10 6
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 6's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,16,6,17 X7,18,8,19 X9,20,10,1 X17,6,18,7 X19,8,20,9 X15,10,16,11 |
| Gauss code | -1, 4, -3, 1, -5, 8, -6, 9, -7, 10, -2, 3, -4, 2, -10, 5, -8, 6, -9, 7 |
| Dowker-Thistlethwaite code | 4 12 16 18 20 14 2 10 6 8 |
| Conway Notation | [532] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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 [{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 1}] |
[edit Notes on presentations of 10 6]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 6"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,16,6,17 X7,18,8,19 X9,20,10,1 X17,6,18,7 X19,8,20,9 X15,10,16,11 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 4, -3, 1, -5, 8, -6, 9, -7, 10, -2, 3, -4, 2, -10, 5, -8, 6, -9, 7 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 16 18 20 14 2 10 6 8 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[532] |
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,-1,-1,-1,-2,1,3,-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[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 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 | 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 -2 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -2 t^{-3} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 z^6-6 z^4-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 | { 37, -4 } |
| 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 1- q^{-1} +3 q^{-2} -4 q^{-3} +5 q^{-4} -6 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} } |
| 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^4 a^8+3 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-a^6-z^6 a^4-4 z^4 a^4-4 z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+3 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 z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+2 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+2 z^7 a^9-4 z^5 a^9+4 z^3 a^9+2 z^8 a^8-7 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+5 z^5 a^7-2 z^3 a^7+3 z^8 a^6-12 z^6 a^6+18 z^4 a^6-10 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+8 z^5 a^5-10 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4-3 z^4 a^4+5 z^2 a^4-2 a^4+z^7 a^3-3 z^5 a^3+2 z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 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^{30}-q^{22}+q^{20}-q^{18}-q^{14}-2 q^{12}+q^{10}+2 q^6+q^4+q^2+1} |
| 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^{162}-q^{160}+2 q^{158}-3 q^{156}+q^{154}-3 q^{150}+5 q^{148}-6 q^{146}+6 q^{144}-4 q^{142}+4 q^{138}-7 q^{136}+9 q^{134}-8 q^{132}+6 q^{130}-4 q^{128}+7 q^{124}-9 q^{122}+13 q^{120}-10 q^{118}+6 q^{116}-7 q^{112}+9 q^{110}-9 q^{108}+5 q^{106}+5 q^{104}-9 q^{102}+7 q^{100}-q^{98}-7 q^{96}+14 q^{94}-17 q^{92}+11 q^{90}-3 q^{88}-7 q^{86}+18 q^{84}-20 q^{82}+18 q^{80}-11 q^{78}-2 q^{76}+9 q^{74}-15 q^{72}+15 q^{70}-14 q^{68}+5 q^{66}+5 q^{64}-11 q^{62}+10 q^{60}-7 q^{58}-4 q^{56}+10 q^{54}-13 q^{52}+5 q^{50}-9 q^{46}+18 q^{44}-17 q^{42}+10 q^{40}-q^{38}-8 q^{36}+14 q^{34}-13 q^{32}+11 q^{30}-4 q^{28}+2 q^{26}+4 q^{24}-5 q^{22}+7 q^{20}-4 q^{18}+4 q^{16}+2 q^{10}-q^8+2 q^6+q^2} |
Further Quantum Invariants
Further quantum knot invariants for 10_6.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{21}-q^{19}+q^{17}-2 q^{15}+q^{13}-q^9+q^7-q^5+2 q^3+ q^{-1} } |
| 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^{58}-q^{56}-q^{54}+2 q^{52}-q^{50}-q^{48}+3 q^{46}-3 q^{44}-3 q^{42}+6 q^{40}-2 q^{38}-3 q^{36}+5 q^{34}+q^{32}-2 q^{30}-q^{28}+2 q^{26}-q^{24}-4 q^{22}+3 q^{20}+q^{18}-5 q^{16}+3 q^{14}+4 q^{12}-4 q^{10}+4 q^6-2 q^4-q^2+2+ q^{-6} } |
| 3 | |
| 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^{30}-q^{22}+q^{20}-q^{18}-q^{14}-2 q^{12}+q^{10}+2 q^6+q^4+q^2+1} |
| 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^{84}-2 q^{82}+4 q^{80}-8 q^{78}+11 q^{76}-14 q^{74}+18 q^{72}-20 q^{70}+23 q^{68}-22 q^{66}+22 q^{64}-28 q^{62}+26 q^{60}-26 q^{58}+24 q^{56}-20 q^{54}+11 q^{52}+6 q^{50}-20 q^{48}+38 q^{46}-54 q^{44}+68 q^{42}-76 q^{40}+82 q^{38}-79 q^{36}+70 q^{34}-58 q^{32}+40 q^{30}-19 q^{28}-4 q^{26}+24 q^{24}-36 q^{22}+43 q^{20}-50 q^{18}+42 q^{16}-38 q^{14}+28 q^{12}-22 q^{10}+18 q^8-8 q^6+10 q^4-2 q^2+4+ q^{-4} } |
| 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^{76}-q^{70}+q^{66}-2 q^{60}-q^{58}-3 q^{52}+4 q^{48}+2 q^{46}+q^{42}+4 q^{40}-q^{38}-3 q^{36}+q^{34}-q^{30}-3 q^{24}-q^{22}+2 q^{20}-q^{18}-3 q^{16}+3 q^{12}-q^{10}-q^8+2 q^6+3 q^4+q^2+1+ q^{-2} + q^{-4} } |
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^{68}-q^{66}+q^{62}-3 q^{60}+3 q^{56}-3 q^{54}-q^{52}+6 q^{50}-2 q^{48}-2 q^{46}+4 q^{44}-q^{42}-q^{40}+q^{38}+2 q^{36}-2 q^{32}+3 q^{30}-5 q^{26}-7 q^{20}-q^{18}+q^{16}-2 q^{14}+4 q^{12}+3 q^{10}+2 q^8+3 q^6+2 q^4+1} |
| 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^{39}+q^{35}-q^{33}+q^{31}-q^{29}+q^{27}-q^{25}-q^{21}-2 q^{19}-q^{17}-2 q^{15}+q^{13}+3 q^9+q^7+2 q^5+q^3+q} |
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^{86}-q^{82}+q^{80}+q^{78}-3 q^{76}-2 q^{74}+2 q^{72}-3 q^{68}+4 q^{64}-2 q^{60}+3 q^{58}+q^{56}-2 q^{54}+2 q^{52}+3 q^{50}-q^{48}+q^{46}+5 q^{44}+q^{42}-q^{40}+q^{38}+3 q^{36}-4 q^{34}-6 q^{32}-4 q^{30}-6 q^{28}-7 q^{26}-5 q^{24}-2 q^{22}+4 q^{18}+4 q^{16}+5 q^{14}+5 q^{12}+5 q^{10}+3 q^8+2 q^6+q^4+q^2} |
| 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^{48}+q^{44}+q^{38}-q^{36}+q^{34}-q^{32}-q^{28}-q^{26}-2 q^{24}-2 q^{22}-q^{20}-2 q^{18}+q^{16}+3 q^{12}+2 q^{10}+2 q^8+2 q^6+q^4+q^2} |
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^{68}-q^{66}+2 q^{64}-3 q^{62}+3 q^{60}-4 q^{58}+5 q^{56}-5 q^{54}+5 q^{52}-4 q^{50}+2 q^{48}-2 q^{44}+5 q^{42}-7 q^{40}+9 q^{38}-10 q^{36}+10 q^{34}-10 q^{32}+7 q^{30}-6 q^{28}+3 q^{26}-2 q^{24}-2 q^{22}+3 q^{20}-5 q^{18}+5 q^{16}-4 q^{14}+6 q^{12}-3 q^{10}+4 q^8-q^6+2 q^4+1} |
| 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^{110}-q^{106}-q^{104}+q^{102}+2 q^{100}-q^{98}-3 q^{96}-2 q^{94}+2 q^{92}+4 q^{90}-4 q^{86}-3 q^{84}+3 q^{82}+6 q^{80}-5 q^{76}-2 q^{74}+4 q^{72}+3 q^{70}-3 q^{68}-3 q^{66}+2 q^{64}+4 q^{62}-3 q^{58}+3 q^{54}+q^{52}-3 q^{50}-q^{48}+2 q^{46}+2 q^{44}-3 q^{42}-5 q^{40}+5 q^{36}+q^{34}-6 q^{32}-6 q^{30}+2 q^{28}+5 q^{26}-4 q^{22}-2 q^{20}+4 q^{18}+3 q^{16}+q^{14}-q^{12}+q^{10}+2 q^8+2 q^6+ q^{-2} } |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{94}-q^{92}+q^{90}-2 q^{88}+2 q^{86}-3 q^{84}+2 q^{82}-3 q^{80}+4 q^{78}-4 q^{76}+3 q^{74}-3 q^{72}+5 q^{70}-2 q^{68}+q^{66}+4 q^{60}-4 q^{58}+5 q^{56}-6 q^{54}+8 q^{52}-7 q^{50}+8 q^{48}-8 q^{46}+7 q^{44}-5 q^{42}+5 q^{40}-5 q^{38}+q^{36}-3 q^{34}-3 q^{32}-2 q^{30}-6 q^{28}+q^{26}-6 q^{24}+3 q^{22}-3 q^{20}+7 q^{18}+6 q^{14}+q^{12}+5 q^{10}+q^8+2 q^6+q^2} |
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^{162}-q^{160}+2 q^{158}-3 q^{156}+q^{154}-3 q^{150}+5 q^{148}-6 q^{146}+6 q^{144}-4 q^{142}+4 q^{138}-7 q^{136}+9 q^{134}-8 q^{132}+6 q^{130}-4 q^{128}+7 q^{124}-9 q^{122}+13 q^{120}-10 q^{118}+6 q^{116}-7 q^{112}+9 q^{110}-9 q^{108}+5 q^{106}+5 q^{104}-9 q^{102}+7 q^{100}-q^{98}-7 q^{96}+14 q^{94}-17 q^{92}+11 q^{90}-3 q^{88}-7 q^{86}+18 q^{84}-20 q^{82}+18 q^{80}-11 q^{78}-2 q^{76}+9 q^{74}-15 q^{72}+15 q^{70}-14 q^{68}+5 q^{66}+5 q^{64}-11 q^{62}+10 q^{60}-7 q^{58}-4 q^{56}+10 q^{54}-13 q^{52}+5 q^{50}-9 q^{46}+18 q^{44}-17 q^{42}+10 q^{40}-q^{38}-8 q^{36}+14 q^{34}-13 q^{32}+11 q^{30}-4 q^{28}+2 q^{26}+4 q^{24}-5 q^{22}+7 q^{20}-4 q^{18}+4 q^{16}+2 q^{10}-q^8+2 q^6+q^2} |
.
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 6"];
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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 -2 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -2 t^{-3} } |
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 -2 z^6-6 z^4-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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{ 37, -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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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- q^{-1} +3 q^{-2} -4 q^{-3} +5 q^{-4} -6 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} } |
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^4 a^8+3 z^2 a^8+a^8-z^6 a^6-4 z^4 a^6-4 z^2 a^6-a^6-z^6 a^4-4 z^4 a^4-4 z^2 a^4-2 a^4+z^4 a^2+4 z^2 a^2+3 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 z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+z a^{11}+2 z^6 a^{10}-3 z^4 a^{10}+z^2 a^{10}+2 z^7 a^9-4 z^5 a^9+4 z^3 a^9+2 z^8 a^8-7 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+5 z^5 a^7-2 z^3 a^7+3 z^8 a^6-12 z^6 a^6+18 z^4 a^6-10 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+8 z^5 a^5-10 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4-3 z^4 a^4+5 z^2 a^4-2 a^4+z^7 a^3-3 z^5 a^3+2 z a^3+z^6 a^2-5 z^4 a^2+7 z^2 a^2-3 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}} ): {}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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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 6"];
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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 -2 t^3+6 t^2-7 t+7-7 t^{-1} +6 t^{-2} -2 t^{-3} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1- q^{-1} +3 q^{-2} -4 q^{-3} +5 q^{-4} -6 q^{-5} +6 q^{-6} -5 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} } } |
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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{} |
Vassiliev invariants
| V2 and V3: | (-1, 4) |
| 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 10 6. 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^2-q+3 q^{-1} -4 q^{-2} - q^{-3} +9 q^{-4} -8 q^{-5} -5 q^{-6} +17 q^{-7} -9 q^{-8} -13 q^{-9} +23 q^{-10} -7 q^{-11} -20 q^{-12} +26 q^{-13} -4 q^{-14} -23 q^{-15} +25 q^{-16} - q^{-17} -19 q^{-18} +17 q^{-19} -11 q^{-21} +8 q^{-22} -5 q^{-24} +4 q^{-25} -2 q^{-27} + q^{-28} } |
| 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-q^5+2 q^2-3 q+2 q^{-1} +4 q^{-2} -8 q^{-3} -2 q^{-4} +7 q^{-5} +12 q^{-6} -15 q^{-7} -12 q^{-8} +10 q^{-9} +24 q^{-10} -12 q^{-11} -26 q^{-12} +2 q^{-13} +34 q^{-14} + q^{-15} -31 q^{-16} -13 q^{-17} +32 q^{-18} +19 q^{-19} -27 q^{-20} -26 q^{-21} +23 q^{-22} +32 q^{-23} -20 q^{-24} -35 q^{-25} +15 q^{-26} +39 q^{-27} -13 q^{-28} -38 q^{-29} +8 q^{-30} +36 q^{-31} -5 q^{-32} -28 q^{-33} - q^{-34} +23 q^{-35} - q^{-36} -11 q^{-37} -2 q^{-38} +6 q^{-39} - q^{-40} - q^{-41} +3 q^{-42} -4 q^{-44} - q^{-45} +5 q^{-46} + q^{-47} -3 q^{-48} -3 q^{-49} +3 q^{-50} + q^{-51} -2 q^{-53} + q^{-54} } |
| 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^{12}-q^{11}-q^8+3 q^7-3 q^6+q^5+2 q^4-4 q^3+5 q^2-8 q+3+10 q^{-1} -6 q^{-2} +7 q^{-3} -22 q^{-4} - q^{-5} +23 q^{-6} + q^{-7} +20 q^{-8} -44 q^{-9} -19 q^{-10} +27 q^{-11} +10 q^{-12} +53 q^{-13} -52 q^{-14} -39 q^{-15} +11 q^{-16} -4 q^{-17} +89 q^{-18} -37 q^{-19} -34 q^{-20} -3 q^{-21} -46 q^{-22} +93 q^{-23} -19 q^{-24} +5 q^{-25} +9 q^{-26} -95 q^{-27} +65 q^{-28} -21 q^{-29} +53 q^{-30} +46 q^{-31} -126 q^{-32} +26 q^{-33} -41 q^{-34} +91 q^{-35} +86 q^{-36} -142 q^{-37} -4 q^{-38} -59 q^{-39} +113 q^{-40} +113 q^{-41} -148 q^{-42} -24 q^{-43} -72 q^{-44} +122 q^{-45} +129 q^{-46} -136 q^{-47} -36 q^{-48} -88 q^{-49} +107 q^{-50} +138 q^{-51} -95 q^{-52} -33 q^{-53} -104 q^{-54} +63 q^{-55} +122 q^{-56} -40 q^{-57} -2 q^{-58} -100 q^{-59} +7 q^{-60} +78 q^{-61} -2 q^{-62} +32 q^{-63} -69 q^{-64} -21 q^{-65} +30 q^{-66} + q^{-67} +45 q^{-68} -31 q^{-69} -19 q^{-70} +3 q^{-71} -8 q^{-72} +34 q^{-73} -9 q^{-74} -7 q^{-75} -3 q^{-76} -11 q^{-77} +17 q^{-78} - q^{-79} - q^{-81} -7 q^{-82} +5 q^{-83} + q^{-85} -2 q^{-87} + q^{-88} } |
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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