10 48: Difference between revisions
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{{Rolfsen Knot Page| |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
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<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15: |
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 48]]</nowiki></pre></td></tr> |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 48]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[20, 15, 1, 16], |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[20, 15, 1, 16], |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>3</nowiki></pre></td></tr> |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>3</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 48]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_48_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 48]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_48_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[10, 48]]&) /@ { |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki> (#[Knot[10, 48]]&) /@ { |
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SymmetryType, UnknottingNumber, ThreeGenus, |
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BridgeIndex, SuperBridgeIndex, NakanishiIndex |
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}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 4, 3, NotAvailable, 1}</nowiki></pre></td></tr> |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 4, 3, NotAvailable, 1}</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 48]][t]</nowiki></pre></td></tr> |
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 48]][t]</nowiki></pre></td></tr> |
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Revision as of 18:42, 31 August 2005
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 48's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X6271 X8493 X14,6,15,5 X20,15,1,16 X16,9,17,10 X18,11,19,12 X10,17,11,18 X12,19,13,20 X2837 X4,14,5,13 |
| Gauss code | 1, -9, 2, -10, 3, -1, 9, -2, 5, -7, 6, -8, 10, -3, 4, -5, 7, -6, 8, -4 |
| Dowker-Thistlethwaite code | 6 8 14 2 16 18 4 20 10 12 |
| Conway Notation | [41,3,2] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||
Length is 10, width is 3, Braid index is 3 |
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 [{5, 13}, {2, 12}, {13, 11}, {12, 6}, {1, 4}, {3, 5}, {4, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 3}, {11, 2}, {10, 1}] |
[edit Notes on presentations of 10 48]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 48"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X6271 X8493 X14,6,15,5 X20,15,1,16 X16,9,17,10 X18,11,19,12 X10,17,11,18 X12,19,13,20 X2837 X4,14,5,13 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -9, 2, -10, 3, -1, 9, -2, 5, -7, 6, -8, 10, -3, 4, -5, 7, -6, 8, -4 |
In[6]:=
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DTCode[K]
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Out[6]=
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6 8 14 2 16 18 4 20 10 12 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[41,3,2] |
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,-1,2,2,-1,2,2,2\})} |
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[{5, 13}, {2, 12}, {13, 11}, {12, 6}, {1, 4}, {3, 5}, {4, 7}, {6, 8}, {7, 9}, {8, 10}, {9, 3}, {11, 2}, {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 | |
| 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+5 z^6+8 z^4+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 | { 49, 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^5+2 q^4-4 q^3+6 q^2-7 q+9-7 q^{-1} +6 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} } |
| 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^8-a^2 z^6-z^6 a^{-2} +7 z^6-5 a^2 z^4-5 z^4 a^{-2} +18 z^4-8 a^2 z^2-8 z^2 a^{-2} +20 z^2-4 a^2-4 a^{-2} +9} |
| 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 z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+2 a^3 z^7+z^7 a^{-1} +3 z^7 a^{-3} +2 a^4 z^6-5 a^2 z^6-11 z^6 a^{-2} +2 z^6 a^{-4} -20 z^6+a^5 z^5-3 a^3 z^5-5 a z^5-11 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+9 a^2 z^4+18 z^4 a^{-2} -5 z^4 a^{-4} +37 z^4-3 a^5 z^3-a^3 z^3+12 a z^3+21 z^3 a^{-1} +8 z^3 a^{-3} -3 z^3 a^{-5} +2 a^4 z^2-11 a^2 z^2-13 z^2 a^{-2} +z^2 a^{-4} -27 z^2+2 a^5 z-7 a z-9 z a^{-1} -3 z a^{-3} +z a^{-5} +4 a^2+4 a^{-2} +9} |
| 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^{14}-2 q^{10}+4 q^2+1+4 q^{-2} -2 q^{-10} - q^{-14} } |
| The G2 invariant |
Further Quantum Invariants
Further quantum knot invariants for 10_48.
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^{11}+q^9-2 q^7+2 q^5-q^3+2 q+2 q^{-1} - q^{-3} +2 q^{-5} -2 q^{-7} + q^{-9} - q^{-11} } |
| 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^{32}-q^{30}-q^{28}+3 q^{26}-2 q^{24}-3 q^{22}+6 q^{20}-3 q^{18}-7 q^{16}+9 q^{14}-10 q^{10}+7 q^8+4 q^6-7 q^4+2 q^2+6+2 q^{-2} -6 q^{-4} +4 q^{-6} +7 q^{-8} -10 q^{-10} + q^{-12} +9 q^{-14} -8 q^{-16} -3 q^{-18} +7 q^{-20} -3 q^{-22} -4 q^{-24} +3 q^{-26} - q^{-30} + q^{-32} } |
| 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^{63}+q^{61}+q^{59}-2 q^{55}+2 q^{51}-2 q^{47}+2 q^{45}+3 q^{43}-4 q^{41}-7 q^{39}+6 q^{37}+13 q^{35}-6 q^{33}-21 q^{31}-2 q^{29}+29 q^{27}+10 q^{25}-29 q^{23}-23 q^{21}+28 q^{19}+29 q^{17}-19 q^{15}-35 q^{13}+11 q^{11}+30 q^9-26 q^5-5 q^3+19 q+16 q^{-1} -7 q^{-3} -23 q^{-5} +5 q^{-7} +28 q^{-9} +5 q^{-11} -34 q^{-13} -10 q^{-15} +35 q^{-17} +20 q^{-19} -33 q^{-21} -26 q^{-23} +22 q^{-25} +30 q^{-27} -12 q^{-29} -28 q^{-31} -2 q^{-33} +23 q^{-35} +7 q^{-37} -15 q^{-39} -11 q^{-41} +6 q^{-43} +10 q^{-45} -2 q^{-47} -5 q^{-49} + q^{-51} +3 q^{-53} - q^{-57} + q^{-61} - q^{-63} } |
| 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^{104}-q^{102}-q^{100}-q^{96}+4 q^{94}-q^{88}-8 q^{86}+4 q^{84}+q^{82}+7 q^{80}+8 q^{78}-13 q^{76}-6 q^{74}-14 q^{72}+10 q^{70}+32 q^{68}+5 q^{66}-6 q^{64}-50 q^{62}-24 q^{60}+40 q^{58}+51 q^{56}+46 q^{54}-62 q^{52}-94 q^{50}-24 q^{48}+61 q^{46}+140 q^{44}+15 q^{42}-116 q^{40}-130 q^{38}-19 q^{36}+174 q^{34}+126 q^{32}-41 q^{30}-168 q^{28}-120 q^{26}+108 q^{24}+156 q^{22}+47 q^{20}-109 q^{18}-140 q^{16}+19 q^{14}+103 q^{12}+77 q^{10}-33 q^8-96 q^6-31 q^4+46 q^2+75+19 q^{-2} -54 q^{-4} -81 q^{-6} +6 q^{-8} +94 q^{-10} +72 q^{-12} -32 q^{-14} -144 q^{-16} -47 q^{-18} +107 q^{-20} +141 q^{-22} +22 q^{-24} -178 q^{-26} -123 q^{-28} +57 q^{-30} +169 q^{-32} +108 q^{-34} -119 q^{-36} -153 q^{-38} -46 q^{-40} +100 q^{-42} +148 q^{-44} -4 q^{-46} -86 q^{-48} -99 q^{-50} -14 q^{-52} +91 q^{-54} +57 q^{-56} +12 q^{-58} -60 q^{-60} -60 q^{-62} +11 q^{-64} +30 q^{-66} +45 q^{-68} -2 q^{-70} -32 q^{-72} -13 q^{-74} -4 q^{-76} +22 q^{-78} +8 q^{-80} -6 q^{-82} -2 q^{-84} -7 q^{-86} +4 q^{-88} +2 q^{-90} -2 q^{-92} + q^{-94} -2 q^{-96} + q^{-98} - q^{-102} + q^{-104} } |
| 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^{155}+q^{153}+q^{151}+q^{147}-q^{145}-4 q^{143}-2 q^{141}+2 q^{139}+3 q^{137}+7 q^{135}+5 q^{133}-7 q^{131}-13 q^{129}-9 q^{127}+15 q^{123}+25 q^{121}+11 q^{119}-17 q^{117}-36 q^{115}-32 q^{113}+3 q^{111}+48 q^{109}+64 q^{107}+24 q^{105}-49 q^{103}-100 q^{101}-73 q^{99}+24 q^{97}+128 q^{95}+151 q^{93}+42 q^{91}-132 q^{89}-232 q^{87}-157 q^{85}+73 q^{83}+293 q^{81}+315 q^{79}+69 q^{77}-290 q^{75}-471 q^{73}-284 q^{71}+170 q^{69}+564 q^{67}+550 q^{65}+53 q^{63}-556 q^{61}-765 q^{59}-357 q^{57}+392 q^{55}+893 q^{53}+664 q^{51}-149 q^{49}-860 q^{47}-878 q^{45}-155 q^{43}+713 q^{41}+962 q^{39}+398 q^{37}-472 q^{35}-901 q^{33}-550 q^{31}+238 q^{29}+735 q^{27}+573 q^{25}-33 q^{23}-536 q^{21}-523 q^{19}-76 q^{17}+347 q^{15}+409 q^{13}+137 q^{11}-204 q^9-329 q^7-150 q^5+123 q^3+271 q+181 q^{-1} -63 q^{-3} -264 q^{-5} -255 q^{-7} +15 q^{-9} +313 q^{-11} +355 q^{-13} +72 q^{-15} -343 q^{-17} -519 q^{-19} -203 q^{-21} +368 q^{-23} +669 q^{-25} +383 q^{-27} -300 q^{-29} -796 q^{-31} -596 q^{-33} +169 q^{-35} +829 q^{-37} +787 q^{-39} +51 q^{-41} -752 q^{-43} -910 q^{-45} -298 q^{-47} +547 q^{-49} +916 q^{-51} +524 q^{-53} -272 q^{-55} -781 q^{-57} -648 q^{-59} -42 q^{-61} +539 q^{-63} +648 q^{-65} +268 q^{-67} -243 q^{-69} -505 q^{-71} -390 q^{-73} -31 q^{-75} +301 q^{-77} +371 q^{-79} +195 q^{-81} -75 q^{-83} -258 q^{-85} -250 q^{-87} -85 q^{-89} +116 q^{-91} +207 q^{-93} +152 q^{-95} +7 q^{-97} -117 q^{-99} -146 q^{-101} -70 q^{-103} +42 q^{-105} +98 q^{-107} +77 q^{-109} +9 q^{-111} -48 q^{-113} -55 q^{-115} -27 q^{-117} +16 q^{-119} +31 q^{-121} +18 q^{-123} - q^{-125} -11 q^{-127} -13 q^{-129} -5 q^{-131} +7 q^{-133} +5 q^{-135} - q^{-141} -2 q^{-143} + q^{-145} +2 q^{-147} - q^{-149} + q^{-153} - 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^{216}-q^{214}-q^{212}-q^{208}+q^{206}+q^{204}+6 q^{202}-4 q^{198}-3 q^{196}-8 q^{194}-4 q^{192}+q^{190}+19 q^{188}+11 q^{186}+2 q^{184}-2 q^{182}-20 q^{180}-25 q^{178}-19 q^{176}+25 q^{174}+30 q^{172}+28 q^{170}+23 q^{168}-18 q^{166}-55 q^{164}-67 q^{162}+3 q^{160}+38 q^{158}+69 q^{156}+83 q^{154}+17 q^{152}-84 q^{150}-156 q^{148}-76 q^{146}+10 q^{144}+130 q^{142}+229 q^{140}+163 q^{138}-49 q^{136}-294 q^{134}-323 q^{132}-235 q^{130}+59 q^{128}+448 q^{126}+602 q^{124}+376 q^{122}-171 q^{120}-633 q^{118}-922 q^{116}-655 q^{114}+174 q^{112}+1036 q^{110}+1402 q^{108}+921 q^{106}-115 q^{104}-1445 q^{102}-2079 q^{100}-1427 q^{98}+215 q^{96}+1980 q^{94}+2741 q^{92}+2044 q^{90}-251 q^{88}-2665 q^{86}-3637 q^{84}-2439 q^{82}+419 q^{80}+3271 q^{78}+4510 q^{76}+2781 q^{74}-794 q^{72}-4072 q^{70}-4962 q^{68}-2786 q^{66}+1209 q^{64}+4754 q^{62}+5147 q^{60}+2357 q^{58}-1920 q^{56}-4923 q^{54}-4810 q^{52}-1698 q^{50}+2545 q^{48}+4784 q^{46}+3960 q^{44}+695 q^{42}-2703 q^{40}-4167 q^{38}-2905 q^{36}+178 q^{34}+2628 q^{32}+3193 q^{30}+1665 q^{28}-609 q^{26}-2218 q^{24}-2198 q^{22}-708 q^{20}+856 q^{18}+1653 q^{16}+1220 q^{14}+152 q^{12}-884 q^{10}-1173 q^8-592 q^6+280 q^4+911 q^2+819+205 q^{-2} -632 q^{-4} -1041 q^{-6} -761 q^{-8} +175 q^{-10} +1145 q^{-12} +1373 q^{-14} +696 q^{-16} -683 q^{-18} -1787 q^{-20} -1894 q^{-22} -502 q^{-24} +1497 q^{-26} +2684 q^{-28} +2193 q^{-30} +17 q^{-32} -2435 q^{-34} -3644 q^{-36} -2256 q^{-38} +828 q^{-40} +3587 q^{-42} +4179 q^{-44} +1947 q^{-46} -1702 q^{-48} -4616 q^{-50} -4393 q^{-52} -1315 q^{-54} +2668 q^{-56} +5038 q^{-58} +4195 q^{-60} +737 q^{-62} -3363 q^{-64} -5078 q^{-66} -3674 q^{-68} -102 q^{-70} +3396 q^{-72} +4701 q^{-74} +3177 q^{-76} -304 q^{-78} -3142 q^{-80} -4005 q^{-82} -2528 q^{-84} +212 q^{-86} +2603 q^{-88} +3315 q^{-90} +1973 q^{-92} -77 q^{-94} -1882 q^{-96} -2452 q^{-98} -1718 q^{-100} -144 q^{-102} +1274 q^{-104} +1701 q^{-106} +1378 q^{-108} +391 q^{-110} -617 q^{-112} -1230 q^{-114} -1091 q^{-116} -476 q^{-118} +167 q^{-120} +751 q^{-122} +870 q^{-124} +585 q^{-126} + q^{-128} -423 q^{-130} -609 q^{-132} -549 q^{-134} -151 q^{-136} +241 q^{-138} +472 q^{-140} +380 q^{-142} +175 q^{-144} -98 q^{-146} -319 q^{-148} -291 q^{-150} -128 q^{-152} +77 q^{-154} +158 q^{-156} +183 q^{-158} +99 q^{-160} -46 q^{-162} -106 q^{-164} -98 q^{-166} -31 q^{-168} +5 q^{-170} +53 q^{-172} +60 q^{-174} +13 q^{-176} -15 q^{-178} -26 q^{-180} -12 q^{-182} -12 q^{-184} +6 q^{-186} +18 q^{-188} +5 q^{-190} -2 q^{-192} -5 q^{-194} + q^{-196} -4 q^{-198} - q^{-200} +5 q^{-202} - q^{-206} -2 q^{-208} + q^{-210} - q^{-214} + q^{-216} } |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | |
| 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^{44}-2 q^{42}+6 q^{40}-12 q^{38}+19 q^{36}-28 q^{34}+40 q^{32}-52 q^{30}+62 q^{28}-74 q^{26}+88 q^{24}-100 q^{22}+100 q^{20}-100 q^{18}+88 q^{16}-64 q^{14}+15 q^{12}+30 q^{10}-88 q^8+142 q^6-188 q^4+228 q^2-232+246 q^{-2} -208 q^{-4} +182 q^{-6} -124 q^{-8} +72 q^{-10} -11 q^{-12} -50 q^{-14} +80 q^{-16} -116 q^{-18} +122 q^{-20} -124 q^{-22} +106 q^{-24} -88 q^{-26} +72 q^{-28} -52 q^{-30} +34 q^{-32} -24 q^{-34} +17 q^{-36} -8 q^{-38} +4 q^{-40} -2 q^{-42} + q^{-44} } |
| 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^{38}+q^{32}-q^{28}+q^{26}-3 q^{22}-3 q^{20}+q^{18}-q^{16}-7 q^{14}-q^{12}+2 q^{10}-3 q^8-2 q^6+7 q^4+6 q^2+4+7 q^{-2} +8 q^{-4} + q^{-6} - q^{-8} +4 q^{-10} - q^{-12} -6 q^{-14} -5 q^{-20} -5 q^{-22} -2 q^{-28} +2 q^{-32} + q^{-34} + q^{-38} } |
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^{34}-q^{32}+q^{30}+2 q^{28}-3 q^{26}+q^{24}+3 q^{22}-7 q^{20}+4 q^{16}-12 q^{14}-4 q^{12}+4 q^{10}-7 q^8+10 q^4+8 q^2+8+7 q^{-2} +10 q^{-4} - q^{-6} -7 q^{-8} +3 q^{-10} -4 q^{-12} -12 q^{-14} +4 q^{-16} -7 q^{-20} +4 q^{-22} + q^{-24} -2 q^{-26} +2 q^{-28} - q^{-32} + q^{-34} } |
| 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^{17}-3 q^{13}-3 q^9+q^7+4 q^3+4 q+4 q^{-1} +4 q^{-3} + q^{-7} -3 q^{-9} -3 q^{-13} - q^{-17} } |
| 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^{56}-2 q^{54}+5 q^{52}-5 q^{50}+2 q^{48}+5 q^{46}-15 q^{44}+20 q^{42}-16 q^{40}+2 q^{38}+16 q^{36}-31 q^{34}+33 q^{32}-10 q^{30}-19 q^{28}+53 q^{26}-58 q^{24}+34 q^{22}+15 q^{20}-86 q^{18}+94 q^{16}-109 q^{14}+20 q^{12}+19 q^{10}-84 q^8+87 q^6-40 q^4+45 q^2+30+39 q^{-2} -2 q^{-4} +65 q^{-6} -33 q^{-8} -7 q^{-10} +50 q^{-12} -121 q^{-14} +91 q^{-16} -82 q^{-18} -10 q^{-20} +50 q^{-22} -83 q^{-24} +69 q^{-26} -32 q^{-28} -2 q^{-30} +32 q^{-32} -33 q^{-34} +25 q^{-36} -8 q^{-38} -8 q^{-40} +14 q^{-42} -12 q^{-44} +7 q^{-46} + q^{-48} -3 q^{-50} +3 q^{-52} -2 q^{-54} + q^{-56} } |
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^{40}+q^{36}+3 q^{34}+q^{32}+4 q^{28}-q^{26}-6 q^{24}-2 q^{22}-3 q^{20}-14 q^{18}-13 q^{16}-6 q^{14}-8 q^{12}-11 q^{10}+3 q^8+16 q^6+10 q^4+20 q^2+29+19 q^{-2} +7 q^{-4} +13 q^{-6} -13 q^{-10} -11 q^{-12} -7 q^{-14} -12 q^{-16} -13 q^{-18} -2 q^{-20} -3 q^{-24} +4 q^{-28} + q^{-32} +2 q^{-34} + q^{-40} } |
| 1,0,0,0 |
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^{34}+q^{32}-3 q^{30}+4 q^{28}-5 q^{26}+7 q^{24}-9 q^{22}+9 q^{20}-10 q^{18}+8 q^{16}-6 q^{14}+2 q^{12}+2 q^{10}-7 q^8+12 q^6-14 q^4+20 q^2-18+21 q^{-2} -14 q^{-4} +13 q^{-6} -7 q^{-8} +3 q^{-10} +2 q^{-12} -6 q^{-14} +8 q^{-16} -10 q^{-18} +9 q^{-20} -10 q^{-22} +7 q^{-24} -6 q^{-26} +4 q^{-28} -2 q^{-30} + q^{-32} - q^{-34} } |
| 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^{56}-q^{52}-q^{50}+2 q^{48}+3 q^{46}-q^{44}-4 q^{42}-2 q^{40}+4 q^{38}+6 q^{36}-3 q^{34}-9 q^{32}-4 q^{30}+7 q^{28}+7 q^{26}-6 q^{24}-12 q^{22}-3 q^{20}+8 q^{18}+4 q^{16}-7 q^{14}-6 q^{12}+5 q^{10}+8 q^8+q^6-4 q^4+5 q^2+10+5 q^{-2} -4 q^{-4} + q^{-6} +7 q^{-8} +5 q^{-10} -6 q^{-12} -7 q^{-14} +3 q^{-16} +8 q^{-18} -2 q^{-20} -12 q^{-22} -7 q^{-24} +7 q^{-26} +8 q^{-28} -4 q^{-30} -10 q^{-32} -3 q^{-34} +7 q^{-36} +4 q^{-38} -2 q^{-40} -4 q^{-42} +3 q^{-46} + q^{-48} - q^{-50} - q^{-52} + q^{-56} } |
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^{46}-q^{44}+2 q^{42}-2 q^{40}+4 q^{38}-4 q^{36}+4 q^{34}-6 q^{32}+7 q^{30}-8 q^{28}+6 q^{26}-8 q^{24}+5 q^{22}-9 q^{20}-2 q^{18}-6 q^{16}-5 q^{14}-10 q^{10}+10 q^8-5 q^6+23 q^4-4 q^2+26-4 q^{-2} +23 q^{-4} -6 q^{-6} +10 q^{-8} -11 q^{-10} -6 q^{-14} -6 q^{-16} -3 q^{-18} -9 q^{-20} +5 q^{-22} -8 q^{-24} +7 q^{-26} -8 q^{-28} +8 q^{-30} -6 q^{-32} +5 q^{-34} -4 q^{-36} +4 q^{-38} -2 q^{-40} + q^{-42} - q^{-44} + q^{-46} } |
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^{80}-q^{78}+3 q^{76}-4 q^{74}+3 q^{72}-q^{70}-3 q^{68}+8 q^{66}-11 q^{64}+13 q^{62}-11 q^{60}+3 q^{58}+4 q^{56}-14 q^{54}+21 q^{52}-25 q^{50}+21 q^{48}-15 q^{46}-3 q^{44}+18 q^{42}-33 q^{40}+35 q^{38}-29 q^{36}+11 q^{34}+8 q^{32}-27 q^{30}+32 q^{28}-22 q^{26}+4 q^{24}+17 q^{22}-28 q^{20}+22 q^{18}-2 q^{16}-21 q^{14}+41 q^{12}-41 q^{10}+34 q^8-4 q^6-24 q^4+51 q^2-55+51 q^{-2} -23 q^{-4} -5 q^{-6} +34 q^{-8} -41 q^{-10} +44 q^{-12} -25 q^{-14} +3 q^{-16} +19 q^{-18} -30 q^{-20} +23 q^{-22} -5 q^{-24} -17 q^{-26} +32 q^{-28} -32 q^{-30} +14 q^{-32} +7 q^{-34} -31 q^{-36} +41 q^{-38} -40 q^{-40} +21 q^{-42} - q^{-44} -20 q^{-46} +30 q^{-48} -31 q^{-50} +23 q^{-52} -10 q^{-54} -2 q^{-56} +8 q^{-58} -14 q^{-60} +12 q^{-62} -9 q^{-64} +6 q^{-66} - q^{-68} -2 q^{-70} +3 q^{-72} -3 q^{-74} +2 q^{-76} - q^{-78} + q^{-80} } |
.
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]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
|
K = Knot["10 48"];
|
In[4]:=
|
Alexander[K][t]
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
In[5]:=
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Conway[K][z]
|
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+5 z^6+8 z^4+4 z^2+1} |
In[6]:=
|
Alexander[K, 2][t]
|
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.
|
Out[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 \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
|
Out[7]=
|
{ 49, 0 } |
In[8]:=
|
Jones[K][q]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[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^5+2 q^4-4 q^3+6 q^2-7 q+9-7 q^{-1} +6 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} } |
In[9]:=
|
HOMFLYPT[K][a, z]
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
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-a^2 z^6-z^6 a^{-2} +7 z^6-5 a^2 z^4-5 z^4 a^{-2} +18 z^4-8 a^2 z^2-8 z^2 a^{-2} +20 z^2-4 a^2-4 a^{-2} +9} |
In[10]:=
|
Kauffman[K][a, z]
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
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 z^9+z^9 a^{-1} +2 a^2 z^8+3 z^8 a^{-2} +5 z^8+2 a^3 z^7+z^7 a^{-1} +3 z^7 a^{-3} +2 a^4 z^6-5 a^2 z^6-11 z^6 a^{-2} +2 z^6 a^{-4} -20 z^6+a^5 z^5-3 a^3 z^5-5 a z^5-11 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} -5 a^4 z^4+9 a^2 z^4+18 z^4 a^{-2} -5 z^4 a^{-4} +37 z^4-3 a^5 z^3-a^3 z^3+12 a z^3+21 z^3 a^{-1} +8 z^3 a^{-3} -3 z^3 a^{-5} +2 a^4 z^2-11 a^2 z^2-13 z^2 a^{-2} +z^2 a^{-4} -27 z^2+2 a^5 z-7 a z-9 z a^{-1} -3 z a^{-3} +z a^{-5} +4 a^2+4 a^{-2} +9} |
"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`
|
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
|
In[3]:=
|
K = Knot["10 48"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[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^5+2 q^4-4 q^3+6 q^2-7 q+9-7 q^{-1} +6 q^{-2} -4 q^{-3} +2 q^{-4} - q^{-5} } } |
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
Out[5]=
|
{} |
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
Out[6]=
|
{} |
Vassiliev invariants
| V2 and V3: | (4, 0) |
| V2,1 through V6,9: |
|
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 48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
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^{15}-2 q^{14}+q^{13}+4 q^{12}-9 q^{11}+2 q^{10}+14 q^9-19 q^8-3 q^7+31 q^6-27 q^5-14 q^4+48 q^3-30 q^2-24 q+56-26 q^{-1} -28 q^{-2} +47 q^{-3} -15 q^{-4} -25 q^{-5} +30 q^{-6} -5 q^{-7} -16 q^{-8} +14 q^{-9} - q^{-10} -7 q^{-11} +5 q^{-12} -2 q^{-14} + q^{-15} } |
| 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^{30}+2 q^{29}-q^{28}-q^{27}+5 q^{25}-3 q^{24}-7 q^{23}+3 q^{22}+17 q^{21}-7 q^{20}-24 q^{19}-q^{18}+39 q^{17}+9 q^{16}-49 q^{15}-27 q^{14}+55 q^{13}+51 q^{12}-57 q^{11}-75 q^{10}+48 q^9+104 q^8-42 q^7-120 q^6+24 q^5+143 q^4-19 q^3-143 q^2-4 q+159+4 q^{-1} -140 q^{-2} -28 q^{-3} +138 q^{-4} +30 q^{-5} -110 q^{-6} -47 q^{-7} +92 q^{-8} +46 q^{-9} -62 q^{-10} -48 q^{-11} +41 q^{-12} +40 q^{-13} -23 q^{-14} -29 q^{-15} +10 q^{-16} +21 q^{-17} -8 q^{-18} -10 q^{-19} +3 q^{-20} +8 q^{-21} -5 q^{-22} -3 q^{-23} +2 q^{-24} +4 q^{-25} -3 q^{-26} - q^{-27} +2 q^{-29} - q^{-30} } |
| 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^{50}-2 q^{49}+q^{48}+q^{47}-3 q^{46}+4 q^{45}-5 q^{44}+5 q^{43}+3 q^{42}-14 q^{41}+9 q^{40}-9 q^{39}+19 q^{38}+17 q^{37}-40 q^{36}-28 q^{34}+49 q^{33}+64 q^{32}-55 q^{31}-19 q^{30}-99 q^{29}+49 q^{28}+136 q^{27}-10 q^{26}+15 q^{25}-204 q^{24}-36 q^{23}+149 q^{22}+72 q^{21}+167 q^{20}-252 q^{19}-182 q^{18}+42 q^{17}+106 q^{16}+394 q^{15}-191 q^{14}-294 q^{13}-138 q^{12}+51 q^{11}+594 q^{10}-72 q^9-328 q^8-292 q^7-46 q^6+706 q^5+32 q^4-306 q^3-380 q^2-133 q+733+105 q^{-1} -250 q^{-2} -409 q^{-3} -210 q^{-4} +668 q^{-5} +168 q^{-6} -140 q^{-7} -383 q^{-8} -294 q^{-9} +509 q^{-10} +199 q^{-11} +16 q^{-12} -274 q^{-13} -342 q^{-14} +281 q^{-15} +151 q^{-16} +143 q^{-17} -107 q^{-18} -294 q^{-19} +88 q^{-20} +40 q^{-21} +157 q^{-22} +24 q^{-23} -169 q^{-24} +9 q^{-25} -45 q^{-26} +87 q^{-27} +56 q^{-28} -61 q^{-29} +14 q^{-30} -56 q^{-31} +23 q^{-32} +30 q^{-33} -17 q^{-34} +25 q^{-35} -29 q^{-36} + q^{-37} +6 q^{-38} -9 q^{-39} +18 q^{-40} -8 q^{-41} -6 q^{-44} +6 q^{-45} - q^{-46} + q^{-47} -2 q^{-49} + q^{-50} } |
| 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^{75}+2 q^{74}-q^{73}-q^{72}+3 q^{71}-q^{70}-4 q^{69}+3 q^{68}-q^{66}+8 q^{65}+q^{64}-16 q^{63}-5 q^{62}+2 q^{61}+9 q^{60}+27 q^{59}+14 q^{58}-31 q^{57}-48 q^{56}-26 q^{55}+16 q^{54}+84 q^{53}+82 q^{52}-10 q^{51}-104 q^{50}-138 q^{49}-60 q^{48}+113 q^{47}+206 q^{46}+135 q^{45}-49 q^{44}-229 q^{43}-261 q^{42}-52 q^{41}+198 q^{40}+318 q^{39}+221 q^{38}-53 q^{37}-331 q^{36}-384 q^{35}-161 q^{34}+203 q^{33}+483 q^{32}+458 q^{31}+49 q^{30}-493 q^{29}-742 q^{28}-403 q^{27}+350 q^{26}+967 q^{25}+845 q^{24}-101 q^{23}-1111 q^{22}-1248 q^{21}-262 q^{20}+1125 q^{19}+1648 q^{18}+635 q^{17}-1069 q^{16}-1908 q^{15}-1027 q^{14}+925 q^{13}+2144 q^{12}+1318 q^{11}-783 q^{10}-2209 q^9-1598 q^8+609 q^7+2320 q^6+1733 q^5-500 q^4-2251 q^3-1896 q^2+339 q+2311+1934 q^{-1} -256 q^{-2} -2161 q^{-3} -2044 q^{-4} +66 q^{-5} +2132 q^{-6} +2059 q^{-7} +85 q^{-8} -1889 q^{-9} -2106 q^{-10} -357 q^{-11} +1685 q^{-12} +2046 q^{-13} +588 q^{-14} -1283 q^{-15} -1944 q^{-16} -854 q^{-17} +897 q^{-18} +1695 q^{-19} +1017 q^{-20} -413 q^{-21} -1380 q^{-22} -1103 q^{-23} +29 q^{-24} +972 q^{-25} +1035 q^{-26} +298 q^{-27} -567 q^{-28} -874 q^{-29} -472 q^{-30} +223 q^{-31} +627 q^{-32} +507 q^{-33} +42 q^{-34} -377 q^{-35} -458 q^{-36} -171 q^{-37} +173 q^{-38} +320 q^{-39} +223 q^{-40} -18 q^{-41} -212 q^{-42} -193 q^{-43} -47 q^{-44} +90 q^{-45} +148 q^{-46} +82 q^{-47} -38 q^{-48} -84 q^{-49} -70 q^{-50} -14 q^{-51} +51 q^{-52} +55 q^{-53} +13 q^{-54} -11 q^{-55} -30 q^{-56} -30 q^{-57} +6 q^{-58} +20 q^{-59} +9 q^{-60} +8 q^{-61} -2 q^{-62} -16 q^{-63} -4 q^{-64} +5 q^{-65} +4 q^{-67} +4 q^{-68} -4 q^{-69} -2 q^{-70} + q^{-71} - q^{-72} +2 q^{-74} - q^{-75} } |
| 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^{105}-2 q^{104}+q^{103}+q^{102}-3 q^{101}+q^{100}+q^{99}+6 q^{98}-8 q^{97}-2 q^{96}+6 q^{95}-9 q^{94}+4 q^{93}+8 q^{92}+19 q^{91}-20 q^{90}-20 q^{89}+6 q^{88}-23 q^{87}+15 q^{86}+36 q^{85}+66 q^{84}-27 q^{83}-68 q^{82}-30 q^{81}-90 q^{80}+7 q^{79}+96 q^{78}+211 q^{77}+57 q^{76}-93 q^{75}-111 q^{74}-295 q^{73}-156 q^{72}+68 q^{71}+432 q^{70}+330 q^{69}+112 q^{68}-19 q^{67}-526 q^{66}-548 q^{65}-330 q^{64}+372 q^{63}+516 q^{62}+536 q^{61}+565 q^{60}-241 q^{59}-667 q^{58}-914 q^{57}-271 q^{56}-99 q^{55}+397 q^{54}+1178 q^{53}+767 q^{52}+320 q^{51}-591 q^{50}-698 q^{49}-1517 q^{48}-1177 q^{47}+444 q^{46}+1337 q^{45}+2125 q^{44}+1459 q^{43}+644 q^{42}-2229 q^{41}-3568 q^{40}-2296 q^{39}-140 q^{38}+2988 q^{37}+4297 q^{36}+4125 q^{35}-705 q^{34}-4873 q^{33}-5794 q^{32}-3712 q^{31}+1584 q^{30}+6012 q^{29}+8225 q^{28}+2753 q^{27}-4030 q^{26}-8164 q^{25}-7695 q^{24}-1494 q^{23}+5789 q^{22}+11139 q^{21}+6402 q^{20}-1798 q^{19}-8756 q^{18}-10454 q^{17}-4578 q^{16}+4401 q^{15}+12348 q^{14}+8854 q^{13}+378 q^{12}-8265 q^{11}-11641 q^{10}-6577 q^9+3009 q^8+12455 q^7+9958 q^6+1757 q^5-7588 q^4-11869 q^3-7547 q^2+2073 q+12175+10367 q^{-1} +2594 q^{-2} -6974 q^{-3} -11777 q^{-4} -8178 q^{-5} +1201 q^{-6} +11594 q^{-7} +10656 q^{-8} +3630 q^{-9} -5906 q^{-10} -11344 q^{-11} -8975 q^{-12} -363 q^{-13} +10104 q^{-14} +10636 q^{-15} +5239 q^{-16} -3632 q^{-17} -9816 q^{-18} -9540 q^{-19} -2813 q^{-20} +7021 q^{-21} +9374 q^{-22} +6703 q^{-23} -234 q^{-24} -6551 q^{-25} -8716 q^{-26} -5052 q^{-27} +2778 q^{-28} +6262 q^{-29} +6590 q^{-30} +2769 q^{-31} -2274 q^{-32} -5926 q^{-33} -5445 q^{-34} -767 q^{-35} +2267 q^{-36} +4414 q^{-37} +3659 q^{-38} +1004 q^{-39} -2351 q^{-40} -3716 q^{-41} -2006 q^{-42} -585 q^{-43} +1556 q^{-44} +2461 q^{-45} +1976 q^{-46} +63 q^{-47} -1421 q^{-48} -1309 q^{-49} -1346 q^{-50} -209 q^{-51} +819 q^{-52} +1324 q^{-53} +697 q^{-54} -91 q^{-55} -273 q^{-56} -865 q^{-57} -575 q^{-58} -43 q^{-59} +495 q^{-60} +430 q^{-61} +198 q^{-62} +189 q^{-63} -318 q^{-64} -349 q^{-65} -197 q^{-66} +106 q^{-67} +136 q^{-68} +110 q^{-69} +218 q^{-70} -73 q^{-71} -137 q^{-72} -131 q^{-73} +7 q^{-74} +16 q^{-75} +24 q^{-76} +138 q^{-77} - q^{-78} -36 q^{-79} -65 q^{-80} -7 q^{-81} -15 q^{-82} -11 q^{-83} +68 q^{-84} +11 q^{-85} + q^{-86} -24 q^{-87} -2 q^{-88} -13 q^{-89} -16 q^{-90} +24 q^{-91} +5 q^{-92} +6 q^{-93} -6 q^{-94} +2 q^{-95} -4 q^{-96} -8 q^{-97} +6 q^{-98} +2 q^{-100} - q^{-101} + q^{-102} -2 q^{-104} + q^{-105} } |
| 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^{140}+2 q^{139}-q^{138}-q^{137}+3 q^{136}-q^{135}-q^{134}-3 q^{133}-q^{132}+10 q^{131}-3 q^{130}-5 q^{129}+5 q^{128}-5 q^{127}-2 q^{126}-7 q^{125}-2 q^{124}+33 q^{123}+6 q^{122}-15 q^{121}-6 q^{120}-30 q^{119}-8 q^{118}-14 q^{117}+2 q^{116}+93 q^{115}+57 q^{114}-38 q^{112}-127 q^{111}-83 q^{110}-65 q^{109}-7 q^{108}+220 q^{107}+237 q^{106}+167 q^{105}+13 q^{104}-302 q^{103}-337 q^{102}-363 q^{101}-225 q^{100}+291 q^{99}+559 q^{98}+685 q^{97}+501 q^{96}-178 q^{95}-589 q^{94}-977 q^{93}-1017 q^{92}-230 q^{91}+444 q^{90}+1187 q^{89}+1506 q^{88}+813 q^{87}+97 q^{86}-988 q^{85}-1804 q^{84}-1439 q^{83}-921 q^{82}+249 q^{81}+1515 q^{80}+1716 q^{79}+1815 q^{78}+921 q^{77}-482 q^{76}-1102 q^{75}-2133 q^{74}-2218 q^{73}-1363 q^{72}-599 q^{71}+1375 q^{70}+2838 q^{69}+3296 q^{68}+3448 q^{67}+1124 q^{66}-1978 q^{65}-4695 q^{64}-6860 q^{63}-5112 q^{62}-943 q^{61}+4263 q^{60}+9705 q^{59}+10291 q^{58}+6218 q^{57}-1433 q^{56}-11012 q^{55}-15404 q^{54}-13089 q^{53}-4249 q^{52}+9591 q^{51}+19270 q^{50}+20798 q^{49}+12334 q^{48}-5240 q^{47}-20874 q^{46}-27868 q^{45}-21771 q^{44}-1941 q^{43}+19532 q^{42}+33263 q^{41}+31503 q^{40}+11041 q^{39}-15545 q^{38}-36280 q^{37}-40119 q^{36}-20861 q^{35}+9319 q^{34}+36757 q^{33}+47008 q^{32}+30303 q^{31}-2078 q^{30}-35127 q^{29}-51586 q^{28}-38441 q^{27}-5362 q^{26}+32074 q^{25}+54193 q^{24}+44726 q^{23}+11958 q^{22}-28330 q^{21}-54978 q^{20}-49189 q^{19}-17473 q^{18}+24703 q^{17}+54826 q^{16}+51883 q^{15}+21366 q^{14}-21480 q^{13}-53812 q^{12}-53399 q^{11}-24259 q^{10}+19016 q^9+52980 q^8+54019 q^7+25786 q^6-17175 q^5-51798 q^4-54289 q^3-27166 q^2+15814 q+51271+54457 q^{-1} +27899 q^{-2} -14603 q^{-3} -50289 q^{-4} -54737 q^{-5} -29332 q^{-6} +13058 q^{-7} +49578 q^{-8} +55178 q^{-9} +30840 q^{-10} -10821 q^{-11} -47832 q^{-12} -55467 q^{-13} -33341 q^{-14} +7352 q^{-15} +45276 q^{-16} +55326 q^{-17} +36057 q^{-18} -2630 q^{-19} -40824 q^{-20} -54014 q^{-21} -39011 q^{-22} -3416 q^{-23} +34612 q^{-24} +51036 q^{-25} +41114 q^{-26} +10162 q^{-27} -26394 q^{-28} -45825 q^{-29} -41831 q^{-30} -16736 q^{-31} +16973 q^{-32} +38327 q^{-33} +40149 q^{-34} +22061 q^{-35} -7075 q^{-36} -29046 q^{-37} -36015 q^{-38} -25090 q^{-39} -1708 q^{-40} +18893 q^{-41} +29386 q^{-42} +25272 q^{-43} +8488 q^{-44} -9186 q^{-45} -21429 q^{-46} -22652 q^{-47} -12248 q^{-48} +1252 q^{-49} +13144 q^{-50} +17871 q^{-51} +13049 q^{-52} +4179 q^{-53} -5889 q^{-54} -12248 q^{-55} -11427 q^{-56} -6715 q^{-57} +624 q^{-58} +6782 q^{-59} +8299 q^{-60} +6977 q^{-61} +2498 q^{-62} -2556 q^{-63} -4992 q^{-64} -5650 q^{-65} -3506 q^{-66} -121 q^{-67} +2082 q^{-68} +3731 q^{-69} +3284 q^{-70} +1360 q^{-71} -242 q^{-72} -1989 q^{-73} -2326 q^{-74} -1505 q^{-75} -724 q^{-76} +703 q^{-77} +1368 q^{-78} +1181 q^{-79} +932 q^{-80} -39 q^{-81} -613 q^{-82} -694 q^{-83} -809 q^{-84} -200 q^{-85} +209 q^{-86} +302 q^{-87} +536 q^{-88} +224 q^{-89} -5 q^{-90} -111 q^{-91} -356 q^{-92} -122 q^{-93} +10 q^{-94} -8 q^{-95} +178 q^{-96} +80 q^{-97} +29 q^{-98} +8 q^{-99} -164 q^{-100} -30 q^{-101} +34 q^{-102} -12 q^{-103} +79 q^{-104} +20 q^{-105} +16 q^{-106} +16 q^{-107} -99 q^{-108} -33 q^{-109} +15 q^{-110} -4 q^{-111} +47 q^{-112} +12 q^{-113} +18 q^{-114} +27 q^{-115} -46 q^{-116} -26 q^{-117} -5 q^{-118} -10 q^{-119} +18 q^{-120} +4 q^{-121} +10 q^{-122} +21 q^{-123} -12 q^{-124} -9 q^{-125} -4 q^{-126} -8 q^{-127} +5 q^{-128} - q^{-129} +2 q^{-130} +8 q^{-131} -2 q^{-132} -2 q^{-133} -2 q^{-135} + q^{-136} - q^{-137} +2 q^{-139} - q^{-140} } |
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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