9 4
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 9 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1627 X3,12,4,13 X7,18,8,1 X9,16,10,17 X15,10,16,11 X17,8,18,9 X5,14,6,15 X11,2,12,3 X13,4,14,5 |
| Gauss code | -1, 8, -2, 9, -7, 1, -3, 6, -4, 5, -8, 2, -9, 7, -5, 4, -6, 3 |
| Dowker-Thistlethwaite code | 6 12 14 18 16 2 4 10 8 |
| Conway Notation | [54] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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 [{11, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 1}] |
[edit Notes on presentations of 9 4]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 4"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1627 X3,12,4,13 X7,18,8,1 X9,16,10,17 X15,10,16,11 X17,8,18,9 X5,14,6,15 X11,2,12,3 X13,4,14,5 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 8, -2, 9, -7, 1, -3, 6, -4, 5, -8, 2, -9, 7, -5, 4, -6, 3 |
In[6]:=
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DTCode[K]
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Out[6]=
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6 12 14 18 16 2 4 10 8 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[54] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{-1,-1,-1,-1,-1,-2,1,-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[{11, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 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 3 t^2-5 t+5-5 t^{-1} +3 t^{-2} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^4+7 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, -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 q^{-2} - q^{-3} +2 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} - q^{-11} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^{10}-2 a^{10}+z^4 a^8+3 z^2 a^8+2 a^8+z^4 a^6+2 z^2 a^6+z^4 a^4+3 z^2 a^4+a^4} |
| 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^5 a^{13}-4 z^3 a^{13}+3 z a^{13}+z^6 a^{12}-3 z^4 a^{12}+z^2 a^{12}+z^7 a^{11}-3 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+z^8 a^{10}-5 z^6 a^{10}+11 z^4 a^{10}-10 z^2 a^{10}+2 a^{10}+2 z^7 a^9-8 z^5 a^9+12 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+11 z^4 a^8-7 z^2 a^8+2 a^8+z^7 a^7-3 z^5 a^7+4 z^3 a^7+z^6 a^6-2 z^4 a^6+z^2 a^6+z^5 a^5-2 z^3 a^5+z^4 a^4-3 z^2 a^4+a^4} |
| The A2 invariant | |
| 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^{176}+q^{172}-q^{170}+q^{168}-q^{164}+2 q^{162}-2 q^{160}+2 q^{158}-2 q^{156}+q^{152}-2 q^{150}+3 q^{148}-5 q^{146}+2 q^{144}-q^{142}-3 q^{140}+2 q^{138}-5 q^{136}+q^{134}+q^{132}-3 q^{130}-3 q^{126}-q^{124}+3 q^{122}-5 q^{120}+2 q^{118}-q^{116}-q^{114}+5 q^{112}-3 q^{110}+4 q^{108}-q^{106}+3 q^{104}+2 q^{102}-3 q^{100}+6 q^{98}-3 q^{96}+4 q^{94}+q^{92}-2 q^{90}+3 q^{88}-q^{86}+q^{82}-3 q^{80}+q^{78}-2 q^{74}+4 q^{72}-3 q^{70}+2 q^{68}-q^{64}+q^{62}-2 q^{60}+3 q^{58}-q^{56}+q^{54}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+q^{34}+q^{30}} |
Further Quantum Invariants
Further quantum knot invariants for 9_4.
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^{23}-q^{19}+q^{17}+q^{13}+q^{11}-q^9+q^7+q^3} |
| 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^{64}+q^{58}-q^{56}-2 q^{54}+q^{52}-2 q^{48}+q^{46}+q^{44}-2 q^{42}+q^{38}-q^{36}+q^{30}-2 q^{28}+3 q^{24}-q^{22}+2 q^{18}+q^{12}+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^{123}+q^{113}+q^{111}-q^{107}+q^{105}+2 q^{103}-3 q^{99}-2 q^{97}+2 q^{95}+2 q^{93}-3 q^{89}+3 q^{85}+2 q^{83}-3 q^{81}-2 q^{79}+2 q^{77}+3 q^{75}-2 q^{73}-2 q^{71}-2 q^{65}-q^{63}-q^{61}+2 q^{57}-2 q^{55}-2 q^{53}+q^{51}+4 q^{49}-4 q^{45}+3 q^{41}+2 q^{39}-q^{37}-q^{35}+q^{31}+q^{29}+q^{21}+q^{19}+q^{17}+q^9} |
| 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^{200}-q^{192}-q^{188}+q^{184}-q^{182}-2 q^{178}-q^{176}+3 q^{174}+2 q^{172}+3 q^{170}-2 q^{168}-4 q^{166}-q^{164}+q^{162}+6 q^{160}+q^{158}-3 q^{156}-4 q^{154}-5 q^{152}+3 q^{150}+4 q^{148}+4 q^{146}-8 q^{142}-3 q^{140}+2 q^{138}+7 q^{136}+6 q^{134}-5 q^{132}-6 q^{130}-2 q^{128}+7 q^{126}+8 q^{124}-2 q^{122}-5 q^{120}-3 q^{118}+3 q^{116}+4 q^{114}-q^{112}-2 q^{110}-2 q^{108}-2 q^{102}-q^{100}-q^{98}-q^{96}+q^{94}+4 q^{92}-q^{90}-3 q^{88}-4 q^{86}-q^{84}+8 q^{82}+2 q^{80}-3 q^{78}-9 q^{76}-5 q^{74}+9 q^{72}+6 q^{70}+2 q^{68}-6 q^{66}-8 q^{64}+q^{62}+4 q^{60}+6 q^{58}+q^{56}-5 q^{54}-2 q^{52}-q^{50}+4 q^{48}+4 q^{46}-q^{42}-3 q^{40}+q^{38}+2 q^{36}+q^{34}+q^{32}-2 q^{30}+q^{26}+q^{24}+2 q^{22}+q^{12}} |
| 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^{295}+q^{287}+q^{285}-q^{277}+2 q^{273}+q^{271}-q^{267}-3 q^{265}-3 q^{263}+3 q^{259}+3 q^{257}+2 q^{255}-2 q^{253}-5 q^{251}-5 q^{249}+5 q^{245}+8 q^{243}+5 q^{241}-q^{239}-6 q^{237}-8 q^{235}-4 q^{233}+3 q^{231}+7 q^{229}+8 q^{227}+3 q^{225}-3 q^{223}-10 q^{221}-11 q^{219}-3 q^{217}+6 q^{215}+13 q^{213}+11 q^{211}-q^{209}-15 q^{207}-16 q^{205}-6 q^{203}+8 q^{201}+19 q^{199}+14 q^{197}-4 q^{195}-18 q^{193}-16 q^{191}-q^{189}+16 q^{187}+19 q^{185}+4 q^{183}-11 q^{181}-16 q^{179}-6 q^{177}+10 q^{175}+12 q^{173}+5 q^{171}-3 q^{169}-8 q^{167}-4 q^{165}+3 q^{163}+5 q^{161}+q^{159}-q^{157}-q^{155}-q^{153}+q^{151}+q^{149}-q^{147}-q^{143}-q^{141}-q^{139}+2 q^{137}+6 q^{135}+2 q^{133}-2 q^{131}-7 q^{129}-11 q^{127}-2 q^{125}+11 q^{123}+14 q^{121}+3 q^{119}-12 q^{117}-22 q^{115}-11 q^{113}+10 q^{111}+25 q^{109}+16 q^{107}-7 q^{105}-21 q^{103}-20 q^{101}-3 q^{99}+16 q^{97}+20 q^{95}+8 q^{93}-8 q^{91}-16 q^{89}-12 q^{87}+10 q^{83}+11 q^{81}+4 q^{79}-4 q^{77}-7 q^{75}-5 q^{73}+4 q^{69}+5 q^{67}+2 q^{65}-q^{63}-3 q^{61}-2 q^{59}+q^{57}+2 q^{55}+3 q^{53}+q^{51}-2 q^{49}-q^{47}+q^{43}+2 q^{41}+2 q^{39}-q^{37}-q^{35}+q^{29}+2 q^{27}+q^{25}+q^{15}} |
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^{34}-q^{32}-q^{30}-q^{28}+q^{26}+q^{24}+q^{22}+q^{20}+q^{16}+q^{10}+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^{92}+2 q^{88}-2 q^{86}+4 q^{84}-4 q^{82}+4 q^{80}-8 q^{78}+5 q^{76}-8 q^{74}+8 q^{72}-6 q^{70}+3 q^{68}+2 q^{66}-2 q^{64}+8 q^{62}-12 q^{60}+12 q^{58}-14 q^{56}+10 q^{54}-15 q^{52}+10 q^{50}-10 q^{48}+6 q^{46}-3 q^{44}-2 q^{42}+2 q^{40}-4 q^{38}+7 q^{36}-4 q^{34}+6 q^{32}+7 q^{28}-2 q^{26}+4 q^{24}-2 q^{22}+4 q^{20}-2 q^{18}+2 q^{16}+q^{12}} |
| 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^{86}+q^{84}+2 q^{82}+q^{80}+q^{78}-q^{76}-2 q^{74}-2 q^{72}-2 q^{70}-2 q^{68}-2 q^{66}+q^{62}+q^{60}-q^{58}-q^{52}-q^{50}-q^{44}-q^{42}+q^{40}+q^{36}+3 q^{34}+2 q^{32}+q^{30}+2 q^{28}+2 q^{26}-q^{22}+q^{20}+q^{18}+q^{12}} |
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^{74}+q^{70}+q^{68}-3 q^{60}-q^{58}-q^{56}-4 q^{54}-q^{52}-q^{48}-q^{46}+q^{44}+q^{40}+q^{38}+3 q^{36}+q^{34}+3 q^{30}-q^{26}+2 q^{24}+q^{22}+q^{18}+q^{16}+q^{12}} |
| 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^{45}-q^{43}-2 q^{41}-q^{39}-q^{37}+q^{35}+q^{33}+2 q^{31}+q^{29}+q^{27}+q^{21}+q^{17}+q^{13}+q^9} |
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^{96}+q^{94}+q^{92}+2 q^{90}+3 q^{88}+q^{86}+q^{84}+q^{82}-q^{80}-5 q^{78}-6 q^{76}-5 q^{74}-6 q^{72}-6 q^{70}-2 q^{68}+2 q^{62}+2 q^{60}+q^{58}+2 q^{54}+2 q^{52}+2 q^{50}+2 q^{48}+4 q^{46}+2 q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+2 q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+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^{56}-q^{54}-2 q^{52}-2 q^{50}-q^{48}-q^{46}+q^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{36}+q^{34}+q^{26}+q^{22}+q^{20}+q^{16}+q^{12}} |
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^{74}-q^{70}+q^{68}-2 q^{66}+2 q^{64}-2 q^{62}+q^{60}-q^{58}+q^{56}-q^{52}+2 q^{50}-3 q^{48}+3 q^{46}-3 q^{44}+4 q^{42}-3 q^{40}+3 q^{38}-q^{36}+q^{34}-q^{30}+2 q^{28}-q^{26}+2 q^{24}-q^{22}+2 q^{20}-q^{18}+q^{16}+q^{12}} |
| 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^{120}+q^{112}+q^{110}-q^{106}+q^{102}+q^{100}-2 q^{98}-3 q^{96}-q^{94}+q^{92}-3 q^{88}-2 q^{86}+q^{82}-q^{78}+q^{74}-q^{70}-q^{68}+q^{66}+q^{64}-q^{60}+q^{58}+2 q^{56}+q^{54}-q^{52}+3 q^{48}+2 q^{46}-q^{44}-q^{42}+2 q^{38}+q^{36}-q^{32}+q^{28}+q^{26}+q^{18}} |
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^{102}+q^{98}+2 q^{94}-q^{92}+q^{90}-2 q^{88}+q^{86}-2 q^{84}-2 q^{80}-q^{78}-2 q^{76}-3 q^{74}-q^{72}-3 q^{70}-4 q^{66}+2 q^{64}-2 q^{62}+4 q^{60}-q^{58}+4 q^{56}+4 q^{52}+q^{50}+2 q^{48}+2 q^{42}-q^{40}+q^{38}-q^{36}+2 q^{34}+2 q^{30}+2 q^{26}+q^{22}+q^{18}} |
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^{176}+q^{172}-q^{170}+q^{168}-q^{164}+2 q^{162}-2 q^{160}+2 q^{158}-2 q^{156}+q^{152}-2 q^{150}+3 q^{148}-5 q^{146}+2 q^{144}-q^{142}-3 q^{140}+2 q^{138}-5 q^{136}+q^{134}+q^{132}-3 q^{130}-3 q^{126}-q^{124}+3 q^{122}-5 q^{120}+2 q^{118}-q^{116}-q^{114}+5 q^{112}-3 q^{110}+4 q^{108}-q^{106}+3 q^{104}+2 q^{102}-3 q^{100}+6 q^{98}-3 q^{96}+4 q^{94}+q^{92}-2 q^{90}+3 q^{88}-q^{86}+q^{82}-3 q^{80}+q^{78}-2 q^{74}+4 q^{72}-3 q^{70}+2 q^{68}-q^{64}+q^{62}-2 q^{60}+3 q^{58}-q^{56}+q^{54}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+q^{34}+q^{30}} |
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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 4"];
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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 3 t^2-5 t+5-5 t^{-1} +3 t^{-2} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^4+7 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, -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 q^{-2} - q^{-3} +2 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} - q^{-11} } |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^{10}-2 a^{10}+z^4 a^8+3 z^2 a^8+2 a^8+z^4 a^6+2 z^2 a^6+z^4 a^4+3 z^2 a^4+a^4} |
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^5 a^{13}-4 z^3 a^{13}+3 z a^{13}+z^6 a^{12}-3 z^4 a^{12}+z^2 a^{12}+z^7 a^{11}-3 z^5 a^{11}+2 z^3 a^{11}-z a^{11}+z^8 a^{10}-5 z^6 a^{10}+11 z^4 a^{10}-10 z^2 a^{10}+2 a^{10}+2 z^7 a^9-8 z^5 a^9+12 z^3 a^9-4 z a^9+z^8 a^8-5 z^6 a^8+11 z^4 a^8-7 z^2 a^8+2 a^8+z^7 a^7-3 z^5 a^7+4 z^3 a^7+z^6 a^6-2 z^4 a^6+z^2 a^6+z^5 a^5-2 z^3 a^5+z^4 a^4-3 z^2 a^4+a^4} |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["9 4"];
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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 3 t^2-5 t+5-5 t^{-1} +3 t^{-2} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} - q^{-3} +2 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +3 q^{-8} -2 q^{-9} + q^{-10} - q^{-11} } } |
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]=
|
{} |
Vassiliev invariants
| V2 and V3: | (7, -19) |
| 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 9 4. 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^{-4} - q^{-5} +2 q^{-7} -2 q^{-8} +4 q^{-10} -4 q^{-11} - q^{-12} +8 q^{-13} -7 q^{-14} -3 q^{-15} +11 q^{-16} -8 q^{-17} -3 q^{-18} +10 q^{-19} -6 q^{-20} -4 q^{-21} +8 q^{-22} -3 q^{-23} -4 q^{-24} +5 q^{-25} - q^{-26} -3 q^{-27} +2 q^{-28} - q^{-30} + q^{-31} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} - q^{-7} +2 q^{-10} - q^{-11} - q^{-13} +2 q^{-14} - q^{-15} + q^{-16} - q^{-17} + q^{-18} -2 q^{-19} + q^{-20} +2 q^{-21} +2 q^{-22} -5 q^{-23} -3 q^{-24} +6 q^{-25} +6 q^{-26} -8 q^{-27} -6 q^{-28} +6 q^{-29} +10 q^{-30} -10 q^{-31} -7 q^{-32} +6 q^{-33} +9 q^{-34} -8 q^{-35} -7 q^{-36} +4 q^{-37} +9 q^{-38} -3 q^{-39} -8 q^{-40} +8 q^{-42} +2 q^{-43} -7 q^{-44} -3 q^{-45} +5 q^{-46} +5 q^{-47} -5 q^{-48} -3 q^{-49} + q^{-50} +4 q^{-51} -2 q^{-52} - q^{-53} +2 q^{-55} - q^{-56} + q^{-59} - q^{-60} } |
| 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^{-8} - q^{-9} +3 q^{-13} -2 q^{-14} - q^{-16} -2 q^{-17} +6 q^{-18} -2 q^{-19} + q^{-20} -2 q^{-21} -6 q^{-22} +8 q^{-23} - q^{-24} +5 q^{-25} -2 q^{-26} -11 q^{-27} +7 q^{-28} -4 q^{-29} +11 q^{-30} +3 q^{-31} -13 q^{-32} +4 q^{-33} -13 q^{-34} +13 q^{-35} +11 q^{-36} -9 q^{-37} +7 q^{-38} -27 q^{-39} +9 q^{-40} +17 q^{-41} -4 q^{-42} +13 q^{-43} -36 q^{-44} +6 q^{-45} +18 q^{-46} -2 q^{-47} +18 q^{-48} -39 q^{-49} +4 q^{-50} +18 q^{-51} -2 q^{-52} +17 q^{-53} -37 q^{-54} +4 q^{-55} +16 q^{-56} -2 q^{-57} +18 q^{-58} -32 q^{-59} +3 q^{-60} +10 q^{-61} -4 q^{-62} +21 q^{-63} -22 q^{-64} +2 q^{-65} + q^{-66} -8 q^{-67} +22 q^{-68} -11 q^{-69} +3 q^{-70} -4 q^{-71} -13 q^{-72} +17 q^{-73} -3 q^{-74} +7 q^{-75} -4 q^{-76} -14 q^{-77} +9 q^{-78} -2 q^{-79} +8 q^{-80} -9 q^{-82} +4 q^{-83} -4 q^{-84} +5 q^{-85} +2 q^{-86} -4 q^{-87} +3 q^{-88} -3 q^{-89} + q^{-90} + q^{-91} -2 q^{-92} +2 q^{-93} - q^{-94} - q^{-97} + q^{-98} } |
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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