10 132: 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^{-1} +2 q^{-3} + q^{-4} -2 q^{-5} - q^{-6} +2 q^{-7} +3 q^{-8} -2 q^{-9} -2 q^{-10} + q^{-11} +3 q^{-12} -2 q^{-13} -3 q^{-14} + q^{-15} +4 q^{-16} - q^{-17} -4 q^{-18} +5 q^{-20} -5 q^{-22} - q^{-23} +5 q^{-24} + q^{-25} -5 q^{-26} - q^{-27} +4 q^{-28} + q^{-29} -3 q^{-30} - q^{-31} +3 q^{-32} -2 q^{-34} +2 q^{-36} -2 q^{-38} + q^{-40} + q^{-41} - q^{-42} </math> |
coloured_jones_3 = <math>- q^{-1} +2 q^{-3} + q^{-4} -2 q^{-5} - q^{-6} +2 q^{-7} +3 q^{-8} -2 q^{-9} -2 q^{-10} + q^{-11} +3 q^{-12} -2 q^{-13} -3 q^{-14} + q^{-15} +4 q^{-16} - q^{-17} -4 q^{-18} +5 q^{-20} -5 q^{-22} - q^{-23} +5 q^{-24} + q^{-25} -5 q^{-26} - q^{-27} +4 q^{-28} + q^{-29} -3 q^{-30} - q^{-31} +3 q^{-32} -2 q^{-34} +2 q^{-36} -2 q^{-38} + q^{-40} + q^{-41} - q^{-42} </math> |
coloured_jones_4 = <math>q^5-q^4-q^3-q^2+5- q^{-2} -6 q^{-3} -2 q^{-4} +9 q^{-5} +2 q^{-6} -9 q^{-8} -3 q^{-9} +10 q^{-10} +3 q^{-11} + q^{-12} -10 q^{-13} -3 q^{-14} +11 q^{-15} +3 q^{-16} - q^{-17} -10 q^{-18} -3 q^{-19} +11 q^{-20} +2 q^{-21} -2 q^{-22} -9 q^{-23} -2 q^{-24} +10 q^{-25} +2 q^{-26} -2 q^{-27} -7 q^{-28} -2 q^{-29} +8 q^{-30} +2 q^{-31} -2 q^{-32} -5 q^{-33} - q^{-34} +6 q^{-35} + q^{-36} -3 q^{-37} -3 q^{-38} + q^{-39} +5 q^{-40} -5 q^{-42} -2 q^{-43} +2 q^{-44} +6 q^{-45} -6 q^{-47} -2 q^{-48} + q^{-49} +6 q^{-50} + q^{-51} -4 q^{-52} -2 q^{-53} - q^{-54} +5 q^{-55} -2 q^{-57} - q^{-58} - q^{-59} +4 q^{-60} - q^{-61} - q^{-62} - q^{-63} - q^{-64} +3 q^{-65} - q^{-68} - q^{-69} + q^{-70} </math> |
coloured_jones_4 = <math>q^5-q^4-q^3-q^2+5- q^{-2} -6 q^{-3} -2 q^{-4} +9 q^{-5} +2 q^{-6} -9 q^{-8} -3 q^{-9} +10 q^{-10} +3 q^{-11} + q^{-12} -10 q^{-13} -3 q^{-14} +11 q^{-15} +3 q^{-16} - q^{-17} -10 q^{-18} -3 q^{-19} +11 q^{-20} +2 q^{-21} -2 q^{-22} -9 q^{-23} -2 q^{-24} +10 q^{-25} +2 q^{-26} -2 q^{-27} -7 q^{-28} -2 q^{-29} +8 q^{-30} +2 q^{-31} -2 q^{-32} -5 q^{-33} - q^{-34} +6 q^{-35} + q^{-36} -3 q^{-37} -3 q^{-38} + q^{-39} +5 q^{-40} -5 q^{-42} -2 q^{-43} +2 q^{-44} +6 q^{-45} -6 q^{-47} -2 q^{-48} + q^{-49} +6 q^{-50} + q^{-51} -4 q^{-52} -2 q^{-53} - q^{-54} +5 q^{-55} -2 q^{-57} - q^{-58} - q^{-59} +4 q^{-60} - q^{-61} - q^{-62} - q^{-63} - q^{-64} +3 q^{-65} - q^{-68} - q^{-69} + q^{-70} </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>
Line 46: Line 49:
<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, 132]]</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, 132]]</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[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[15, 18, 16, 19],
<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[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 12, 6, 13], X[15, 18, 16, 19],
Line 66: Line 69:
<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>4</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>4</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, 132]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_132_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, 132]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_132_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, 132]]&) /@ {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, 132]]&) /@ {
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, 1, 2, 3, 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, 1, 2, 3, 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, 132]][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, 132]][t]</nowiki></pre></td></tr>

Revision as of 18:54, 31 August 2005

10 131.gif

10_131

10 133.gif

10_133

10 132.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 132's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 132 at Knotilus!

[edit 10 132 Quick Notes]
[edit 10 132 Further Notes and Views]


Knot presentations

Planar diagram presentation X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837
Gauss code 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7
Dowker-Thistlethwaite code 4 8 -12 2 -16 -6 -20 -18 -10 -14
Conway Notation [23,3,2-]


Minimum Braid Representative A Morse Link Presentation An Arc Presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif

Length is 11, width is 4,

Braid index is 4

10 132 ML.gif 10 132 AP.gif
[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 8}, {7, 9}, {8, 5}, {4, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}]

[edit Notes on presentations of 10 132]


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["10 132"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X8493 X5,12,6,13 X15,18,16,19 X9,16,10,17 X17,10,18,11 X13,20,14,1 X19,14,20,15 X11,6,12,7 X2837
In[5]:= GaussCode[K]
Out[5]= 1, -10, 2, -1, -3, 9, 10, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7
In[6]:= DTCode[K]
Out[6]= 4 8 -12 2 -16 -6 -20 -18 -10 -14

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [23,3,2-]
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]= 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,-2,-1,-1,-2,-3,2,-3,-3\})}
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]= { 4, 11, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.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."
10 132 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 10}, {2, 4}, {1, 3}, {13, 11}, {10, 12}, {11, 8}, {7, 9}, {8, 5}, {4, 6}, {5, 7}, {6, 13}, {12, 2}, {9, 1}]
In[14]:= Draw[ap]
10 132 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index 1
Maximal Thurston-Bennequin number [-8][-1]
Hyperbolic Volume 4.05686
A-Polynomial See Data:10 132/A-polynomial

[edit Notes for 10 132's three dimensional invariants] 10 132 is a very interesting knot from the point of view of contact geometry. In particular, it is a transversely nonsimple knot, and it was the last knot with at most 10 crossings for which the maximal Thurston-Bennequin number was calculated.

Four dimensional invariants

Smooth 4 genus
Topological 4 genus 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}
Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}
Rasmussen s-Invariant 2

[edit Notes for 10 132's four dimensional invariants]

Polynomial invariants

Alexander polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^2-t+1- t^{-1} + 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 z^4+3 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 { 5, 0 }
Jones polynomial
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^6-2 a^6+z^4 a^4+4 z^2 a^4+3 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 a^6 z^8+a^4 z^8+a^7 z^7+2 a^5 z^7+a^3 z^7-6 a^6 z^6-6 a^4 z^6-6 a^7 z^5-12 a^5 z^5-6 a^3 z^5+10 a^6 z^4+10 a^4 z^4+10 a^7 z^3+19 a^5 z^3+9 a^3 z^3-6 a^6 z^2-7 a^4 z^2-a^2 z^2-5 a^7 z-8 a^5 z-4 a^3 z-a z+2 a^6+3 a^4}
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^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6}
The G2 invariant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{108}+q^{104}-q^{100}-q^{92}-q^{90}-q^{86}-q^{84}-q^{82}-2 q^{80}-q^{78}-q^{76}-2 q^{74}-q^{68}-q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+2 q^{52}+3 q^{50}+q^{48}+q^{46}+2 q^{44}+q^{42}+2 q^{40}+q^{38}+q^{34}+q^{32}-q^{28}+q^{26}-q^{24}-q^{18}+q^{16}-q^{12}+q^4-q^2+1+ q^{-6} - q^{-8} }
Further Quantum Invariants
Further quantum knot invariants for 10_132.

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^{15}+q^7+q^5+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^{44}-q^{40}-q^{30}-q^{24}+q^{16}+q^{14}+q^{10}+q^6+2- q^{-4} }
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^{87}+q^{83}+q^{81}-q^{77}+q^{67}-q^{63}+q^{59}+q^{57}-q^{55}-q^{53}-q^{45}-q^{43}+q^{39}-q^{35}+q^{31}-q^{27}-q^{25}+q^{17}+2 q^{15}+2 q^{13}+q^7+2 q^5+q^3-q- q^{-1} }

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^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2 q^{10}+q^8+q^6}
1,1
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^{58}+q^{56}+q^{54}-q^{48}-q^{46}-2 q^{44}-2 q^{42}-2 q^{40}-q^{38}+q^{36}-q^{34}+q^{28}+q^{24}+2 q^{22}+2 q^{20}+q^{18}+q^{16}+q^{14}+q^{10}+q^8+q^4+q^2+1- q^{-2} - q^{-4} }

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^{46}+q^{42}+q^{40}-2 q^{36}-2 q^{34}-4 q^{32}-5 q^{30}-2 q^{28}-q^{26}+2 q^{24}+3 q^{22}+6 q^{20}+4 q^{18}+3 q^{16}+2 q^{14}+q^{12}-q^{10}-q^8-q^4+1}
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^{29}-q^{27}-2 q^{25}-q^{23}+q^{19}+2 q^{17}+2 q^{15}+2 q^{13}+q^{11}+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^{60}+q^{58}+2 q^{56}+2 q^{54}+2 q^{52}+q^{50}-q^{48}-4 q^{46}-6 q^{44}-8 q^{42}-9 q^{40}-8 q^{38}-5 q^{36}+4 q^{32}+8 q^{30}+11 q^{28}+11 q^{26}+8 q^{24}+6 q^{22}+q^{20}-q^{18}-3 q^{16}-2 q^{14}-2 q^{12}-q^{10}+q^6+1}
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^{36}-q^{34}-2 q^{32}-2 q^{30}-q^{28}+q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+q^{14}+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^{46}-q^{42}-q^{40}+q^{30}+q^{26}+q^{22}+q^{16}+q^{12}+q^{10}+q^8+q^4-1}
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^{76}+q^{68}-q^{58}-2 q^{56}-q^{54}-q^{52}-2 q^{50}-2 q^{48}-q^{46}-q^{44}+q^{38}+q^{36}+2 q^{34}+2 q^{32}+2 q^{30}+q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{18}-q^{12}-q^4+1}

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^{62}+q^{58}+q^{56}+q^{54}-q^{50}-2 q^{48}-4 q^{46}-4 q^{44}-5 q^{42}-4 q^{40}-3 q^{38}+q^{34}+4 q^{32}+5 q^{30}+6 q^{28}+5 q^{26}+4 q^{24}+3 q^{22}+q^{20}+q^{18}-q^{16}-q^{14}-q^{12}-q^8+1}

G2 Invariants.

Weight Invariant
1,0

.

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["10 132"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
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 t^2-t+1- t^{-1} + t^{-2} }
In[5]:= Conway[K][z]
Out[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 z^4+3 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]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 5, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]=
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^2 a^6-2 a^6+z^4 a^4+4 z^2 a^4+3 a^4}
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^6 z^8+a^4 z^8+a^7 z^7+2 a^5 z^7+a^3 z^7-6 a^6 z^6-6 a^4 z^6-6 a^7 z^5-12 a^5 z^5-6 a^3 z^5+10 a^6 z^4+10 a^4 z^4+10 a^7 z^3+19 a^5 z^3+9 a^3 z^3-6 a^6 z^2-7 a^4 z^2-a^2 z^2-5 a^7 z-8 a^5 z-4 a^3 z-a z+2 a^6+3 a^4}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {5_1,}

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}} ): {5_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.

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

In[3]:= K = Knot["10 132"];
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 t^2-t+1- t^{-1} + 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^{-4} - q^{-5} + q^{-6} - q^{-7} } }
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]= {5_1,}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {5_1,}

Vassiliev invariants

V2 and V3: (3, -5)
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
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 12} 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 -40} 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 72} 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 174} 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 26} 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 -\frac{2608}{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 -\frac{352}{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 -168} 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 288} 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 800} 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 2088} 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 312} 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 \frac{42751}{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 \frac{10034}{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 \frac{203}{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 \frac{2751}{10}}

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 10 132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-10χ
-1      110
-3       11
-5    12  1
-7   1    1
-9   11   0
-11 11     0
-13        0
-151       -1
Integral Khovanov Homology

(db, data source)

  
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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} 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 i=-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 i=-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 i=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 {\mathbb Z}}
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=-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 {\mathbb Z}_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 {\mathbb Z}}
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=-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 {\mathbb Z}}
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=-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 {\mathbb Z}\oplus{\mathbb Z}_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 {\mathbb Z}}
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=-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 {\mathbb Z}} 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 {\mathbb Z}_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 {\mathbb Z}}
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=-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 {\mathbb Z}_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 {\mathbb Z}^{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 r=-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 {\mathbb Z}_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 {\mathbb Z}}
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=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 {\mathbb Z}} 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 {\mathbb Z}}

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+1+2 q^{-1} -3 q^{-2} + q^{-3} +3 q^{-4} -4 q^{-5} +2 q^{-6} +2 q^{-7} -3 q^{-8} +2 q^{-9} + q^{-10} -3 q^{-11} +2 q^{-12} -2 q^{-14} +2 q^{-15} - q^{-16} - q^{-17} +2 q^{-18} - q^{-19} - q^{-20} + q^{-21} }
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^{-1} +2 q^{-3} + q^{-4} -2 q^{-5} - q^{-6} +2 q^{-7} +3 q^{-8} -2 q^{-9} -2 q^{-10} + q^{-11} +3 q^{-12} -2 q^{-13} -3 q^{-14} + q^{-15} +4 q^{-16} - q^{-17} -4 q^{-18} +5 q^{-20} -5 q^{-22} - q^{-23} +5 q^{-24} + q^{-25} -5 q^{-26} - q^{-27} +4 q^{-28} + q^{-29} -3 q^{-30} - q^{-31} +3 q^{-32} -2 q^{-34} +2 q^{-36} -2 q^{-38} + q^{-40} + q^{-41} - q^{-42} }
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-q^4-q^3-q^2+5- q^{-2} -6 q^{-3} -2 q^{-4} +9 q^{-5} +2 q^{-6} -9 q^{-8} -3 q^{-9} +10 q^{-10} +3 q^{-11} + q^{-12} -10 q^{-13} -3 q^{-14} +11 q^{-15} +3 q^{-16} - q^{-17} -10 q^{-18} -3 q^{-19} +11 q^{-20} +2 q^{-21} -2 q^{-22} -9 q^{-23} -2 q^{-24} +10 q^{-25} +2 q^{-26} -2 q^{-27} -7 q^{-28} -2 q^{-29} +8 q^{-30} +2 q^{-31} -2 q^{-32} -5 q^{-33} - q^{-34} +6 q^{-35} + q^{-36} -3 q^{-37} -3 q^{-38} + q^{-39} +5 q^{-40} -5 q^{-42} -2 q^{-43} +2 q^{-44} +6 q^{-45} -6 q^{-47} -2 q^{-48} + q^{-49} +6 q^{-50} + q^{-51} -4 q^{-52} -2 q^{-53} - q^{-54} +5 q^{-55} -2 q^{-57} - q^{-58} - q^{-59} +4 q^{-60} - q^{-61} - q^{-62} - q^{-63} - q^{-64} +3 q^{-65} - q^{-68} - q^{-69} + q^{-70} }
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 \textrm{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}

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.

10 131.gif

10_131

10 133.gif

10_133

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