-17} -150 q^{-18} +8 q^{-19} +119 q^{-20} +3 q^{-21} -89 q^{-22} -6 q^{-23} +61 q^{-24} +4 q^{-25} -38 q^{-26} -2 q^{-27} +25 q^{-28} -2 q^{-29} -15 q^{-30} +2 q^{-31} +11 q^{-32} -3 q^{-33} -7 q^{-34} +2 q^{-35} +3 q^{-36} + q^{-37} -3 q^{-38} + q^{-39} } |
| 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^{36}-2 q^{35}+2 q^{33}-q^{32}+4 q^{31}-9 q^{30}+10 q^{28}-2 q^{27}+12 q^{26}-31 q^{25}-8 q^{24}+31 q^{23}+9 q^{22}+37 q^{21}-77 q^{20}-46 q^{19}+48 q^{18}+40 q^{17}+118 q^{16}-120 q^{15}-127 q^{14}+10 q^{13}+48 q^{12}+267 q^{11}-91 q^{10}-187 q^9-101 q^8-52 q^7+407 q^6+29 q^5-127 q^4-202 q^3-275 q^2+425 q+158+74 q^{-1} -195 q^{-2} -534 q^{-3} +307 q^{-4} +205 q^{-5} +328 q^{-6} -75 q^{-7} -732 q^{-8} +127 q^{-9} +168 q^{-10} +548 q^{-11} +84 q^{-12} -846 q^{-13} -43 q^{-14} +95 q^{-15} +696 q^{-16} +230 q^{-17} -874 q^{-18} -184 q^{-19} +2 q^{-20} +756 q^{-21} +355 q^{-22} -790 q^{-23} -273 q^{-24} -125 q^{-25} +683 q^{-26} +439 q^{-27} -574 q^{-28} -266 q^{-29} -253 q^{-30} +474 q^{-31} +422 q^{-32} -305 q^{-33} -143 q^{-34} -295 q^{-35} +217 q^{-36} +291 q^{-37} -109 q^{-38} +8 q^{-39} -225 q^{-40} +47 q^{-41} +130 q^{-42} -39 q^{-43} +83 q^{-44} -112 q^{-45} -5 q^{-46} +32 q^{-47} -33 q^{-48} +70 q^{-49} -36 q^{-50} +2 q^{-52} -28 q^{-53} +32 q^{-54} -8 q^{-55} +5 q^{-56} + q^{-57} -13 q^{-58} +7 q^{-59} -2 q^{-60} +3 q^{-61} + q^{-62} -3 q^{-63} + q^{-64} } |
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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