10 4: Difference between revisions

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{{Rolfsen Knot Page|
{{Rolfsen Knot Page|
n = 10 |
n = 10 |
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coloured_jones_3 = <math>q^{33}-q^{32}-q^{31}+3 q^{29}-3 q^{27}-3 q^{26}+5 q^{25}+3 q^{24}-3 q^{23}-7 q^{22}+4 q^{21}+6 q^{20}-q^{19}-8 q^{18}+2 q^{17}+7 q^{16}-7 q^{14}+q^{13}+6 q^{12}-q^{11}-6 q^{10}+q^9+7 q^8-3 q^7-6 q^6+2 q^5+8 q^4-5 q^3-7 q^2+5 q+9-8 q^{-1} -9 q^{-2} +10 q^{-3} +10 q^{-4} -10 q^{-5} -12 q^{-6} +11 q^{-7} +11 q^{-8} -7 q^{-9} -13 q^{-10} +8 q^{-11} +9 q^{-12} -3 q^{-13} -10 q^{-14} +3 q^{-15} +7 q^{-16} - q^{-17} -6 q^{-18} + q^{-19} +4 q^{-20} -4 q^{-22} + q^{-23} + q^{-24} + q^{-25} -2 q^{-26} + q^{-27} </math> |
coloured_jones_3 = <math>q^{33}-q^{32}-q^{31}+3 q^{29}-3 q^{27}-3 q^{26}+5 q^{25}+3 q^{24}-3 q^{23}-7 q^{22}+4 q^{21}+6 q^{20}-q^{19}-8 q^{18}+2 q^{17}+7 q^{16}-7 q^{14}+q^{13}+6 q^{12}-q^{11}-6 q^{10}+q^9+7 q^8-3 q^7-6 q^6+2 q^5+8 q^4-5 q^3-7 q^2+5 q+9-8 q^{-1} -9 q^{-2} +10 q^{-3} +10 q^{-4} -10 q^{-5} -12 q^{-6} +11 q^{-7} +11 q^{-8} -7 q^{-9} -13 q^{-10} +8 q^{-11} +9 q^{-12} -3 q^{-13} -10 q^{-14} +3 q^{-15} +7 q^{-16} - q^{-17} -6 q^{-18} + q^{-19} +4 q^{-20} -4 q^{-22} + q^{-23} + q^{-24} + q^{-25} -2 q^{-26} + q^{-27} </math> |
coloured_jones_4 = <math>q^{56}-q^{55}-q^{54}+4 q^{51}-q^{50}-2 q^{49}-2 q^{48}-4 q^{47}+8 q^{46}+2 q^{45}-3 q^{43}-11 q^{42}+8 q^{41}+3 q^{40}+5 q^{39}+q^{38}-16 q^{37}+6 q^{36}+7 q^{34}+6 q^{33}-17 q^{32}+8 q^{31}-4 q^{30}+5 q^{29}+7 q^{28}-19 q^{27}+12 q^{26}-3 q^{25}+4 q^{24}+7 q^{23}-24 q^{22}+13 q^{21}+6 q^{19}+10 q^{18}-30 q^{17}+9 q^{16}+q^{15}+10 q^{14}+16 q^{13}-32 q^{12}+2 q^{11}-q^{10}+13 q^9+24 q^8-33 q^7-4 q^6-3 q^5+14 q^4+30 q^3-34 q^2-9 q-4+17 q^{-1} +36 q^{-2} -36 q^{-3} -17 q^{-4} -6 q^{-5} +20 q^{-6} +43 q^{-7} -33 q^{-8} -23 q^{-9} -12 q^{-10} +19 q^{-11} +47 q^{-12} -26 q^{-13} -22 q^{-14} -16 q^{-15} +14 q^{-16} +41 q^{-17} -19 q^{-18} -15 q^{-19} -14 q^{-20} +9 q^{-21} +31 q^{-22} -16 q^{-23} -7 q^{-24} -9 q^{-25} +5 q^{-26} +20 q^{-27} -14 q^{-28} -5 q^{-30} +3 q^{-31} +10 q^{-32} -11 q^{-33} +3 q^{-34} -2 q^{-35} +2 q^{-36} +4 q^{-37} -6 q^{-38} +2 q^{-39} - q^{-40} + q^{-41} + q^{-42} -2 q^{-43} + q^{-44} </math> |
coloured_jones_4 = <math>q^{56}-q^{55}-q^{54}+4 q^{51}-q^{50}-2 q^{49}-2 q^{48}-4 q^{47}+8 q^{46}+2 q^{45}-3 q^{43}-11 q^{42}+8 q^{41}+3 q^{40}+5 q^{39}+q^{38}-16 q^{37}+6 q^{36}+7 q^{34}+6 q^{33}-17 q^{32}+8 q^{31}-4 q^{30}+5 q^{29}+7 q^{28}-19 q^{27}+12 q^{26}-3 q^{25}+4 q^{24}+7 q^{23}-24 q^{22}+13 q^{21}+6 q^{19}+10 q^{18}-30 q^{17}+9 q^{16}+q^{15}+10 q^{14}+16 q^{13}-32 q^{12}+2 q^{11}-q^{10}+13 q^9+24 q^8-33 q^7-4 q^6-3 q^5+14 q^4+30 q^3-34 q^2-9 q-4+17 q^{-1} +36 q^{-2} -36 q^{-3} -17 q^{-4} -6 q^{-5} +20 q^{-6} +43 q^{-7} -33 q^{-8} -23 q^{-9} -12 q^{-10} +19 q^{-11} +47 q^{-12} -26 q^{-13} -22 q^{-14} -16 q^{-15} +14 q^{-16} +41 q^{-17} -19 q^{-18} -15 q^{-19} -14 q^{-20} +9 q^{-21} +31 q^{-22} -16 q^{-23} -7 q^{-24} -9 q^{-25} +5 q^{-26} +20 q^{-27} -14 q^{-28} -5 q^{-30} +3 q^{-31} +10 q^{-32} -11 q^{-33} +3 q^{-34} -2 q^{-35} +2 q^{-36} +4 q^{-37} -6 q^{-38} +2 q^{-39} - q^{-40} + q^{-41} + q^{-42} -2 q^{-43} + q^{-44} </math> |
coloured_jones_5 = |
coloured_jones_5 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_6 = |
coloured_jones_6 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_7 = |
coloured_jones_7 = <math>\textrm{NotAvailable}(q)</math> |
computer_talk =
computer_talk =
<table>
<table>
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<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
</tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</td></tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</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, 4]]</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, 4]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[6, 2, 7, 1], X[16, 12, 17, 11], X[12, 3, 13, 4], X[2, 15, 3, 16],
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[6, 2, 7, 1], X[16, 12, 17, 11], X[12, 3, 13, 4], X[2, 15, 3, 16],
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></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, 4]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_4_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, 4]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_4_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[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[10, 4]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr>
<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, 4]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 2, 2, NotAvailable, 1}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 2, 2, NotAvailable, 1}</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, 4]][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, 4]][t]</nowiki></pre></td></tr>

Revision as of 18:47, 31 August 2005

10 3.gif

10_3

10 5.gif

10_5

10 4.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 4 at Knotilus!

[edit 10 4 Quick Notes]
[edit 10 4 Further Notes and Views]


Knot presentations

Planar diagram presentation X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X20,8,1,7 X18,10,19,9 X4,13,5,14 X10,18,11,17 X8,20,9,19
Gauss code 1, -4, 3, -8, 5, -1, 6, -10, 7, -9, 2, -3, 8, -5, 4, -2, 9, -7, 10, -6
Dowker-Thistlethwaite code 6 12 14 20 18 16 4 2 10 8
Conway Notation [613]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Length is 12, width is 5,

Braid index is 5

10 4 ML.gif 10 4 AP.gif
[{2, 12}, {1, 3}, {4, 2}, {3, 5}, {6, 4}, {5, 7}, {11, 6}, {12, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 1}]

[edit Notes on presentations of 10 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]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["10 4"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X20,8,1,7 X18,10,19,9 X4,13,5,14 X10,18,11,17 X8,20,9,19
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -8, 5, -1, 6, -10, 7, -9, 2, -3, 8, -5, 4, -2, 9, -7, 10, -6
In[6]:= DTCode[K]
Out[6]= 6 12 14 20 18 16 4 2 10 8

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [613]
In[9]:= br = BR[K]
KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
Out[9]= [math]\displaystyle{ \textrm{BR}(5,\{-1,-1,-1,2,-1,2,3,-2,3,4,-3,4\}) }[/math]
In[10]:= {First[br], Crossings[br], BraidIndex[K]}
KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
KnotTheory::loading: Loading precomputed data in IndianaData`.
Out[10]= { 5, 12, 5 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
Out[11]= -Graphics-
In[12]:= Show[DrawMorseLink[K]]
KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
10 4 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{2, 12}, {1, 3}, {4, 2}, {3, 5}, {6, 4}, {5, 7}, {11, 6}, {12, 8}, {7, 9}, {8, 10}, {9, 11}, {10, 1}]
In[14]:= Draw[ap]
10 4 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-7][-5]
Hyperbolic Volume 5.81713
A-Polynomial See Data:10 4/A-polynomial

[edit Notes for 10 4's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 2 }[/math]
Rasmussen s-Invariant 2

[edit Notes for 10 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -3 t^2+7 t-7+7 t^{-1} -3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -3 z^4-5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 27, -2 }
Jones polynomial [math]\displaystyle{ q^5-q^4+2 q^3-3 q^2+3 q-4+4 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-z^4-2 z^2-z^4 a^{-2} -3 z^2 a^{-2} -2 a^{-2} +z^2 a^{-4} +2 a^{-4} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +3 z^8 a^{-2} +z^8 a^{-4} +2 z^8+3 a z^7-3 z^7 a^{-1} -6 z^7 a^{-3} +3 a^2 z^6-17 z^6 a^{-2} -7 z^6 a^{-4} -7 z^6+3 a^3 z^5-10 a z^5-2 z^5 a^{-1} +11 z^5 a^{-3} +3 a^4 z^4-6 a^2 z^4+29 z^4 a^{-2} +16 z^4 a^{-4} +4 z^4+2 a^5 z^3-4 a^3 z^3+8 a z^3+7 z^3 a^{-1} -7 z^3 a^{-3} +a^6 z^2-3 a^4 z^2-16 z^2 a^{-2} -13 z^2 a^{-4} +z^2-3 a z-z a^{-1} +2 z a^{-3} +a^4+2 a^{-2} +2 a^{-4} }[/math]
The A2 invariant [math]\displaystyle{ q^{16}+q^{10}+q^6- q^{-2} - q^{-4} - q^{-6} - q^{-8} + q^{-10} + q^{-12} + q^{-14} + q^{-16} }[/math]
The G2 invariant [math]\displaystyle{ q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+q^{40}+2 q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{26}-q^{24}+q^{22}-q^{20}-q^{14}-q^{10}+2 q^4-4 q^2+2-3 q^{-4} +4 q^{-6} -5 q^{-8} +2 q^{-10} - q^{-12} - q^{-14} +2 q^{-16} -3 q^{-18} +2 q^{-20} -2 q^{-22} - q^{-26} - q^{-28} +2 q^{-36} -2 q^{-38} + q^{-40} + q^{-42} -2 q^{-44} +5 q^{-46} -5 q^{-48} +3 q^{-50} + q^{-52} -2 q^{-54} +5 q^{-56} -4 q^{-58} +3 q^{-60} + q^{-66} -2 q^{-68} +2 q^{-70} + q^{-74} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_4.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{11}-q^9+q^7+q^3- q^{-1} - q^{-5} + q^{-7} + q^{-11} }[/math]
2 [math]\displaystyle{ q^{30}-q^{28}+2 q^{24}-q^{22}-q^{16}+q^6-q^4+1+ q^{-6} +2 q^{-8} + q^{-14} - q^{-16} -2 q^{-18} + q^{-20} -2 q^{-24} + q^{-26} + q^{-28} - q^{-30} + q^{-34} }[/math]
3 [math]\displaystyle{ q^{57}-q^{55}+q^{51}+q^{49}-q^{47}-2 q^{45}+q^{43}+q^{41}-q^{39}-2 q^{37}+q^{35}+3 q^{33}-q^{31}-3 q^{29}-q^{27}+4 q^{25}+q^{23}-3 q^{21}-q^{19}+2 q^{17}+3 q^{15}-q^{11}-2 q^9+q^7+3 q^5+2 q^3-3 q- q^{-1} +2 q^{-3} + q^{-5} -2 q^{-7} - q^{-9} + q^{-11} - q^{-15} - q^{-17} + q^{-19} - q^{-25} + q^{-29} +2 q^{-31} + q^{-33} - q^{-37} + q^{-39} +2 q^{-41} -3 q^{-45} -2 q^{-47} +2 q^{-49} +2 q^{-51} - q^{-53} -3 q^{-55} +2 q^{-59} + q^{-61} - q^{-63} - q^{-65} + q^{-69} }[/math]
4 [math]\displaystyle{ q^{92}-q^{90}+q^{86}+q^{82}-3 q^{80}+q^{76}+q^{72}-4 q^{70}+2 q^{68}+3 q^{66}-3 q^{62}-6 q^{60}+4 q^{58}+6 q^{56}+2 q^{54}-5 q^{52}-7 q^{50}+4 q^{48}+8 q^{46}+3 q^{44}-5 q^{42}-8 q^{40}+2 q^{38}+7 q^{36}+5 q^{34}-2 q^{32}-9 q^{30}-3 q^{28}+2 q^{26}+6 q^{24}+5 q^{22}-2 q^{20}-6 q^{18}-5 q^{16}+q^{14}+7 q^{12}+4 q^{10}-3 q^8-6 q^6-4 q^4+4 q^2+6-3 q^{-4} -2 q^{-6} +3 q^{-8} +4 q^{-10} -2 q^{-12} -3 q^{-14} - q^{-16} +5 q^{-18} +6 q^{-20} -2 q^{-22} -5 q^{-24} -3 q^{-26} +4 q^{-28} +6 q^{-30} -4 q^{-34} -5 q^{-36} - q^{-38} +5 q^{-40} +2 q^{-42} -3 q^{-46} -4 q^{-48} + q^{-50} + q^{-52} +2 q^{-54} + q^{-56} -3 q^{-58} - q^{-60} -2 q^{-62} +4 q^{-66} +2 q^{-68} +3 q^{-70} -2 q^{-72} -4 q^{-74} - q^{-76} + q^{-78} +6 q^{-80} +2 q^{-82} -3 q^{-84} -4 q^{-86} -4 q^{-88} +3 q^{-90} +4 q^{-92} +2 q^{-94} - q^{-96} -5 q^{-98} - q^{-100} + q^{-102} +2 q^{-104} +2 q^{-106} - q^{-108} - q^{-110} - q^{-112} + q^{-116} }[/math]
5 [math]\displaystyle{ q^{135}-q^{133}+q^{129}-q^{123}-q^{121}+q^{117}-q^{113}+q^{109}+2 q^{107}-q^{105}-4 q^{103}-4 q^{101}+2 q^{99}+7 q^{97}+5 q^{95}-q^{93}-8 q^{91}-6 q^{89}+2 q^{87}+10 q^{85}+6 q^{83}-3 q^{81}-8 q^{79}-4 q^{77}+3 q^{75}+7 q^{73}+2 q^{71}-6 q^{69}-8 q^{67}+q^{65}+10 q^{63}+8 q^{61}-11 q^{57}-13 q^{55}-2 q^{53}+11 q^{51}+13 q^{49}+4 q^{47}-8 q^{45}-12 q^{43}-9 q^{41}+q^{39}+10 q^{37}+11 q^{35}+6 q^{33}-q^{31}-10 q^{29}-13 q^{27}-5 q^{25}+7 q^{23}+14 q^{21}+11 q^{19}-q^{17}-12 q^{15}-12 q^{13}-2 q^{11}+8 q^9+11 q^7+3 q^5-3 q^3-4 q- q^{-1} +4 q^{-3} +4 q^{-5} -2 q^{-7} -5 q^{-9} -4 q^{-11} +3 q^{-13} +7 q^{-15} +6 q^{-17} -2 q^{-19} -11 q^{-21} -10 q^{-23} - q^{-25} +9 q^{-27} +12 q^{-29} +5 q^{-31} -9 q^{-33} -15 q^{-35} -9 q^{-37} +5 q^{-39} +15 q^{-41} +14 q^{-43} -14 q^{-47} -15 q^{-49} -2 q^{-51} +11 q^{-53} +17 q^{-55} +8 q^{-57} -6 q^{-59} -15 q^{-61} -11 q^{-63} + q^{-65} +11 q^{-67} +12 q^{-69} +5 q^{-71} -5 q^{-73} -10 q^{-75} -8 q^{-77} +6 q^{-81} +6 q^{-83} +4 q^{-85} -5 q^{-89} -5 q^{-91} -2 q^{-93} - q^{-95} + q^{-97} +3 q^{-99} +3 q^{-101} + q^{-103} -3 q^{-107} -5 q^{-109} -4 q^{-111} + q^{-113} +5 q^{-115} +7 q^{-117} +5 q^{-119} - q^{-121} -7 q^{-123} -8 q^{-125} -3 q^{-127} +4 q^{-129} +8 q^{-131} +7 q^{-133} + q^{-135} -5 q^{-137} -8 q^{-139} -5 q^{-141} + q^{-143} +5 q^{-145} +6 q^{-147} +3 q^{-149} -2 q^{-151} -5 q^{-153} -3 q^{-155} - q^{-157} + q^{-159} +3 q^{-161} +2 q^{-163} - q^{-167} - q^{-169} - q^{-171} + q^{-175} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{16}+q^{10}+q^6- q^{-2} - q^{-4} - q^{-6} - q^{-8} + q^{-10} + q^{-12} + q^{-14} + q^{-16} }[/math]
1,1 [math]\displaystyle{ q^{44}-2 q^{42}+2 q^{40}-2 q^{38}+5 q^{36}-4 q^{34}+4 q^{32}-4 q^{30}+5 q^{28}-4 q^{26}+4 q^{24}-2 q^{22}+4 q^{20}-4 q^{18}-q^{12}-4 q^8-4 q^4+2 q^2-4+10 q^{-2} -7 q^{-4} +20 q^{-6} -10 q^{-8} +18 q^{-10} -14 q^{-12} +12 q^{-14} -12 q^{-16} +2 q^{-18} -5 q^{-20} -2 q^{-22} +6 q^{-24} -10 q^{-26} +11 q^{-28} -12 q^{-30} +12 q^{-32} -10 q^{-34} +6 q^{-36} -4 q^{-38} +4 q^{-40} + q^{-44} }[/math]
2,0 [math]\displaystyle{ q^{40}+q^{34}+q^{32}+q^{26}-2 q^{22}-2 q^{16}-q^{14}+q^{12}-q^8-q^6+1+2 q^{-2} + q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} +2 q^{-14} + q^{-16} + q^{-18} - q^{-20} -2 q^{-22} -2 q^{-24} -2 q^{-26} -2 q^{-28} -2 q^{-30} + q^{-34} + q^{-36} + q^{-40} + q^{-42} + q^{-44} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{36}-q^{34}-q^{32}+2 q^{30}-q^{26}+2 q^{24}+q^{22}+q^{18}+q^{16}-q^{14}-q^6+q^2-1- q^{-2} +2 q^{-10} + q^{-14} -2 q^{-18} - q^{-20} + q^{-26} +2 q^{-28} + q^{-32} }[/math]
1,0,0 [math]\displaystyle{ q^{21}+q^{17}+q^{13}+q^9- q^{-3} - q^{-5} -2 q^{-7} - q^{-9} - q^{-11} + q^{-13} + q^{-15} +2 q^{-17} + q^{-19} + q^{-21} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{36}-q^{34}+q^{32}-2 q^{30}+2 q^{28}-q^{26}+2 q^{24}-q^{22}+2 q^{20}-q^{18}+q^{16}+q^{14}+2 q^{10}-2 q^8+3 q^6-4 q^4+3 q^2-5+3 q^{-2} -4 q^{-4} +2 q^{-6} -2 q^{-8} - q^{-14} +2 q^{-16} -2 q^{-18} +3 q^{-20} -2 q^{-22} +2 q^{-24} - q^{-26} +2 q^{-28} + q^{-32} }[/math]
1,0 [math]\displaystyle{ q^{58}-q^{54}-q^{52}+2 q^{48}+q^{46}-q^{44}-q^{42}+2 q^{38}+q^{36}-q^{32}+q^{28}+q^{26}-q^{24}-q^{22}+2 q^{18}-q^{14}-q^{12}+q^8-q^4-q^2+1- q^{-2} - q^{-4} +2 q^{-8} + q^{-10} - q^{-12} - q^{-14} +2 q^{-16} +3 q^{-18} + q^{-20} -2 q^{-22} - q^{-24} + q^{-26} + q^{-28} - q^{-30} -3 q^{-32} - q^{-34} + q^{-36} + q^{-38} - q^{-40} - q^{-42} + q^{-44} +2 q^{-46} + q^{-54} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{86}-q^{84}+q^{82}-q^{80}-q^{74}+3 q^{72}-2 q^{70}+2 q^{68}-q^{66}+q^{62}-2 q^{60}+3 q^{58}-2 q^{56}+2 q^{48}-q^{46}+2 q^{44}-q^{42}+q^{40}+2 q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{26}-q^{24}+q^{22}-q^{20}-q^{14}-q^{10}+2 q^4-4 q^2+2-3 q^{-4} +4 q^{-6} -5 q^{-8} +2 q^{-10} - q^{-12} - q^{-14} +2 q^{-16} -3 q^{-18} +2 q^{-20} -2 q^{-22} - q^{-26} - q^{-28} +2 q^{-36} -2 q^{-38} + q^{-40} + q^{-42} -2 q^{-44} +5 q^{-46} -5 q^{-48} +3 q^{-50} + q^{-52} -2 q^{-54} +5 q^{-56} -4 q^{-58} +3 q^{-60} + q^{-66} -2 q^{-68} +2 q^{-70} + q^{-74} }[/math]

.

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 4"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -3 t^2+7 t-7+7 t^{-1} -3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -3 z^4-5 z^2+1 }[/math]
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]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 27, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^5-q^4+2 q^3-3 q^2+3 q-4+4 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^4+a^4-z^4 a^2-2 z^2 a^2-z^4-2 z^2-z^4 a^{-2} -3 z^2 a^{-2} -2 a^{-2} +z^2 a^{-4} +2 a^{-4} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^9 a^{-1} +z^9 a^{-3} +3 z^8 a^{-2} +z^8 a^{-4} +2 z^8+3 a z^7-3 z^7 a^{-1} -6 z^7 a^{-3} +3 a^2 z^6-17 z^6 a^{-2} -7 z^6 a^{-4} -7 z^6+3 a^3 z^5-10 a z^5-2 z^5 a^{-1} +11 z^5 a^{-3} +3 a^4 z^4-6 a^2 z^4+29 z^4 a^{-2} +16 z^4 a^{-4} +4 z^4+2 a^5 z^3-4 a^3 z^3+8 a z^3+7 z^3 a^{-1} -7 z^3 a^{-3} +a^6 z^2-3 a^4 z^2-16 z^2 a^{-2} -13 z^2 a^{-4} +z^2-3 a z-z a^{-1} +2 z a^{-3} +a^4+2 a^{-2} +2 a^{-4} }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

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.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["10 4"];
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]= { [math]\displaystyle{ -3 t^2+7 t-7+7 t^{-1} -3 t^{-2} }[/math], [math]\displaystyle{ q^5-q^4+2 q^3-3 q^2+3 q-4+4 q^{-1} -3 q^{-2} +3 q^{-3} -2 q^{-4} + q^{-5} }[/math] }
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: (-5, -1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ -20 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{890}{3} }[/math] [math]\displaystyle{ \frac{382}{3} }[/math] [math]\displaystyle{ 160 }[/math] [math]\displaystyle{ \frac{880}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ 120 }[/math] [math]\displaystyle{ -\frac{4000}{3} }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ -\frac{17800}{3} }[/math] [math]\displaystyle{ -\frac{7640}{3} }[/math] [math]\displaystyle{ -\frac{32095}{6} }[/math] [math]\displaystyle{ 1330 }[/math] [math]\displaystyle{ -\frac{47702}{9} }[/math] [math]\displaystyle{ \frac{11867}{18} }[/math] [math]\displaystyle{ -\frac{7423}{6} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of 10 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-10123456χ
11          11
9           0
7        21 1
5       1   -1
3      22   0
1     21    -1
-1    22     0
-3   23      1
-5  11       0
-7 12        1
-9 1         -1
-111          1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

 [math]\displaystyle{ n }[/math]  [math]\displaystyle{ J_n }[/math]
2 [math]\displaystyle{ q^{16}-q^{15}-q^{14}+3 q^{13}-q^{12}-4 q^{11}+5 q^{10}-7 q^8+6 q^7+2 q^6-8 q^5+6 q^4+4 q^3-9 q^2+5 q+4-8 q^{-1} +4 q^{-2} +3 q^{-3} -6 q^{-4} +3 q^{-5} +3 q^{-6} -6 q^{-7} +3 q^{-8} +2 q^{-9} -5 q^{-10} +3 q^{-11} + q^{-12} -2 q^{-13} + q^{-14} }[/math]
3 [math]\displaystyle{ q^{33}-q^{32}-q^{31}+3 q^{29}-3 q^{27}-3 q^{26}+5 q^{25}+3 q^{24}-3 q^{23}-7 q^{22}+4 q^{21}+6 q^{20}-q^{19}-8 q^{18}+2 q^{17}+7 q^{16}-7 q^{14}+q^{13}+6 q^{12}-q^{11}-6 q^{10}+q^9+7 q^8-3 q^7-6 q^6+2 q^5+8 q^4-5 q^3-7 q^2+5 q+9-8 q^{-1} -9 q^{-2} +10 q^{-3} +10 q^{-4} -10 q^{-5} -12 q^{-6} +11 q^{-7} +11 q^{-8} -7 q^{-9} -13 q^{-10} +8 q^{-11} +9 q^{-12} -3 q^{-13} -10 q^{-14} +3 q^{-15} +7 q^{-16} - q^{-17} -6 q^{-18} + q^{-19} +4 q^{-20} -4 q^{-22} + q^{-23} + q^{-24} + q^{-25} -2 q^{-26} + q^{-27} }[/math]
4 [math]\displaystyle{ q^{56}-q^{55}-q^{54}+4 q^{51}-q^{50}-2 q^{49}-2 q^{48}-4 q^{47}+8 q^{46}+2 q^{45}-3 q^{43}-11 q^{42}+8 q^{41}+3 q^{40}+5 q^{39}+q^{38}-16 q^{37}+6 q^{36}+7 q^{34}+6 q^{33}-17 q^{32}+8 q^{31}-4 q^{30}+5 q^{29}+7 q^{28}-19 q^{27}+12 q^{26}-3 q^{25}+4 q^{24}+7 q^{23}-24 q^{22}+13 q^{21}+6 q^{19}+10 q^{18}-30 q^{17}+9 q^{16}+q^{15}+10 q^{14}+16 q^{13}-32 q^{12}+2 q^{11}-q^{10}+13 q^9+24 q^8-33 q^7-4 q^6-3 q^5+14 q^4+30 q^3-34 q^2-9 q-4+17 q^{-1} +36 q^{-2} -36 q^{-3} -17 q^{-4} -6 q^{-5} +20 q^{-6} +43 q^{-7} -33 q^{-8} -23 q^{-9} -12 q^{-10} +19 q^{-11} +47 q^{-12} -26 q^{-13} -22 q^{-14} -16 q^{-15} +14 q^{-16} +41 q^{-17} -19 q^{-18} -15 q^{-19} -14 q^{-20} +9 q^{-21} +31 q^{-22} -16 q^{-23} -7 q^{-24} -9 q^{-25} +5 q^{-26} +20 q^{-27} -14 q^{-28} -5 q^{-30} +3 q^{-31} +10 q^{-32} -11 q^{-33} +3 q^{-34} -2 q^{-35} +2 q^{-36} +4 q^{-37} -6 q^{-38} +2 q^{-39} - q^{-40} + q^{-41} + q^{-42} -2 q^{-43} + q^{-44} }[/math]
5 [math]\displaystyle{ \textrm{NotAvailable}(q) }[/math]
6 [math]\displaystyle{ \textrm{NotAvailable}(q) }[/math]
7 [math]\displaystyle{ \textrm{NotAvailable}(q) }[/math]

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.

10 3.gif

10_3

10 5.gif

10_5

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