9 1
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
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9_1 should perhaps be called "The Nonafoil Knot", following the trefoil knot, the cinquefoil knot and (maybe) the septafoil knot. The next in the series is K11a367. See also T(9,2). |
Knot presentations
| Planar diagram presentation | X1,10,2,11 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X11,2,12,3 X13,4,14,5 X15,6,16,7 X17,8,18,9 |
| Gauss code | -1, 6, -2, 7, -3, 8, -4, 9, -5, 1, -6, 2, -7, 3, -8, 4, -9, 5 |
| Dowker-Thistlethwaite code | 10 12 14 16 18 2 4 6 8 |
| Conway Notation | [9] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||
Length is 9, width is 2, Braid index is 2 |
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 [{11, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 1}] |
[edit Notes on presentations of 9 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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 1"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1,10,2,11 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X11,2,12,3 X13,4,14,5 X15,6,16,7 X17,8,18,9 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 6, -2, 7, -3, 8, -4, 9, -5, 1, -6, 2, -7, 3, -8, 4, -9, 5 |
In[6]:=
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DTCode[K]
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Out[6]=
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10 12 14 16 18 2 4 6 8 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[9] |
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}(2,\{-1,-1,-1,-1,-1,-1,-1,-1,-1\})} |
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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{ 2, 9, 2 } |
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, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 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 t^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} } |
| 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^8+7 z^6+15 z^4+10 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 | { 9, -8 } |
| 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^{-4} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} + q^{-12} - q^{-13} } |
| 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^6 a^{10}-6 z^4 a^{10}-10 z^2 a^{10}-4 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+20 z^2 a^8+5 a^8} |
| 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 a^{17}+z^2 a^{16}+z^3 a^{15}-z a^{15}+z^4 a^{14}-2 z^2 a^{14}+z^5 a^{13}-3 z^3 a^{13}+z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-z a^{11}+z^8 a^{10}-7 z^6 a^{10}+16 z^4 a^{10}-14 z^2 a^{10}+4 a^{10}+z^7 a^9-6 z^5 a^9+10 z^3 a^9-4 z a^9+z^8 a^8-8 z^6 a^8+21 z^4 a^8-20 z^2 a^8+5 a^8} |
| 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^{38}-q^{36}-q^{34}+q^{22}+q^{20}+2 q^{18}+q^{16}+q^{14}} |
| 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^{216}-q^{172}-q^{170}-q^{164}-q^{162}-q^{160}-q^{154}-q^{152}-q^{126}-q^{120}-q^{118}-q^{116}-q^{114}-q^{108}+q^{100}+q^{98}+q^{94}+2 q^{92}+2 q^{90}+2 q^{88}+q^{86}+q^{84}+2 q^{82}+2 q^{80}+q^{78}+q^{74}+q^{72}+q^{70}} |
Further Quantum Invariants
Further quantum knot invariants for 9_1.
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^{27}+q^{11}+q^9+q^7} |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{72}-q^{56}-q^{54}-q^{52}+q^{22}+q^{20}+q^{18}+q^{16}+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^{135}+q^{119}+q^{117}+q^{115}-q^{85}-q^{83}-q^{81}-q^{79}-q^{77}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}+q^{23}+q^{21}} |
| 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^{216}-q^{200}-q^{198}-q^{196}+q^{166}+q^{164}+q^{162}+q^{160}+q^{158}-q^{114}-q^{112}-q^{110}-q^{108}-q^{106}-q^{104}-q^{102}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+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^{315}+q^{299}+q^{297}+q^{295}-q^{265}-q^{263}-q^{261}-q^{259}-q^{257}+q^{213}+q^{211}+q^{209}+q^{207}+q^{205}+q^{203}+q^{201}-q^{143}-q^{141}-q^{139}-q^{137}-q^{135}-q^{133}-q^{131}-q^{129}-q^{127}+q^{55}+q^{53}+q^{51}+q^{49}+q^{47}+q^{45}+q^{43}+q^{41}+q^{39}+q^{37}+q^{35}} |
| 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^{432}-q^{416}-q^{414}-q^{412}+q^{382}+q^{380}+q^{378}+q^{376}+q^{374}-q^{330}-q^{328}-q^{326}-q^{324}-q^{322}-q^{320}-q^{318}+q^{260}+q^{258}+q^{256}+q^{254}+q^{252}+q^{250}+q^{248}+q^{246}+q^{244}-q^{172}-q^{170}-q^{168}-q^{166}-q^{164}-q^{162}-q^{160}-q^{158}-q^{156}-q^{154}-q^{152}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}} |
| 8 | 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^{720}-q^{704}-q^{702}-q^{700}+q^{670}+q^{668}+q^{666}+q^{664}+q^{662}-q^{618}-q^{616}-q^{614}-q^{612}-q^{610}-q^{608}-q^{606}+q^{548}+q^{546}+q^{544}+q^{542}+q^{540}+q^{538}+q^{536}+q^{534}+q^{532}-q^{460}-q^{458}-q^{456}-q^{454}-q^{452}-q^{450}-q^{448}-q^{446}-q^{444}-q^{442}-q^{440}+q^{354}+q^{352}+q^{350}+q^{348}+q^{346}+q^{344}+q^{342}+q^{340}+q^{338}+q^{336}+q^{334}+q^{332}+q^{330}-q^{230}-q^{228}-q^{226}-q^{224}-q^{222}-q^{220}-q^{218}-q^{216}-q^{214}-q^{212}-q^{210}-q^{208}-q^{206}-q^{204}-q^{202}+q^{88}+q^{86}+q^{84}+q^{82}+q^{80}+q^{78}+q^{76}+q^{74}+q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56}} |
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^{38}-q^{36}-q^{34}+q^{22}+q^{20}+2 q^{18}+q^{16}+q^{14}} |
| 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^{108}-2 q^{60}-2 q^{58}-4 q^{56}-4 q^{54}-4 q^{52}-2 q^{50}-2 q^{48}+q^{44}+2 q^{42}+4 q^{40}+4 q^{38}+5 q^{36}+4 q^{34}+4 q^{32}+2 q^{30}+q^{28}} |
| 2,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{94}+q^{92}+2 q^{90}+q^{88}+q^{86}-q^{78}-2 q^{76}-3 q^{74}-3 q^{72}-3 q^{70}-2 q^{68}-q^{66}+q^{44}+q^{42}+2 q^{40}+2 q^{38}+3 q^{36}+2 q^{34}+2 q^{32}+q^{30}+q^{28}} |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{90}-q^{60}-2 q^{58}-3 q^{56}-3 q^{54}-3 q^{52}-2 q^{50}-q^{48}+q^{44}+q^{42}+3 q^{40}+3 q^{38}+4 q^{36}+3 q^{34}+3 q^{32}+q^{30}+q^{28}} |
| 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^{49}-q^{47}-2 q^{45}-q^{43}-q^{41}+q^{33}+q^{31}+2 q^{29}+2 q^{27}+2 q^{25}+q^{23}+q^{21}} |
| 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^{144}+q^{98}+q^{96}+3 q^{94}+3 q^{92}+4 q^{90}+3 q^{88}+3 q^{86}+q^{84}-q^{82}-4 q^{80}-8 q^{78}-10 q^{76}-14 q^{74}-14 q^{72}-14 q^{70}-10 q^{68}-7 q^{66}-2 q^{64}+3 q^{62}+7 q^{60}+10 q^{58}+11 q^{56}+12 q^{54}+11 q^{52}+10 q^{50}+7 q^{48}+5 q^{46}+2 q^{44}+q^{42}} |
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^{112}+q^{110}+q^{108}+q^{106}+q^{104}-q^{82}-2 q^{80}-4 q^{78}-5 q^{76}-7 q^{74}-7 q^{72}-7 q^{70}-5 q^{68}-3 q^{66}-q^{64}+2 q^{62}+4 q^{60}+6 q^{58}+6 q^{56}+8 q^{54}+6 q^{52}+6 q^{50}+4 q^{48}+3 q^{46}+q^{44}+q^{42}} |
| 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^{60}-q^{58}-2 q^{56}-2 q^{54}-2 q^{52}-q^{50}-q^{48}+q^{44}+q^{42}+2 q^{40}+2 q^{38}+3 q^{36}+2 q^{34}+2 q^{32}+q^{30}+q^{28}} |
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^{90}-q^{60}-q^{56}-q^{54}-q^{52}-q^{48}+q^{44}+q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+q^{32}+q^{30}+q^{28}} |
| 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^{144}-q^{98}-q^{96}-q^{94}-q^{92}-2 q^{90}-q^{88}-q^{86}-q^{84}-q^{82}+q^{66}+q^{62}+q^{60}+2 q^{58}+q^{56}+2 q^{54}+q^{52}+2 q^{50}+q^{48}+q^{46}+q^{42}} |
D4 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^{216}-q^{172}-q^{170}-3 q^{168}-3 q^{166}-4 q^{164}-4 q^{162}-4 q^{160}-3 q^{158}+2 q^{154}+5 q^{152}+9 q^{150}+12 q^{148}+12 q^{146}+15 q^{144}+12 q^{142}+12 q^{140}+9 q^{138}+6 q^{136}+3 q^{134}+3 q^{132}-q^{126}-3 q^{124}-6 q^{122}-10 q^{120}-16 q^{118}-22 q^{116}-28 q^{114}-33 q^{112}-36 q^{110}-37 q^{108}-33 q^{106}-27 q^{104}-18 q^{102}-8 q^{100}+4 q^{98}+12 q^{96}+22 q^{94}+26 q^{92}+29 q^{90}+29 q^{88}+28 q^{86}+22 q^{84}+20 q^{82}+14 q^{80}+10 q^{78}+6 q^{76}+4 q^{74}+q^{72}+q^{70}} |
| 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^{126}-q^{82}-q^{80}-3 q^{78}-3 q^{76}-4 q^{74}-4 q^{72}-4 q^{70}-3 q^{68}-2 q^{66}+q^{62}+3 q^{60}+4 q^{58}+4 q^{56}+5 q^{54}+4 q^{52}+4 q^{50}+3 q^{48}+2 q^{46}+q^{44}+q^{42}} |
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^{216}-q^{172}-q^{170}-q^{164}-q^{162}-q^{160}-q^{154}-q^{152}-q^{126}-q^{120}-q^{118}-q^{116}-q^{114}-q^{108}+q^{100}+q^{98}+q^{94}+2 q^{92}+2 q^{90}+2 q^{88}+q^{86}+q^{84}+2 q^{82}+2 q^{80}+q^{78}+q^{74}+q^{72}+q^{70}} |
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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["9 1"];
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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^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-4} } |
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^8+7 z^6+15 z^4+10 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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{ 9, -8 } |
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^{-4} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} + q^{-12} - q^{-13} } |
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^6 a^{10}-6 z^4 a^{10}-10 z^2 a^{10}-4 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+20 z^2 a^8+5 a^8} |
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 a^{17}+z^2 a^{16}+z^3 a^{15}-z a^{15}+z^4 a^{14}-2 z^2 a^{14}+z^5 a^{13}-3 z^3 a^{13}+z a^{13}+z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}+z^7 a^{11}-5 z^5 a^{11}+6 z^3 a^{11}-z a^{11}+z^8 a^{10}-7 z^6 a^{10}+16 z^4 a^{10}-14 z^2 a^{10}+4 a^{10}+z^7 a^9-6 z^5 a^9+10 z^3 a^9-4 z a^9+z^8 a^8-8 z^6 a^8+21 z^4 a^8-20 z^2 a^8+5 a^8} |
"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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 1"];
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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^4-t^3+t^2-t+1- t^{-1} + t^{-2} - t^{-3} + t^{-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^{-4} + q^{-6} - q^{-7} + q^{-8} - q^{-9} + q^{-10} - q^{-11} + q^{-12} - q^{-13} } } |
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: | (10, -30) |
| 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=} -8 is the signature of 9 1. 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^{-8} + q^{-11} - q^{-13} + q^{-14} - q^{-16} + q^{-17} - q^{-19} + q^{-20} - q^{-22} + q^{-23} - q^{-25} + q^{-26} - q^{-27} - q^{-28} + q^{-29} - q^{-31} + q^{-32} - q^{-34} + q^{-35} } |
| 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^{-12} + q^{-16} - q^{-19} + q^{-20} - q^{-23} + q^{-24} - q^{-27} + q^{-28} - q^{-31} + q^{-32} - q^{-35} + q^{-36} - q^{-39} - q^{-43} + q^{-45} - q^{-47} + q^{-49} - q^{-51} + q^{-53} - q^{-55} + q^{-57} + q^{-61} - q^{-62} + q^{-65} - q^{-66} } |
| 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^{-16} + q^{-21} - q^{-25} + q^{-26} - q^{-30} + q^{-31} - q^{-35} + q^{-36} - q^{-40} + q^{-41} - q^{-45} + q^{-46} - q^{-50} + q^{-51} - q^{-53} - q^{-55} + q^{-56} - q^{-58} + q^{-61} - q^{-63} + q^{-66} - q^{-68} + q^{-71} - q^{-73} + q^{-76} - q^{-78} +2 q^{-81} - q^{-83} + q^{-86} - q^{-88} + q^{-91} - q^{-93} + q^{-96} - q^{-98} - q^{-100} + q^{-101} - q^{-105} + q^{-106} } |
| 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^{-20} + q^{-26} - q^{-31} + q^{-32} - q^{-37} + q^{-38} - q^{-43} + q^{-44} - q^{-49} + q^{-50} - q^{-55} + q^{-56} - q^{-61} + q^{-62} - q^{-66} - q^{-67} + q^{-68} - q^{-72} - q^{-73} + q^{-74} + q^{-75} - q^{-78} - q^{-79} + q^{-80} + q^{-81} - q^{-84} - q^{-85} + q^{-86} + q^{-87} - q^{-90} - q^{-91} + q^{-92} + q^{-93} - q^{-96} - q^{-97} + q^{-98} + q^{-99} - q^{-102} + q^{-104} + q^{-105} - q^{-108} + q^{-111} - q^{-114} + q^{-117} - q^{-120} + q^{-123} - q^{-126} + q^{-129} - q^{-131} - q^{-132} + q^{-135} + q^{-136} - q^{-137} - q^{-138} + q^{-141} + q^{-142} - q^{-143} - q^{-144} + q^{-147} + q^{-148} - q^{-149} + q^{-154} - q^{-155} } |
| 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^{-24} + q^{-31} - q^{-37} + q^{-38} - q^{-44} + q^{-45} - q^{-51} + q^{-52} - q^{-58} + q^{-59} - q^{-65} + q^{-66} - q^{-72} + q^{-73} -2 q^{-79} + q^{-80} -2 q^{-86} + q^{-87} + q^{-90} -2 q^{-93} + q^{-94} + q^{-97} -2 q^{-100} + q^{-101} + q^{-104} -2 q^{-107} + q^{-108} + q^{-111} -2 q^{-114} + q^{-115} + q^{-118} -2 q^{-121} + q^{-122} +2 q^{-125} -2 q^{-128} + q^{-129} +2 q^{-132} - q^{-134} -2 q^{-135} + q^{-136} +2 q^{-139} - q^{-141} -2 q^{-142} + q^{-143} +2 q^{-146} - q^{-148} -2 q^{-149} + q^{-150} +2 q^{-153} - q^{-155} -2 q^{-156} + q^{-157} +2 q^{-160} -2 q^{-162} -2 q^{-163} + q^{-164} +2 q^{-167} - q^{-169} -2 q^{-170} + q^{-171} +2 q^{-174} - q^{-176} -2 q^{-177} + q^{-178} +2 q^{-181} - q^{-183} -2 q^{-184} + q^{-185} +2 q^{-188} -2 q^{-191} + q^{-192} + q^{-195} -2 q^{-198} + q^{-199} + q^{-202} -2 q^{-205} + q^{-206} - q^{-212} + q^{-213} } |
| 7 | 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^{-36} - q^{-43} + q^{-44} - q^{-51} + q^{-52} - q^{-59} + q^{-60} - q^{-67} + q^{-68} - q^{-75} + q^{-76} - q^{-83} + q^{-84} - q^{-91} - q^{-99} + q^{-105} - q^{-107} + q^{-113} - q^{-115} + q^{-121} - q^{-123} + q^{-129} - q^{-131} + q^{-137} - q^{-139} + q^{-145} + q^{-153} - q^{-158} + q^{-161} - q^{-166} + q^{-169} - q^{-174} + q^{-177} - q^{-182} + q^{-185} - q^{-190} - q^{-198} + q^{-202} - q^{-206} + q^{-210} - q^{-214} + q^{-218} - q^{-222} + q^{-226} + q^{-234} - q^{-237} + q^{-242} - q^{-245} + q^{-250} - q^{-253} - q^{-261} + q^{-263} - q^{-269} + q^{-271} + q^{-279} - q^{-280} } |
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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