10 141
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 141's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X3,10,4,11 X14,6,15,5 X16,8,17,7 X6,16,7,15 X17,20,18,1 X11,18,12,19 X19,12,20,13 X8,14,9,13 X9,2,10,3 |
| Gauss code | -1, 10, -2, 1, 3, -5, 4, -9, -10, 2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6 |
| Dowker-Thistlethwaite code | 4 10 -14 -16 2 18 -8 -6 20 12 |
| Conway Notation | [4,21,21-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 10, width is 3, Braid index is 3 |
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 [{11, 5}, {2, 9}, {8, 10}, {9, 11}, {6, 4}, {5, 3}, {4, 1}, {3, 8}, {7, 2}, {10, 6}, {1, 7}] |
[edit Notes on presentations of 10 141]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 141"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,10,4,11 X14,6,15,5 X16,8,17,7 X6,16,7,15 X17,20,18,1 X11,18,12,19 X19,12,20,13 X8,14,9,13 X9,2,10,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 10, -2, 1, 3, -5, 4, -9, -10, 2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 -14 -16 2 18 -8 -6 20 12 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[4,21,21-] |
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}(3,\{1,1,1,1,-2,-1,-1,-1,-2,-2\})} |
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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{ 3, 10, 3 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{11, 5}, {2, 9}, {8, 10}, {9, 11}, {6, 4}, {5, 3}, {4, 1}, {3, 8}, {7, 2}, {10, 6}, {1, 7}] |
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 -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-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 -z^6-3 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 | { 21, 0 } |
| 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^2-2 q+3-3 q^{-1} +4 q^{-2} -3 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} } |
| 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 -a^2 z^6+a^4 z^4-5 a^2 z^4+z^4+3 a^4 z^2-7 a^2 z^2+3 z^2+a^4-2 a^2+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^4 z^8+a^2 z^8+2 a^5 z^7+3 a^3 z^7+a z^7+a^6 z^6-3 a^4 z^6-4 a^2 z^6-9 a^5 z^5-12 a^3 z^5-3 a z^5-4 a^6 z^4+a^4 z^4+8 a^2 z^4+3 z^4+10 a^5 z^3+13 a^3 z^3+5 a z^3+2 z^3 a^{-1} +3 a^6 z^2-a^4 z^2-9 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^4+2 a^2+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^{18}-q^{12}-q^{10}+q^8+q^4+ q^{-2} + q^{-6} } |
| 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^{94}-q^{92}+2 q^{90}-3 q^{88}+q^{86}-q^{84}-2 q^{82}+6 q^{80}-7 q^{78}+6 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+4 q^{66}-3 q^{62}+7 q^{60}-q^{58}-2 q^{56}+4 q^{54}-8 q^{52}+7 q^{50}-7 q^{46}+3 q^{44}-4 q^{42}+9 q^{40}-4 q^{38}-2 q^{36}-q^{34}-3 q^{32}+7 q^{30}-6 q^{28}-q^{26}+q^{22}+5 q^{20}-3 q^{18}-3 q^{16}+5 q^{14}-5 q^{12}+4 q^{10}-6 q^6+8 q^4-3 q^2+2+2 q^{-2} -3 q^{-4} +2 q^{-6} - q^{-8} + q^{-10} +2 q^{-12} +2 q^{-18} +2 q^{-24} -2 q^{-26} + q^{-28} - q^{-30} - q^{-32} + q^{-34} - q^{-36} + q^{-38} } |
Further Quantum Invariants
Further quantum knot invariants for 10_141.
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^{13}-q^{11}-q^7+q^5+q^3+ q^{-1} - q^{-3} + q^{-5} } |
| 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^{38}-q^{36}-2 q^{34}+2 q^{32}+q^{30}-2 q^{28}+q^{26}+3 q^{24}-q^{22}-2 q^{20}-q^{16}-2 q^{14}+q^{12}+2 q^{10}-q^8+q^6+3 q^4+q^2-2+ q^{-2} + q^{-4} -2 q^{-6} + q^{-14} } |
| 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^{75}-q^{73}-2 q^{71}+3 q^{67}+3 q^{65}-3 q^{63}-4 q^{61}+q^{59}+5 q^{57}+2 q^{55}-5 q^{53}-6 q^{51}+q^{49}+5 q^{47}+3 q^{45}-3 q^{43}-4 q^{41}+3 q^{39}+7 q^{37}+2 q^{35}-7 q^{33}-4 q^{31}+4 q^{29}+3 q^{27}-6 q^{25}-3 q^{23}+4 q^{21}+2 q^{19}-4 q^{17}-2 q^{15}+4 q^{13}+4 q^{11}-3 q^7-2 q^5+4 q^3+7 q-2 q^{-1} -8 q^{-3} -2 q^{-5} +8 q^{-7} +5 q^{-9} -5 q^{-11} -5 q^{-13} + q^{-15} +5 q^{-17} + q^{-19} -3 q^{-21} -2 q^{-23} +2 q^{-27} } |
| 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^{185}-q^{183}-2 q^{181}+q^{177}+3 q^{175}+3 q^{173}+q^{171}-5 q^{169}-7 q^{167}-4 q^{165}+q^{163}+8 q^{161}+12 q^{159}+7 q^{157}-8 q^{155}-15 q^{153}-14 q^{151}-5 q^{149}+12 q^{147}+23 q^{145}+19 q^{143}+q^{141}-16 q^{139}-27 q^{137}-23 q^{135}-q^{133}+20 q^{131}+30 q^{129}+22 q^{127}+3 q^{125}-22 q^{123}-37 q^{121}-30 q^{119}-3 q^{117}+31 q^{115}+50 q^{113}+40 q^{111}-2 q^{109}-47 q^{107}-65 q^{105}-36 q^{103}+26 q^{101}+71 q^{99}+68 q^{97}+12 q^{95}-64 q^{93}-91 q^{91}-48 q^{89}+33 q^{87}+88 q^{85}+76 q^{83}-77 q^{79}-87 q^{77}-23 q^{75}+57 q^{73}+87 q^{71}+46 q^{69}-34 q^{67}-79 q^{65}-50 q^{63}+19 q^{61}+64 q^{59}+51 q^{57}-7 q^{55}-52 q^{53}-43 q^{51}+4 q^{49}+35 q^{47}+29 q^{45}-7 q^{43}-30 q^{41}-18 q^{39}+11 q^{37}+24 q^{35}+11 q^{33}-15 q^{31}-24 q^{29}-16 q^{27}+9 q^{25}+31 q^{23}+30 q^{21}+9 q^{19}-25 q^{17}-45 q^{15}-36 q^{13}+9 q^{11}+59 q^9+71 q^7+22 q^5-55 q^3-96 q-63 q^{-1} +29 q^{-3} +110 q^{-5} +101 q^{-7} +6 q^{-9} -97 q^{-11} -115 q^{-13} -43 q^{-15} +62 q^{-17} +111 q^{-19} +63 q^{-21} -28 q^{-23} -82 q^{-25} -61 q^{-27} -2 q^{-29} +45 q^{-31} +48 q^{-33} +13 q^{-35} -19 q^{-37} -25 q^{-39} -11 q^{-41} +4 q^{-43} +9 q^{-45} +7 q^{-47} -2 q^{-49} -3 q^{-51} + q^{-53} +3 q^{-55} + q^{-57} - q^{-59} -3 q^{-61} -3 q^{-63} + q^{-65} + q^{-67} + q^{-69} } |
| 6 | 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^{258}-q^{256}-2 q^{254}+q^{250}+3 q^{248}+q^{246}+3 q^{244}-q^{242}-7 q^{240}-6 q^{238}-4 q^{236}+2 q^{234}+5 q^{232}+15 q^{230}+11 q^{228}+q^{226}-10 q^{224}-19 q^{222}-18 q^{220}-15 q^{218}+12 q^{216}+27 q^{214}+31 q^{212}+22 q^{210}-24 q^{206}-48 q^{204}-39 q^{202}-17 q^{200}+17 q^{198}+43 q^{196}+56 q^{194}+45 q^{192}+4 q^{190}-32 q^{188}-61 q^{186}-61 q^{184}-41 q^{182}+q^{180}+51 q^{178}+80 q^{176}+79 q^{174}+43 q^{172}-14 q^{170}-82 q^{168}-121 q^{166}-102 q^{164}-32 q^{162}+62 q^{160}+142 q^{158}+165 q^{156}+94 q^{154}-39 q^{152}-158 q^{150}-211 q^{148}-158 q^{146}-6 q^{144}+169 q^{142}+259 q^{140}+209 q^{138}+49 q^{136}-157 q^{134}-289 q^{132}-257 q^{130}-74 q^{128}+154 q^{126}+300 q^{124}+289 q^{122}+95 q^{120}-158 q^{118}-319 q^{116}-293 q^{114}-97 q^{112}+154 q^{110}+327 q^{108}+286 q^{106}+68 q^{104}-183 q^{102}-312 q^{100}-249 q^{98}-34 q^{96}+211 q^{94}+301 q^{92}+189 q^{90}-35 q^{88}-217 q^{86}-249 q^{84}-119 q^{82}+97 q^{80}+220 q^{78}+179 q^{76}+28 q^{74}-122 q^{72}-174 q^{70}-104 q^{68}+37 q^{66}+126 q^{64}+106 q^{62}+19 q^{60}-64 q^{58}-87 q^{56}-42 q^{54}+28 q^{52}+59 q^{50}+33 q^{48}-19 q^{46}-53 q^{44}-41 q^{42}+6 q^{40}+53 q^{38}+67 q^{36}+37 q^{34}-23 q^{32}-76 q^{30}-91 q^{28}-52 q^{26}+22 q^{24}+106 q^{22}+147 q^{20}+102 q^{18}-17 q^{16}-150 q^{14}-215 q^{12}-166 q^{10}+10 q^8+212 q^6+306 q^4+218 q^2-19-267 q^{-2} -389 q^{-4} -259 q^{-6} +44 q^{-8} +329 q^{-10} +419 q^{-12} +255 q^{-14} -60 q^{-16} -354 q^{-18} -407 q^{-20} -213 q^{-22} +91 q^{-24} +309 q^{-26} +335 q^{-28} +170 q^{-30} -90 q^{-32} -241 q^{-34} -237 q^{-36} -104 q^{-38} +50 q^{-40} +152 q^{-42} +156 q^{-44} +63 q^{-46} -23 q^{-48} -77 q^{-50} -78 q^{-52} -50 q^{-54} +35 q^{-58} +33 q^{-60} +28 q^{-62} +9 q^{-64} -4 q^{-66} -18 q^{-68} -15 q^{-70} -7 q^{-72} -3 q^{-74} +7 q^{-76} +8 q^{-78} +9 q^{-80} + q^{-82} -2 q^{-84} -4 q^{-86} -5 q^{-88} - q^{-90} +2 q^{-94} + q^{-96} } |
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^{18}-q^{12}-q^{10}+q^8+q^4+ q^{-2} + q^{-6} } |
| 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^{52}-2 q^{50}+4 q^{48}-8 q^{46}+11 q^{44}-16 q^{42}+18 q^{40}-14 q^{38}+12 q^{36}-4 q^{34}-4 q^{32}+14 q^{30}-23 q^{28}+22 q^{26}-26 q^{24}+24 q^{22}-21 q^{20}+18 q^{18}-8 q^{16}+4 q^{14}+3 q^{12}-10 q^{10}+12 q^8-14 q^6+14 q^4-4 q^2+4+2 q^{-2} - q^{-4} -2 q^{-8} -4 q^{-10} +5 q^{-12} -2 q^{-14} +4 q^{-16} -2 q^{-18} + q^{-20} } |
| 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^{48}-q^{44}-q^{42}-q^{36}+q^{34}+3 q^{32}+4 q^{30}-q^{26}-3 q^{24}-4 q^{22}-3 q^{20}-3 q^{18}+q^{16}+q^{14}+4 q^{12}+4 q^{10}+3 q^8+2 q^6+2 q^4-3-2 q^{-2} - q^{-4} - q^{-8} +2 q^{-12} + q^{-14} + q^{-16} } |
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^{40}-q^{38}-2 q^{32}+q^{30}+q^{26}+2 q^{24}+q^{22}+q^{18}-2 q^{14}-2 q^{10}-q^8-2 q^6+2 q^2+1+3 q^{-2} +3 q^{-4} + q^{-10} - q^{-12} - q^{-14} + q^{-16} } |
| 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^{23}+q^{19}-q^{17}-q^{13}+q+2 q^{-3} + q^{-7} } |
| 1,0,1 | 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^{66}-2 q^{64}+3 q^{62}-3 q^{60}+q^{58}+q^{56}-6 q^{54}+8 q^{52}-7 q^{50}+9 q^{48}-4 q^{46}+4 q^{44}-5 q^{40}+8 q^{38}-15 q^{36}+12 q^{34}-18 q^{32}+13 q^{30}-8 q^{28}+3 q^{26}+11 q^{24}-9 q^{22}+17 q^{20}-13 q^{18}+16 q^{16}-17 q^{14}+8 q^{12}-7 q^{10}-7 q^8+8 q^6-8 q^4+15 q^2-5+12 q^{-2} -2 q^{-4} + q^{-6} -2 q^{-8} -3 q^{-10} -3 q^{-14} +7 q^{-16} -2 q^{-18} +3 q^{-20} + q^{-22} -2 q^{-24} + q^{-26} } |
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^{50}-q^{42}-q^{40}-q^{38}-2 q^{36}-q^{34}+2 q^{32}+2 q^{30}+2 q^{28}+4 q^{26}+6 q^{24}+3 q^{22}-q^{20}-3 q^{16}-8 q^{14}-6 q^{12}-4 q^{10}-4 q^8+5 q^4+5 q^2+4+5 q^{-2} +4 q^{-4} -2 q^{-8} + q^{-10} - q^{-14} + q^{-18} } |
| 1,0,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}+q^{24}-q^{16}-q^{12}-q^8+q^2+1+ q^{-2} +2 q^{-4} + q^{-8} } |
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^{40}-q^{38}+2 q^{36}-2 q^{34}+2 q^{32}-3 q^{30}+2 q^{28}-q^{26}+q^{22}-2 q^{20}+3 q^{18}-4 q^{16}+4 q^{14}-4 q^{12}+4 q^{10}-3 q^8+2 q^6+1- q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-8} - q^{-10} + q^{-12} - q^{-14} + q^{-16} } |
| 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^{66}-q^{62}-q^{60}+q^{58}+q^{56}-2 q^{54}-2 q^{52}+q^{50}+3 q^{48}-2 q^{44}+3 q^{40}+2 q^{38}-q^{36}-2 q^{34}+q^{30}-q^{26}-q^{24}+q^{22}+q^{20}-q^{18}-2 q^{16}+q^{14}+2 q^{12}-q^{10}-3 q^8+2 q^4+q^2-1+2 q^{-4} +3 q^{-6} - q^{-10} + q^{-14} + q^{-16} - q^{-20} - q^{-22} + q^{-26} } |
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^{54}-q^{52}+q^{50}-2 q^{48}+2 q^{46}-3 q^{44}+q^{42}-2 q^{40}+2 q^{38}+q^{34}+2 q^{32}+q^{30}+4 q^{28}-q^{26}+3 q^{24}-3 q^{22}+2 q^{20}-5 q^{18}+q^{16}-6 q^{14}-4 q^{10}+q^4+3 q^2+2+4 q^{-2} + q^{-4} +4 q^{-6} - q^{-8} + q^{-10} - q^{-12} + q^{-14} - q^{-16} - q^{-20} + q^{-22} } |
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^{94}-q^{92}+2 q^{90}-3 q^{88}+q^{86}-q^{84}-2 q^{82}+6 q^{80}-7 q^{78}+6 q^{76}-2 q^{74}-3 q^{72}+6 q^{70}-5 q^{68}+4 q^{66}-3 q^{62}+7 q^{60}-q^{58}-2 q^{56}+4 q^{54}-8 q^{52}+7 q^{50}-7 q^{46}+3 q^{44}-4 q^{42}+9 q^{40}-4 q^{38}-2 q^{36}-q^{34}-3 q^{32}+7 q^{30}-6 q^{28}-q^{26}+q^{22}+5 q^{20}-3 q^{18}-3 q^{16}+5 q^{14}-5 q^{12}+4 q^{10}-6 q^6+8 q^4-3 q^2+2+2 q^{-2} -3 q^{-4} +2 q^{-6} - q^{-8} + q^{-10} +2 q^{-12} +2 q^{-18} +2 q^{-24} -2 q^{-26} + q^{-28} - q^{-30} - q^{-32} + q^{-34} - q^{-36} + q^{-38} } |
.
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["10 141"];
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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 -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-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 -z^6-3 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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{ 21, 0 } |
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^2-2 q+3-3 q^{-1} +4 q^{-2} -3 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} } |
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 -a^2 z^6+a^4 z^4-5 a^2 z^4+z^4+3 a^4 z^2-7 a^2 z^2+3 z^2+a^4-2 a^2+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^4 z^8+a^2 z^8+2 a^5 z^7+3 a^3 z^7+a z^7+a^6 z^6-3 a^4 z^6-4 a^2 z^6-9 a^5 z^5-12 a^3 z^5-3 a z^5-4 a^6 z^4+a^4 z^4+8 a^2 z^4+3 z^4+10 a^5 z^3+13 a^3 z^3+5 a z^3+2 z^3 a^{-1} +3 a^6 z^2-a^4 z^2-9 a^2 z^2+z^2 a^{-2} -4 z^2-2 a^5 z-4 a^3 z-3 a z-z a^{-1} +a^4+2 a^2+2} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {8_5,}
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["10 141"];
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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 -t^3+3 t^2-4 t+5-4 t^{-1} +3 t^{-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 q^2-2 q+3-3 q^{-1} +4 q^{-2} -3 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} } } |
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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{8_5,} |
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, 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=} 0 is the signature of 10 141. 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^6-q^5+q^3-3 q^2+3 q+1-6 q^{-1} +6 q^{-2} +3 q^{-3} -8 q^{-4} +4 q^{-5} +6 q^{-6} -9 q^{-7} + q^{-8} +7 q^{-9} -8 q^{-10} - q^{-11} +8 q^{-12} -4 q^{-13} -3 q^{-14} +5 q^{-15} - q^{-16} -2 q^{-17} + q^{-18} } |
| 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 2 q^{12}-2 q^{11}-2 q^{10}-q^9+6 q^8+2 q^7-6 q^6-7 q^5+6 q^4+12 q^3-3 q^2-17 q+18 q^{-1} +6 q^{-2} -20 q^{-3} -6 q^{-4} +17 q^{-5} +9 q^{-6} -16 q^{-7} -6 q^{-8} +11 q^{-9} +7 q^{-10} -10 q^{-11} -4 q^{-12} +4 q^{-13} +4 q^{-14} - q^{-15} -3 q^{-16} -4 q^{-17} + q^{-18} +8 q^{-19} +2 q^{-20} -8 q^{-21} -6 q^{-22} +9 q^{-23} +8 q^{-24} -6 q^{-25} -10 q^{-26} +2 q^{-27} +9 q^{-28} + q^{-29} -7 q^{-30} -2 q^{-31} +4 q^{-32} +2 q^{-33} - q^{-34} -2 q^{-35} + q^{-36} } |
| 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^{21}+q^{20}-4 q^{19}-q^{18}+q^{17}+6 q^{16}+5 q^{15}-13 q^{14}-7 q^{13}-q^{12}+21 q^{11}+21 q^{10}-23 q^9-29 q^8-20 q^7+40 q^6+57 q^5-16 q^4-56 q^3-57 q^2+43 q+91+7 q^{-1} -63 q^{-2} -87 q^{-3} +28 q^{-4} +104 q^{-5} +26 q^{-6} -54 q^{-7} -95 q^{-8} +16 q^{-9} +100 q^{-10} +25 q^{-11} -42 q^{-12} -87 q^{-13} +8 q^{-14} +89 q^{-15} +21 q^{-16} -30 q^{-17} -75 q^{-18} -4 q^{-19} +75 q^{-20} +20 q^{-21} -12 q^{-22} -57 q^{-23} -19 q^{-24} +51 q^{-25} +15 q^{-26} +9 q^{-27} -30 q^{-28} -24 q^{-29} +23 q^{-30} - q^{-31} +18 q^{-32} -3 q^{-33} -14 q^{-34} +9 q^{-35} -21 q^{-36} +8 q^{-37} +8 q^{-38} +3 q^{-39} +16 q^{-40} -25 q^{-41} -6 q^{-42} +5 q^{-44} +24 q^{-45} -10 q^{-46} -7 q^{-47} -9 q^{-48} -4 q^{-49} +17 q^{-50} -5 q^{-53} -6 q^{-54} +5 q^{-55} + q^{-56} +2 q^{-57} - q^{-58} -2 q^{-59} + q^{-60} } |
| 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^{32}-4 q^{29}+2 q^{27}+3 q^{26}+2 q^{25}-2 q^{24}-8 q^{23}+q^{22}+11 q^{21}+5 q^{20}-3 q^{19}-17 q^{18}-22 q^{17}+7 q^{16}+43 q^{15}+40 q^{14}-6 q^{13}-64 q^{12}-81 q^{11}-14 q^{10}+97 q^9+131 q^8+42 q^7-113 q^6-186 q^5-86 q^4+115 q^3+234 q^2+137 q-104-267 q^{-1} -178 q^{-2} +82 q^{-3} +275 q^{-4} +214 q^{-5} -55 q^{-6} -279 q^{-7} -228 q^{-8} +37 q^{-9} +266 q^{-10} +234 q^{-11} -21 q^{-12} -258 q^{-13} -227 q^{-14} +15 q^{-15} +241 q^{-16} +226 q^{-17} -12 q^{-18} -232 q^{-19} -214 q^{-20} +2 q^{-21} +212 q^{-22} +214 q^{-23} +11 q^{-24} -196 q^{-25} -208 q^{-26} -29 q^{-27} +165 q^{-28} +205 q^{-29} +56 q^{-30} -138 q^{-31} -195 q^{-32} -74 q^{-33} +96 q^{-34} +176 q^{-35} +101 q^{-36} -58 q^{-37} -154 q^{-38} -104 q^{-39} +16 q^{-40} +112 q^{-41} +111 q^{-42} +19 q^{-43} -78 q^{-44} -92 q^{-45} -39 q^{-46} +31 q^{-47} +68 q^{-48} +46 q^{-49} -2 q^{-50} -36 q^{-51} -36 q^{-52} -14 q^{-53} +6 q^{-54} +17 q^{-55} +16 q^{-56} +9 q^{-57} +6 q^{-58} -4 q^{-59} -13 q^{-60} -17 q^{-61} -11 q^{-62} +2 q^{-63} +21 q^{-64} +21 q^{-65} +6 q^{-66} -9 q^{-67} -21 q^{-68} -19 q^{-69} - q^{-70} +17 q^{-71} +17 q^{-72} +8 q^{-73} -3 q^{-74} -15 q^{-75} -12 q^{-76} +8 q^{-78} +7 q^{-79} +4 q^{-80} -7 q^{-82} -4 q^{-83} + q^{-84} +2 q^{-85} + q^{-86} +2 q^{-87} - q^{-88} -2 q^{-89} + q^{-90} } |
| 6 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{45}+q^{44}-2 q^{43}-q^{42}-4 q^{41}+q^{40}+2 q^{39}+4 q^{38}+9 q^{37}-3 q^{36}-2 q^{35}-14 q^{34}-3 q^{33}-6 q^{32}+q^{31}+23 q^{30}+10 q^{29}+17 q^{28}-9 q^{27}-q^{26}-41 q^{25}-49 q^{24}-5 q^{23}+11 q^{22}+71 q^{21}+77 q^{20}+92 q^{19}-45 q^{18}-151 q^{17}-159 q^{16}-122 q^{15}+67 q^{14}+228 q^{13}+352 q^{12}+120 q^{11}-177 q^{10}-377 q^9-426 q^8-127 q^7+281 q^6+646 q^5+435 q^4-13 q^3-467 q^2-711 q-430+151 q^{-1} +768 q^{-2} +683 q^{-3} +224 q^{-4} -379 q^{-5} -805 q^{-6} -632 q^{-7} -25 q^{-8} +719 q^{-9} +748 q^{-10} +357 q^{-11} -260 q^{-12} -760 q^{-13} -673 q^{-14} -109 q^{-15} +645 q^{-16} +709 q^{-17} +372 q^{-18} -207 q^{-19} -700 q^{-20} -643 q^{-21} -123 q^{-22} +598 q^{-23} +662 q^{-24} +360 q^{-25} -173 q^{-26} -648 q^{-27} -617 q^{-28} -154 q^{-29} +528 q^{-30} +617 q^{-31} +383 q^{-32} -90 q^{-33} -561 q^{-34} -597 q^{-35} -243 q^{-36} +387 q^{-37} +547 q^{-38} +435 q^{-39} +60 q^{-40} -410 q^{-41} -556 q^{-42} -366 q^{-43} +171 q^{-44} +417 q^{-45} +467 q^{-46} +242 q^{-47} -186 q^{-48} -444 q^{-49} -456 q^{-50} -74 q^{-51} +202 q^{-52} +404 q^{-53} +371 q^{-54} +65 q^{-55} -226 q^{-56} -415 q^{-57} -247 q^{-58} -49 q^{-59} +208 q^{-60} +345 q^{-61} +226 q^{-62} +27 q^{-63} -221 q^{-64} -236 q^{-65} -195 q^{-66} -20 q^{-67} +162 q^{-68} +194 q^{-69} +159 q^{-70} -15 q^{-71} -76 q^{-72} -145 q^{-73} -110 q^{-74} -13 q^{-75} +42 q^{-76} +106 q^{-77} +38 q^{-78} +43 q^{-79} -12 q^{-80} -39 q^{-81} -36 q^{-82} -38 q^{-83} +12 q^{-84} -32 q^{-85} +24 q^{-86} +27 q^{-87} +29 q^{-88} +21 q^{-89} -2 q^{-90} +13 q^{-91} -61 q^{-92} -26 q^{-93} -15 q^{-94} +9 q^{-95} +21 q^{-96} +27 q^{-97} +49 q^{-98} -20 q^{-99} -15 q^{-100} -28 q^{-101} -17 q^{-102} -13 q^{-103} +5 q^{-104} +40 q^{-105} +4 q^{-106} +9 q^{-107} -6 q^{-108} -8 q^{-109} -17 q^{-110} -10 q^{-111} +13 q^{-112} + q^{-113} +8 q^{-114} +3 q^{-115} +3 q^{-116} -7 q^{-117} -6 q^{-118} +3 q^{-119} -2 q^{-120} +2 q^{-121} + q^{-122} +2 q^{-123} - q^{-124} -2 q^{-125} + q^{-126} } |
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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