10 21
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 21's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X7,16,8,17 X11,20,12,1 X15,6,16,7 X19,8,20,9 X9,18,10,19 X17,10,18,11 |
| Gauss code | -1, 4, -3, 1, -2, 7, -5, 8, -9, 10, -6, 3, -4, 2, -7, 5, -10, 9, -8, 6 |
| Dowker-Thistlethwaite code | 4 12 14 16 18 20 2 6 10 8 |
| Conway Notation | [3412] |
| 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, 5}, {1, 10}, {11, 6}, {5, 7}, {10, 12}, {6, 8}, {7, 9}, {4, 11}, {8, 3}, {2, 4}, {3, 1}, {9, 2}] |
[edit Notes on presentations of 10 21]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 21"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X5,14,6,15 X3,13,4,12 X13,3,14,2 X7,16,8,17 X11,20,12,1 X15,6,16,7 X19,8,20,9 X9,18,10,19 X17,10,18,11 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 4, -3, 1, -2, 7, -5, 8, -9, 10, -6, 3, -4, 2, -7, 5, -10, 9, -8, 6 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 14 16 18 20 2 6 10 8 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[3412] |
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,-2,1,-2,-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[{12, 5}, {1, 10}, {11, 6}, {5, 7}, {10, 12}, {6, 8}, {7, 9}, {4, 11}, {8, 3}, {2, 4}, {3, 1}, {9, 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
| 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+7 t^2-9 t+9-9 t^{-1} +7 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^6-5 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 | { 45, -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 1-2 q^{-1} +4 q^{-2} -5 q^{-3} +7 q^{-4} -7 q^{-5} +7 q^{-6} -6 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^6-5 z^2 a^6-3 a^6-z^6 a^4-3 z^4 a^4+2 a^4+z^4 a^2+3 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 z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+2 z a^{11}+2 z^6 a^{10}-2 z^4 a^{10}+2 z^7 a^9-2 z^5 a^9+3 z a^9+2 z^8 a^8-5 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-z^7 a^7+2 z^3 a^7-z a^7+4 z^8 a^6-14 z^6 a^6+20 z^4 a^6-14 z^2 a^6+3 a^6+z^9 a^5-z^7 a^5-3 z^5 a^5+3 z^3 a^5-2 z a^5+2 z^8 a^4-6 z^6 a^4+4 z^4 a^4-3 z^2 a^4+2 a^4+2 z^7 a^3-7 z^5 a^3+5 z^3 a^3+z^6 a^2-4 z^4 a^2+4 z^2 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^{30}-2 q^{22}-2 q^{18}+q^{14}+3 q^{10}+q^6+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}-q^{152}-3 q^{150}+6 q^{148}-7 q^{146}+7 q^{144}-6 q^{142}+3 q^{140}+4 q^{138}-9 q^{136}+12 q^{134}-14 q^{132}+12 q^{130}-7 q^{128}+q^{126}+6 q^{124}-12 q^{122}+24 q^{120}-18 q^{118}+14 q^{116}-7 q^{114}-4 q^{112}+15 q^{110}-20 q^{108}+17 q^{106}-9 q^{104}-3 q^{102}+12 q^{100}-15 q^{98}+5 q^{96}+6 q^{94}-20 q^{92}+21 q^{90}-20 q^{88}+2 q^{86}+16 q^{84}-33 q^{82}+38 q^{80}-32 q^{78}+13 q^{76}+8 q^{74}-28 q^{72}+36 q^{70}-34 q^{68}+22 q^{66}-4 q^{64}-13 q^{62}+25 q^{60}-21 q^{58}+13 q^{56}+4 q^{54}-15 q^{52}+19 q^{50}-13 q^{48}-q^{46}+19 q^{44}-28 q^{42}+30 q^{40}-16 q^{38}-2 q^{36}+21 q^{34}-29 q^{32}+30 q^{30}-21 q^{28}+7 q^{26}+6 q^{24}-16 q^{22}+18 q^{20}-13 q^{18}+9 q^{16}-q^{14}-2 q^{12}+4 q^{10}-4 q^8+3 q^6-q^4+q^2} |
Further Quantum Invariants
Further quantum knot invariants for 10_21.
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^{21}-q^{19}+q^{17}-3 q^{15}+q^{13}+2 q^7-q^5+2 q^3-q+ 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}-2 q^{50}-q^{48}+4 q^{46}-4 q^{44}-q^{42}+9 q^{40}-5 q^{38}-3 q^{36}+7 q^{34}-3 q^{32}-5 q^{30}+q^{28}+3 q^{26}-3 q^{24}-3 q^{22}+6 q^{20}-7 q^{16}+6 q^{14}+4 q^{12}-7 q^{10}+3 q^8+6 q^6-5 q^4-q^2+4- q^{-2} - q^{-4} + q^{-6} } |
| 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^{111}-q^{109}-q^{107}+2 q^{103}-2 q^{99}-q^{97}+q^{95}+2 q^{93}+q^{89}-3 q^{87}-2 q^{85}+4 q^{83}+9 q^{81}-6 q^{79}-14 q^{77}-q^{75}+21 q^{73}+2 q^{71}-24 q^{69}-6 q^{67}+20 q^{65}+16 q^{63}-15 q^{61}-14 q^{59}+5 q^{57}+17 q^{55}+2 q^{53}-13 q^{51}-12 q^{49}+12 q^{47}+15 q^{45}-11 q^{43}-21 q^{41}+7 q^{39}+22 q^{37}-3 q^{35}-23 q^{33}-3 q^{31}+23 q^{29}+9 q^{27}-18 q^{25}-16 q^{23}+13 q^{21}+20 q^{19}-4 q^{17}-19 q^{15}-3 q^{13}+17 q^{11}+10 q^9-11 q^7-10 q^5+4 q^3+10 q-6 q^{-3} - q^{-5} +3 q^{-7} + q^{-9} - q^{-11} - q^{-13} + q^{-15} } |
| 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^{180}-q^{178}-q^{176}+4 q^{170}-2 q^{168}-2 q^{166}-2 q^{164}-3 q^{162}+10 q^{160}+2 q^{158}-q^{156}-7 q^{154}-12 q^{152}+13 q^{150}+10 q^{148}+9 q^{146}-10 q^{144}-31 q^{142}+2 q^{140}+17 q^{138}+36 q^{136}+5 q^{134}-55 q^{132}-34 q^{130}+6 q^{128}+75 q^{126}+54 q^{124}-56 q^{122}-79 q^{120}-44 q^{118}+83 q^{116}+113 q^{114}-12 q^{112}-94 q^{110}-101 q^{108}+34 q^{106}+115 q^{104}+45 q^{102}-41 q^{100}-98 q^{98}-28 q^{96}+58 q^{94}+61 q^{92}+22 q^{90}-48 q^{88}-54 q^{86}-8 q^{84}+45 q^{82}+57 q^{80}-5 q^{78}-63 q^{76}-47 q^{74}+36 q^{72}+75 q^{70}+23 q^{68}-73 q^{66}-78 q^{64}+30 q^{62}+89 q^{60}+51 q^{58}-69 q^{56}-102 q^{54}+3 q^{52}+79 q^{50}+86 q^{48}-27 q^{46}-100 q^{44}-42 q^{42}+29 q^{40}+93 q^{38}+32 q^{36}-50 q^{34}-58 q^{32}-38 q^{30}+50 q^{28}+57 q^{26}+16 q^{24}-21 q^{22}-62 q^{20}-9 q^{18}+25 q^{16}+38 q^{14}+25 q^{12}-33 q^{10}-25 q^8-12 q^6+14 q^4+30 q^2-1-7 q^{-2} -15 q^{-4} -4 q^{-6} +12 q^{-8} +2 q^{-10} +2 q^{-12} -4 q^{-14} -3 q^{-16} +3 q^{-18} + q^{-22} - q^{-24} - q^{-26} + q^{-28} } |
| 5 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{265}-q^{263}-q^{261}+2 q^{255}+2 q^{253}-2 q^{251}-4 q^{249}-q^{247}+5 q^{243}+8 q^{241}-9 q^{237}-9 q^{235}-4 q^{233}+8 q^{231}+18 q^{229}+10 q^{227}-12 q^{225}-25 q^{223}-17 q^{221}+9 q^{219}+31 q^{217}+33 q^{215}-3 q^{213}-50 q^{211}-51 q^{209}-5 q^{207}+59 q^{205}+89 q^{203}+36 q^{201}-78 q^{199}-139 q^{197}-79 q^{195}+77 q^{193}+209 q^{191}+162 q^{189}-63 q^{187}-273 q^{185}-273 q^{183}-5 q^{181}+327 q^{179}+402 q^{177}+96 q^{175}-324 q^{173}-507 q^{171}-250 q^{169}+271 q^{167}+574 q^{165}+384 q^{163}-155 q^{161}-554 q^{159}-477 q^{157}-q^{155}+466 q^{153}+518 q^{151}+126 q^{149}-322 q^{147}-461 q^{145}-231 q^{143}+152 q^{141}+378 q^{139}+265 q^{137}-22 q^{135}-241 q^{133}-264 q^{131}-91 q^{129}+138 q^{127}+235 q^{125}+146 q^{123}-45 q^{121}-209 q^{119}-193 q^{117}+q^{115}+201 q^{113}+226 q^{111}+34 q^{109}-218 q^{107}-266 q^{105}-52 q^{103}+238 q^{101}+320 q^{99}+86 q^{97}-261 q^{95}-383 q^{93}-135 q^{91}+264 q^{89}+437 q^{87}+213 q^{85}-230 q^{83}-477 q^{81}-299 q^{79}+155 q^{77}+474 q^{75}+376 q^{73}-40 q^{71}-421 q^{69}-439 q^{67}-87 q^{65}+316 q^{63}+438 q^{61}+207 q^{59}-167 q^{57}-386 q^{55}-285 q^{53}+18 q^{51}+273 q^{49}+297 q^{47}+108 q^{45}-128 q^{43}-246 q^{41}-178 q^{39}-4 q^{37}+148 q^{35}+177 q^{33}+94 q^{31}-31 q^{29}-124 q^{27}-128 q^{25}-49 q^{23}+49 q^{21}+101 q^{19}+91 q^{17}+25 q^{15}-56 q^{13}-84 q^{11}-55 q^9+4 q^7+50 q^5+60 q^3+27 q-18 q^{-1} -40 q^{-3} -32 q^{-5} -3 q^{-7} +17 q^{-9} +23 q^{-11} +13 q^{-13} -6 q^{-15} -12 q^{-17} -7 q^{-19} +2 q^{-23} +5 q^{-25} +2 q^{-27} -3 q^{-29} - q^{-31} + q^{-33} + q^{-39} - q^{-41} - q^{-43} + q^{-45} } |
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}-2 q^{22}-2 q^{18}+q^{14}+3 q^{10}+q^6+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}+13 q^{76}-18 q^{74}+22 q^{72}-30 q^{70}+37 q^{68}-40 q^{66}+42 q^{64}-48 q^{62}+57 q^{60}-54 q^{58}+52 q^{56}-56 q^{54}+47 q^{52}-30 q^{50}+8 q^{48}+24 q^{46}-56 q^{44}+96 q^{42}-128 q^{40}+154 q^{38}-174 q^{36}+170 q^{34}-160 q^{32}+132 q^{30}-102 q^{28}+54 q^{26}-6 q^{24}-32 q^{22}+67 q^{20}-90 q^{18}+106 q^{16}-98 q^{14}+88 q^{12}-72 q^{10}+56 q^8-34 q^6+23 q^4-12 q^2+6-2 q^{-2} + 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^{64}-q^{62}-2 q^{60}+2 q^{56}+3 q^{54}-2 q^{52}+3 q^{50}+6 q^{48}+2 q^{46}-5 q^{44}-3 q^{42}+2 q^{40}-4 q^{38}-5 q^{36}+q^{32}-3 q^{30}+q^{28}+q^{26}-q^{24}+q^{22}+5 q^{20}+q^{18}-3 q^{16}+2 q^{14}+5 q^{12}-q^{10}-3 q^8+2 q^6+3 q^4-1+ 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}-4 q^{60}-q^{58}+4 q^{56}-3 q^{54}-q^{52}+10 q^{50}-q^{48}-3 q^{46}+6 q^{44}-2 q^{42}-5 q^{40}+q^{38}-3 q^{34}-4 q^{32}+q^{30}-7 q^{26}+3 q^{24}+5 q^{22}-4 q^{20}+3 q^{18}+6 q^{16}-3 q^{14}+3 q^{12}+3 q^{10}-2 q^8+2 q^6+q^4-q^2+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}-2 q^{29}-3 q^{25}-q^{23}-q^{21}+2 q^{17}+q^{15}+3 q^{13}+2 q^9-q^7+q^5+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}-4 q^{76}-4 q^{74}+q^{72}-4 q^{68}+9 q^{64}+5 q^{62}+q^{60}+7 q^{58}+6 q^{56}-3 q^{54}-2 q^{52}-6 q^{48}-8 q^{46}-2 q^{44}-3 q^{42}-9 q^{40}-4 q^{38}+4 q^{36}-q^{34}-4 q^{32}+5 q^{30}+5 q^{28}+q^{26}+q^{24}+4 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+2 q^{14}+q^{12}+q^{10}+q^8+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}-2 q^{36}-3 q^{32}-2 q^{30}-2 q^{28}-q^{26}+q^{22}+3 q^{20}+q^{18}+3 q^{16}+2 q^{12}+q^6+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}+4 q^{60}-5 q^{58}+6 q^{56}-7 q^{54}+7 q^{52}-8 q^{50}+5 q^{48}-3 q^{46}+4 q^{42}-7 q^{40}+11 q^{38}-14 q^{36}+15 q^{34}-16 q^{32}+13 q^{30}-12 q^{28}+9 q^{26}-5 q^{24}+q^{22}+4 q^{20}-5 q^{18}+8 q^{16}-7 q^{14}+9 q^{12}-7 q^{10}+6 q^8-4 q^6+3 q^4-q^2+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}-4 q^{96}-3 q^{94}+2 q^{92}+5 q^{90}+q^{88}-5 q^{86}-4 q^{84}+4 q^{82}+10 q^{80}+2 q^{78}-6 q^{76}-4 q^{74}+5 q^{72}+6 q^{70}-3 q^{68}-7 q^{66}-q^{64}+5 q^{62}+q^{60}-6 q^{58}-4 q^{56}+3 q^{54}+3 q^{52}-4 q^{50}-4 q^{48}+2 q^{46}+4 q^{44}-2 q^{42}-7 q^{40}+8 q^{36}+5 q^{34}-6 q^{32}-7 q^{30}+3 q^{28}+10 q^{26}+3 q^{24}-6 q^{22}-5 q^{20}+5 q^{18}+6 q^{16}-4 q^{12}-q^{10}+3 q^8+2 q^6-q^4-q^2+ 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}-4 q^{84}+2 q^{82}-4 q^{80}+5 q^{78}-5 q^{76}+5 q^{74}-4 q^{72}+9 q^{70}-3 q^{68}+4 q^{66}-2 q^{64}+2 q^{62}+2 q^{60}-4 q^{58}+4 q^{56}-8 q^{54}+9 q^{52}-11 q^{50}+10 q^{48}-15 q^{46}+9 q^{44}-13 q^{42}+7 q^{40}-11 q^{38}+5 q^{36}-4 q^{34}+3 q^{32}+2 q^{30}+7 q^{26}-3 q^{24}+7 q^{22}-5 q^{20}+9 q^{18}-5 q^{16}+6 q^{14}-4 q^{12}+5 q^{10}-2 q^8+2 q^6-q^4+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}-q^{152}-3 q^{150}+6 q^{148}-7 q^{146}+7 q^{144}-6 q^{142}+3 q^{140}+4 q^{138}-9 q^{136}+12 q^{134}-14 q^{132}+12 q^{130}-7 q^{128}+q^{126}+6 q^{124}-12 q^{122}+24 q^{120}-18 q^{118}+14 q^{116}-7 q^{114}-4 q^{112}+15 q^{110}-20 q^{108}+17 q^{106}-9 q^{104}-3 q^{102}+12 q^{100}-15 q^{98}+5 q^{96}+6 q^{94}-20 q^{92}+21 q^{90}-20 q^{88}+2 q^{86}+16 q^{84}-33 q^{82}+38 q^{80}-32 q^{78}+13 q^{76}+8 q^{74}-28 q^{72}+36 q^{70}-34 q^{68}+22 q^{66}-4 q^{64}-13 q^{62}+25 q^{60}-21 q^{58}+13 q^{56}+4 q^{54}-15 q^{52}+19 q^{50}-13 q^{48}-q^{46}+19 q^{44}-28 q^{42}+30 q^{40}-16 q^{38}-2 q^{36}+21 q^{34}-29 q^{32}+30 q^{30}-21 q^{28}+7 q^{26}+6 q^{24}-16 q^{22}+18 q^{20}-13 q^{18}+9 q^{16}-q^{14}-2 q^{12}+4 q^{10}-4 q^8+3 q^6-q^4+q^2} |
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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 21"];
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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+7 t^2-9 t+9-9 t^{-1} +7 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^6-5 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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{ 45, -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 1-2 q^{-1} +4 q^{-2} -5 q^{-3} +7 q^{-4} -7 q^{-5} +7 q^{-6} -6 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^6-5 z^2 a^6-3 a^6-z^6 a^4-3 z^4 a^4+2 a^4+z^4 a^2+3 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 z^4 a^{12}-2 z^2 a^{12}+2 z^5 a^{11}-4 z^3 a^{11}+2 z a^{11}+2 z^6 a^{10}-2 z^4 a^{10}+2 z^7 a^9-2 z^5 a^9+3 z a^9+2 z^8 a^8-5 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-z^7 a^7+2 z^3 a^7-z a^7+4 z^8 a^6-14 z^6 a^6+20 z^4 a^6-14 z^2 a^6+3 a^6+z^9 a^5-z^7 a^5-3 z^5 a^5+3 z^3 a^5-2 z a^5+2 z^8 a^4-6 z^6 a^4+4 z^4 a^4-3 z^2 a^4+2 a^4+2 z^7 a^3-7 z^5 a^3+5 z^3 a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^2} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n69,}
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 21"];
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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+7 t^2-9 t+9-9 t^{-1} +7 t^{-2} -2 t^{-3} } , } |
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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{K11n69,} |
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, 0) |
| 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 21. 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^2-2 q+6 q^{-1} -7 q^{-2} -4 q^{-3} +17 q^{-4} -10 q^{-5} -14 q^{-6} +28 q^{-7} -8 q^{-8} -27 q^{-9} +35 q^{-10} -2 q^{-11} -36 q^{-12} +35 q^{-13} +4 q^{-14} -38 q^{-15} +29 q^{-16} +6 q^{-17} -28 q^{-18} +19 q^{-19} +4 q^{-20} -14 q^{-21} +9 q^{-22} + q^{-23} -6 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^6-2 q^5+2 q^3+3 q^2-6 q-5+8 q^{-1} +13 q^{-2} -12 q^{-3} -19 q^{-4} +7 q^{-5} +34 q^{-6} -5 q^{-7} -39 q^{-8} -9 q^{-9} +49 q^{-10} +19 q^{-11} -46 q^{-12} -38 q^{-13} +47 q^{-14} +46 q^{-15} -32 q^{-16} -64 q^{-17} +27 q^{-18} +66 q^{-19} -7 q^{-20} -79 q^{-21} - q^{-22} +76 q^{-23} +19 q^{-24} -82 q^{-25} -25 q^{-26} +75 q^{-27} +34 q^{-28} -67 q^{-29} -37 q^{-30} +56 q^{-31} +33 q^{-32} -36 q^{-33} -33 q^{-34} +30 q^{-35} +15 q^{-36} -10 q^{-37} -14 q^{-38} +8 q^{-39} +2 q^{-40} -2 q^{-41} + q^{-42} +3 q^{-43} -4 q^{-44} -3 q^{-45} +5 q^{-46} +2 q^{-47} -2 q^{-48} -4 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}-2 q^{11}+2 q^9-q^8+4 q^7-8 q^6-q^5+8 q^4-q^3+14 q^2-24 q-12+16 q^{-1} +5 q^{-2} +45 q^{-3} -40 q^{-4} -38 q^{-5} +3 q^{-6} -3 q^{-7} +103 q^{-8} -27 q^{-9} -51 q^{-10} -31 q^{-11} -56 q^{-12} +144 q^{-13} +10 q^{-14} -10 q^{-15} -38 q^{-16} -144 q^{-17} +124 q^{-18} +18 q^{-19} +72 q^{-20} +23 q^{-21} -208 q^{-22} +53 q^{-23} -40 q^{-24} +145 q^{-25} +136 q^{-26} -215 q^{-27} -23 q^{-28} -145 q^{-29} +178 q^{-30} +256 q^{-31} -177 q^{-32} -82 q^{-33} -253 q^{-34} +183 q^{-35} +352 q^{-36} -125 q^{-37} -121 q^{-38} -336 q^{-39} +167 q^{-40} +410 q^{-41} -63 q^{-42} -133 q^{-43} -389 q^{-44} +121 q^{-45} +416 q^{-46} +7 q^{-47} -94 q^{-48} -392 q^{-49} +35 q^{-50} +346 q^{-51} +64 q^{-52} -8 q^{-53} -322 q^{-54} -46 q^{-55} +211 q^{-56} +71 q^{-57} +74 q^{-58} -197 q^{-59} -76 q^{-60} +84 q^{-61} +36 q^{-62} +97 q^{-63} -87 q^{-64} -55 q^{-65} +15 q^{-66} -4 q^{-67} +76 q^{-68} -27 q^{-69} -24 q^{-70} -4 q^{-71} -19 q^{-72} +43 q^{-73} -6 q^{-74} -5 q^{-75} -3 q^{-76} -16 q^{-77} +18 q^{-78} - q^{-79} + q^{-80} -8 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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