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<!-- WARNING! WARNING! WARNING!
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<!-- <math>\text{Null}</math> -->
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{{Rolfsen Knot Page|
{{Rolfsen Knot Page|
n = 10 |
n = 10 |
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coloured_jones_3 = <math>2 q^{-1} -2 q^{-2} + q^{-3} -2 q^{-4} +7 q^{-5} -5 q^{-6} -6 q^{-7} +3 q^{-8} +18 q^{-9} -10 q^{-10} -25 q^{-11} +6 q^{-12} +43 q^{-13} -9 q^{-14} -50 q^{-15} - q^{-16} +63 q^{-17} +4 q^{-18} -63 q^{-19} -15 q^{-20} +63 q^{-21} +22 q^{-22} -57 q^{-23} -27 q^{-24} +46 q^{-25} +35 q^{-26} -38 q^{-27} -37 q^{-28} +24 q^{-29} +43 q^{-30} -16 q^{-31} -40 q^{-32} + q^{-33} +41 q^{-34} +6 q^{-35} -33 q^{-36} -15 q^{-37} +25 q^{-38} +19 q^{-39} -15 q^{-40} -18 q^{-41} +5 q^{-42} +15 q^{-43} -9 q^{-45} -3 q^{-46} +5 q^{-47} +2 q^{-48} - q^{-49} -2 q^{-50} + q^{-51} </math> |
coloured_jones_3 = <math>2 q^{-1} -2 q^{-2} + q^{-3} -2 q^{-4} +7 q^{-5} -5 q^{-6} -6 q^{-7} +3 q^{-8} +18 q^{-9} -10 q^{-10} -25 q^{-11} +6 q^{-12} +43 q^{-13} -9 q^{-14} -50 q^{-15} - q^{-16} +63 q^{-17} +4 q^{-18} -63 q^{-19} -15 q^{-20} +63 q^{-21} +22 q^{-22} -57 q^{-23} -27 q^{-24} +46 q^{-25} +35 q^{-26} -38 q^{-27} -37 q^{-28} +24 q^{-29} +43 q^{-30} -16 q^{-31} -40 q^{-32} + q^{-33} +41 q^{-34} +6 q^{-35} -33 q^{-36} -15 q^{-37} +25 q^{-38} +19 q^{-39} -15 q^{-40} -18 q^{-41} +5 q^{-42} +15 q^{-43} -9 q^{-45} -3 q^{-46} +5 q^{-47} +2 q^{-48} - q^{-49} -2 q^{-50} + q^{-51} </math> |
coloured_jones_4 = <math>1+ q^{-1} -2 q^{-2} - q^{-3} +4 q^{-4} -2 q^{-5} +2 q^{-6} -7 q^{-7} -4 q^{-8} +22 q^{-9} -5 q^{-11} -37 q^{-12} -17 q^{-13} +73 q^{-14} +34 q^{-15} -11 q^{-16} -108 q^{-17} -72 q^{-18} +132 q^{-19} +115 q^{-20} +21 q^{-21} -184 q^{-22} -170 q^{-23} +150 q^{-24} +192 q^{-25} +92 q^{-26} -207 q^{-27} -256 q^{-28} +119 q^{-29} +211 q^{-30} +154 q^{-31} -174 q^{-32} -282 q^{-33} +76 q^{-34} +172 q^{-35} +181 q^{-36} -117 q^{-37} -262 q^{-38} +39 q^{-39} +112 q^{-40} +186 q^{-41} -57 q^{-42} -222 q^{-43} + q^{-44} +44 q^{-45} +180 q^{-46} +12 q^{-47} -168 q^{-48} -38 q^{-49} -29 q^{-50} +151 q^{-51} +74 q^{-52} -89 q^{-53} -50 q^{-54} -94 q^{-55} +85 q^{-56} +95 q^{-57} -5 q^{-58} -17 q^{-59} -112 q^{-60} +8 q^{-61} +58 q^{-62} +36 q^{-63} +32 q^{-64} -73 q^{-65} -27 q^{-66} +5 q^{-67} +21 q^{-68} +46 q^{-69} -21 q^{-70} -16 q^{-71} -15 q^{-72} -3 q^{-73} +25 q^{-74} -7 q^{-77} -7 q^{-78} +6 q^{-79} + q^{-80} +2 q^{-81} - q^{-82} -2 q^{-83} + q^{-84} </math> |
coloured_jones_4 = <math>1+ q^{-1} -2 q^{-2} - q^{-3} +4 q^{-4} -2 q^{-5} +2 q^{-6} -7 q^{-7} -4 q^{-8} +22 q^{-9} -5 q^{-11} -37 q^{-12} -17 q^{-13} +73 q^{-14} +34 q^{-15} -11 q^{-16} -108 q^{-17} -72 q^{-18} +132 q^{-19} +115 q^{-20} +21 q^{-21} -184 q^{-22} -170 q^{-23} +150 q^{-24} +192 q^{-25} +92 q^{-26} -207 q^{-27} -256 q^{-28} +119 q^{-29} +211 q^{-30} +154 q^{-31} -174 q^{-32} -282 q^{-33} +76 q^{-34} +172 q^{-35} +181 q^{-36} -117 q^{-37} -262 q^{-38} +39 q^{-39} +112 q^{-40} +186 q^{-41} -57 q^{-42} -222 q^{-43} + q^{-44} +44 q^{-45} +180 q^{-46} +12 q^{-47} -168 q^{-48} -38 q^{-49} -29 q^{-50} +151 q^{-51} +74 q^{-52} -89 q^{-53} -50 q^{-54} -94 q^{-55} +85 q^{-56} +95 q^{-57} -5 q^{-58} -17 q^{-59} -112 q^{-60} +8 q^{-61} +58 q^{-62} +36 q^{-63} +32 q^{-64} -73 q^{-65} -27 q^{-66} +5 q^{-67} +21 q^{-68} +46 q^{-69} -21 q^{-70} -16 q^{-71} -15 q^{-72} -3 q^{-73} +25 q^{-74} -7 q^{-77} -7 q^{-78} +6 q^{-79} + q^{-80} +2 q^{-81} - q^{-82} -2 q^{-83} + q^{-84} </math> |
coloured_jones_5 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_5 = |
coloured_jones_6 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_6 = |
coloured_jones_7 = <math>\textrm{NotAvailable}(q)</math> |
coloured_jones_7 = |
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, 15:27:48)...</td></tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<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, 131]]</nowiki></pre></td></tr>
<table><tr align=left>
<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[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1],
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[10, 131]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[14, 6, 15, 5], X[15, 20, 16, 1],
X[9, 16, 10, 17], X[19, 10, 20, 11], X[11, 18, 12, 19],
X[9, 16, 10, 17], X[19, 10, 20, 11], X[11, 18, 12, 19],
X[17, 12, 18, 13], X[6, 14, 7, 13], X[7, 2, 8, 3]]</nowiki></pre></td></tr>
X[17, 12, 18, 13], X[6, 14, 7, 13], X[7, 2, 8, 3]]</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 131]]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 6, -7, 8, 9, -3, -4, 5, -8,
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[10, 131]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[-1, 10, -2, 1, 3, -9, -10, 2, -5, 6, -7, 8, 9, -3, -4, 5, -8,
7, -6, 4]</nowiki></pre></td></tr>
7, -6, 4]</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[10, 131]]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 8, -14, 2, 16, 18, -6, 20, 12, 10]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[10, 131]]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[4, {-1, -1, -1, -2, 1, 1, -2, -2, -3, 2, -3}]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[10, 131]]</nowiki></code></td></tr>
<tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{First[br], Crossings[br]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{4, 11}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BraidIndex[Knot[10, 131]]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 8, -14, 2, 16, 18, -6, 20, 12, 10]</nowiki></code></td></tr>
</table>
<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>
<table><tr align=left>
<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, 131]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_131_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, 131]]&) /@ {
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[10, 131]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[4, {-1, -1, -1, -2, 1, 1, -2, -2, -3, 2, -3}]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{First[br], Crossings[br]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{4, 11}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[10, 131]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>4</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[10, 131]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:10_131_ML.gif]]</td></tr><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[10, 131]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></pre></td></tr>
}</nowiki></code></td></tr>
<tr align=left>
<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, 131]][t]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 8 2
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 1, 2, 3, NotAvailable, 1}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[10, 131]][t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 8 2
-11 - -- + - + 8 t - 2 t
-11 - -- + - + 8 t - 2 t
2 t
2 t
t</nowiki></pre></td></tr>
t</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 131]][z]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 4
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
1 - 2 z</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[10, 131]][z]</nowiki></code></td></tr>
<tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[8, 14], Knot[9, 8], Knot[10, 131]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[10, 131]], KnotSignature[Knot[10, 131]]}</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{31, -2}</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 4
1 - 2 z</nowiki></code></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[10, 131]][q]</nowiki></pre></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -9 2 3 5 5 5 5 3 2
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[8, 14], Knot[9, 8], Knot[10, 131]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[10, 131]], KnotSignature[Knot[10, 131]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{31, -2}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[10, 131]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -9 2 3 5 5 5 5 3 2
q - -- + -- - -- + -- - -- + -- - -- + -
q - -- + -- - -- + -- - -- + -- - -- + -
8 7 6 5 4 3 2 q
8 7 6 5 4 3 2 q
q q q q q q q</nowiki></pre></td></tr>
q q q q q q q</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 131]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 131]][q]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -28 -22 2 -18 -16 -14 -12 2 -6 2
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 131]}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[10, 131]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -28 -22 2 -18 -16 -14 -12 2 -6 2
q + q - --- - q - q - q + q + -- + q + --
q + q - --- - q - q - q + q + -- + q + --
20 8 2
20 8 2
q q q</nowiki></pre></td></tr>
q q q</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[17]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Knot[10, 131]][a, z]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 6 8 2 2 4 2 6 2 8 2 4 4 6 4
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
2 a - 2 a + a + 2 a z - a z - 2 a z + a z - a z - a z</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 131]][a, z]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 131]][a, z]</nowiki></code></td></tr>
<tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 6 8 3 5 7 9 2 2 4 2
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 6 8 2 2 4 2 6 2 8 2 4 4 6 4
2 a - 2 a + a + 2 a z - a z - 2 a z + a z - a z - a z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[10, 131]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 6 8 3 5 7 9 2 2 4 2
-2 a + 2 a + a + a z - a z - 5 a z - 3 a z + 3 a z + 2 a z -
-2 a + 2 a + a + a z - a z - 5 a z - 3 a z + 3 a z + 2 a z -
Line 117: Line 203:
6 8 8 8
6 8 8 8
a z + a z</nowiki></pre></td></tr>
a z + a z</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[19]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 131]], Vassiliev[3][Knot[10, 131]]}</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 2}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[20]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[10, 131]][q, t]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -3 2 1 1 1 2 1 3
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 131]], Vassiliev[3][Knot[10, 131]]}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{0, 2}</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[10, 131]][q, t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -3 2 1 1 1 2 1 3
q + - + ------ + ------ + ------ + ------ + ------ + ------ +
q + - + ------ + ------ + ------ + ------ + ------ + ------ +
q 19 8 17 7 15 7 15 6 13 6 13 5
q 19 8 17 7 15 7 15 6 13 6 13 5
Line 129: Line 225:
------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ----
------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ----
11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 3
11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 3
q t q t q t q t q t q t q t q t q t</nowiki></pre></td></tr>
q t q t q t q t q t q t q t q t q t</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[21]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>ColouredJones[Knot[10, 131], 2][q]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -26 2 -24 6 4 7 13 2 16 18 3
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[10, 131], 2][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -26 2 -24 6 4 7 13 2 16 18 3
q - --- - q + --- - --- - --- + --- - --- - --- + --- + --- -
q - --- - q + --- - --- - --- + --- - --- - --- + --- + --- -
25 23 22 21 20 19 18 17 16
25 23 22 21 20 19 18 17 16
Line 144: Line 245:
-- + q + -
-- + q + -
3 q
3 q
q</nowiki></pre></td></tr>
q</nowiki></code></td></tr>
</table> }}
</table> }}

Latest revision as of 18:05, 1 September 2005

10 130.gif

10_130

10 132.gif

10_132

10 131.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 131's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 131 at Knotilus!

[edit 10 131 Quick Notes]
[edit 10 131 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X3849 X14,6,15,5 X15,20,16,1 X9,16,10,17 X19,10,20,11 X11,18,12,19 X17,12,18,13 X6,14,7,13 X7283
Gauss code -1, 10, -2, 1, 3, -9, -10, 2, -5, 6, -7, 8, 9, -3, -4, 5, -8, 7, -6, 4
Dowker-Thistlethwaite code 4 8 -14 2 16 18 -6 20 12 10
Conway Notation [311,21,2-]


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

Length is 11, width is 4,

Braid index is 4

10 131 ML.gif 10 131 AP.gif
[{11, 6}, {5, 9}, {8, 10}, {9, 11}, {7, 1}, {6, 8}, {10, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 7}]

[edit Notes on presentations of 10 131]


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["10 131"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X3849 X14,6,15,5 X15,20,16,1 X9,16,10,17 X19,10,20,11 X11,18,12,19 X17,12,18,13 X6,14,7,13 X7283
In[5]:= GaussCode[K]
Out[5]= -1, 10, -2, 1, 3, -9, -10, 2, -5, 6, -7, 8, 9, -3, -4, 5, -8, 7, -6, 4
In[6]:= DTCode[K]
Out[6]= 4 8 -14 2 16 18 -6 20 12 10

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [311,21,2-]
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]= [math]\displaystyle{ \textrm{BR}(4,\{-1,-1,-1,-2,1,1,-2,-2,-3,2,-3\}) }[/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]= { 4, 11, 4 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.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."
10 131 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{11, 6}, {5, 9}, {8, 10}, {9, 11}, {7, 1}, {6, 8}, {10, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 7}]
In[14]:= Draw[ap]
10 131 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 [-11][1]
Hyperbolic Volume 9.46502
A-Polynomial See Data:10 131/A-polynomial

[edit Notes for 10 131'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 10 131's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ 1-2 z^4 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 31, -2 }
Jones polynomial [math]\displaystyle{ 2 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +5 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4-z^2 a^4+2 z^2 a^2+2 a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+9 z^3 a^9-3 z a^9+z^8 a^8-z^6 a^8-4 z^4 a^8+2 z^2 a^8+a^8+4 z^7 a^7-12 z^5 a^7+10 z^3 a^7-5 z a^7+z^8 a^6-2 z^4 a^6-3 z^2 a^6+2 a^6+2 z^7 a^5-3 z^5 a^5+2 z^3 a^5-z a^5+2 z^6 a^4-2 z^4 a^4+2 z^2 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 a^2 }[/math]
The A2 invariant [math]\displaystyle{ q^{28}+q^{22}-2 q^{20}-q^{18}-q^{16}-q^{14}+q^{12}+2 q^8+q^6+2 q^2 }[/math]
The G2 invariant [math]\displaystyle{ q^{142}-q^{140}+3 q^{138}-5 q^{136}+3 q^{134}-2 q^{132}-4 q^{130}+10 q^{128}-14 q^{126}+14 q^{124}-7 q^{122}-4 q^{120}+14 q^{118}-19 q^{116}+18 q^{114}-8 q^{112}-2 q^{110}+13 q^{108}-15 q^{106}+12 q^{104}+q^{102}-8 q^{100}+14 q^{98}-11 q^{96}+2 q^{94}+7 q^{92}-15 q^{90}+19 q^{88}-18 q^{86}+9 q^{84}+2 q^{82}-17 q^{80}+21 q^{78}-25 q^{76}+14 q^{74}-4 q^{72}-11 q^{70}+16 q^{68}-18 q^{66}+11 q^{64}+q^{62}-12 q^{60}+14 q^{58}-9 q^{56}-q^{54}+11 q^{52}-15 q^{50}+14 q^{48}-4 q^{46}-3 q^{44}+10 q^{42}-14 q^{40}+14 q^{38}-7 q^{36}+2 q^{34}+4 q^{32}-7 q^{30}+8 q^{28}-5 q^{26}+6 q^{24}-q^{22}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_131.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{19}-q^{17}+q^{15}-2 q^{13}+2 q^5-q^3+2 q }[/math]
2 [math]\displaystyle{ q^{54}-q^{52}-2 q^{50}+3 q^{48}+q^{46}-5 q^{44}+2 q^{42}+4 q^{40}-5 q^{38}+5 q^{34}-2 q^{32}-3 q^{30}+4 q^{28}+q^{26}-4 q^{24}+3 q^{20}-2 q^{18}-5 q^{16}+5 q^{14}+q^{12}-5 q^{10}+4 q^8+2 q^6-2 q^4+2 q^2+1 }[/math]
3 [math]\displaystyle{ q^{105}-q^{103}-2 q^{101}+4 q^{97}+3 q^{95}-5 q^{93}-7 q^{91}+3 q^{89}+11 q^{87}+2 q^{85}-13 q^{83}-9 q^{81}+11 q^{79}+14 q^{77}-4 q^{75}-17 q^{73}-q^{71}+15 q^{69}+8 q^{67}-14 q^{65}-12 q^{63}+11 q^{61}+14 q^{59}-8 q^{57}-16 q^{55}+6 q^{53}+16 q^{51}-3 q^{49}-16 q^{47}+q^{45}+13 q^{43}+7 q^{41}-11 q^{39}-11 q^{37}+3 q^{35}+16 q^{33}+3 q^{31}-17 q^{29}-10 q^{27}+15 q^{25}+14 q^{23}-11 q^{21}-14 q^{19}+5 q^{17}+10 q^{15}-q^{13}-6 q^{11}+q^9+4 q^7-q^5+q^3+2 q^{-1} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{28}+q^{22}-2 q^{20}-q^{18}-q^{16}-q^{14}+q^{12}+2 q^8+q^6+2 q^2 }[/math]
1,1 [math]\displaystyle{ q^{76}-2 q^{74}+6 q^{72}-14 q^{70}+21 q^{68}-32 q^{66}+44 q^{64}-48 q^{62}+47 q^{60}-40 q^{58}+28 q^{56}-6 q^{54}-22 q^{52}+38 q^{50}-60 q^{48}+76 q^{46}-84 q^{44}+88 q^{42}-74 q^{40}+66 q^{38}-40 q^{36}+18 q^{34}+4 q^{32}-26 q^{30}+36 q^{28}-46 q^{26}+36 q^{24}-38 q^{22}+28 q^{20}-24 q^{18}+14 q^{16}-8 q^{14}+13 q^{12}+4 q^8+2 q^4+2 q^2 }[/math]
2,0 [math]\displaystyle{ q^{72}-q^{68}-q^{66}+q^{64}+2 q^{62}-2 q^{60}-2 q^{58}+q^{56}+q^{54}-2 q^{52}-3 q^{50}+2 q^{48}+4 q^{46}+2 q^{40}+2 q^{38}-q^{30}-2 q^{28}-q^{26}-4 q^{24}-6 q^{22}+q^{20}+2 q^{18}-q^{16}+5 q^{12}+5 q^{10}-q^6+3 q^4+q^2 }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{60}-q^{58}+q^{56}-4 q^{52}+q^{50}-q^{48}-2 q^{46}+5 q^{44}+3 q^{42}+6 q^{38}+q^{36}-4 q^{34}-2 q^{32}-3 q^{30}-3 q^{28}-4 q^{26}-q^{24}+2 q^{22}-3 q^{20}+5 q^{16}-2 q^{14}+2 q^{12}+6 q^{10}+3 q^4 }[/math]
1,0,0 [math]\displaystyle{ q^{37}+q^{33}+q^{29}-2 q^{27}-q^{25}-2 q^{23}-q^{21}-q^{19}+q^{15}+2 q^{11}+q^9+2 q^7+2 q^3 }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{78}+2 q^{72}-4 q^{68}-2 q^{66}-q^{64}-5 q^{62}-5 q^{60}+3 q^{58}+8 q^{56}+3 q^{54}+6 q^{52}+12 q^{50}+6 q^{48}-3 q^{46}-q^{44}-4 q^{42}-10 q^{40}-8 q^{38}-4 q^{36}-4 q^{34}-6 q^{32}+q^{30}+3 q^{28}-3 q^{26}-q^{24}+5 q^{22}+3 q^{20}+3 q^{16}+6 q^{14}+4 q^{12}+q^{10}+q^8+3 q^6 }[/math]
1,0,0,0 [math]\displaystyle{ q^{46}+q^{42}+q^{40}+q^{36}-2 q^{34}-q^{32}-2 q^{30}-2 q^{28}-q^{26}-q^{24}+q^{18}+2 q^{14}+q^{12}+2 q^{10}+2 q^8+2 q^4 }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{60}-q^{58}+3 q^{56}-4 q^{54}+4 q^{52}-5 q^{50}+5 q^{48}-4 q^{46}+3 q^{44}-q^{42}-2 q^{40}+4 q^{38}-7 q^{36}+8 q^{34}-10 q^{32}+9 q^{30}-9 q^{28}+6 q^{26}-5 q^{24}+2 q^{22}+q^{20}-2 q^{18}+5 q^{16}-4 q^{14}+6 q^{12}-4 q^{10}+4 q^8-2 q^6+3 q^4 }[/math]
1,0 [math]\displaystyle{ q^{98}-q^{94}-q^{92}+2 q^{90}+2 q^{88}-3 q^{86}-4 q^{84}+4 q^{80}+q^{78}-5 q^{76}-3 q^{74}+4 q^{72}+6 q^{70}+q^{68}-4 q^{66}+5 q^{62}+4 q^{60}-2 q^{58}-3 q^{56}+q^{54}+2 q^{52}-3 q^{50}-5 q^{48}-q^{46}+3 q^{44}-q^{42}-5 q^{40}-3 q^{38}+3 q^{36}+3 q^{34}-3 q^{32}-4 q^{30}+q^{28}+6 q^{26}+q^{24}-3 q^{22}-2 q^{20}+4 q^{18}+5 q^{16}+q^{14}-2 q^{12}-q^{10}+q^8+3 q^6 }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{82}-q^{80}+2 q^{78}-3 q^{76}+3 q^{74}-5 q^{72}+2 q^{70}-5 q^{68}+4 q^{66}-3 q^{64}+3 q^{62}+q^{60}+4 q^{58}+5 q^{56}+6 q^{52}-5 q^{50}+5 q^{48}-10 q^{46}+4 q^{44}-11 q^{42}+3 q^{40}-9 q^{38}+3 q^{36}-4 q^{34}+2 q^{32}-q^{30}-q^{28}+2 q^{26}-2 q^{24}+5 q^{22}-3 q^{20}+5 q^{18}-q^{16}+7 q^{14}+3 q^{10}-q^8+3 q^6 }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{142}-q^{140}+3 q^{138}-5 q^{136}+3 q^{134}-2 q^{132}-4 q^{130}+10 q^{128}-14 q^{126}+14 q^{124}-7 q^{122}-4 q^{120}+14 q^{118}-19 q^{116}+18 q^{114}-8 q^{112}-2 q^{110}+13 q^{108}-15 q^{106}+12 q^{104}+q^{102}-8 q^{100}+14 q^{98}-11 q^{96}+2 q^{94}+7 q^{92}-15 q^{90}+19 q^{88}-18 q^{86}+9 q^{84}+2 q^{82}-17 q^{80}+21 q^{78}-25 q^{76}+14 q^{74}-4 q^{72}-11 q^{70}+16 q^{68}-18 q^{66}+11 q^{64}+q^{62}-12 q^{60}+14 q^{58}-9 q^{56}-q^{54}+11 q^{52}-15 q^{50}+14 q^{48}-4 q^{46}-3 q^{44}+10 q^{42}-14 q^{40}+14 q^{38}-7 q^{36}+2 q^{34}+4 q^{32}-7 q^{30}+8 q^{28}-5 q^{26}+6 q^{24}-q^{22}+2 q^{18}-2 q^{16}+3 q^{14}-q^{12}+2 q^{10}+q^8 }[/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["10 131"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 1-2 z^4 }[/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]= { 31, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 2 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +5 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^2 a^8+a^8-z^4 a^6-2 z^2 a^6-2 a^6-z^4 a^4-z^2 a^4+2 z^2 a^2+2 a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+4 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+9 z^3 a^9-3 z a^9+z^8 a^8-z^6 a^8-4 z^4 a^8+2 z^2 a^8+a^8+4 z^7 a^7-12 z^5 a^7+10 z^3 a^7-5 z a^7+z^8 a^6-2 z^4 a^6-3 z^2 a^6+2 a^6+2 z^7 a^5-3 z^5 a^5+2 z^3 a^5-z a^5+2 z^6 a^4-2 z^4 a^4+2 z^2 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 a^2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_14, 9_8,}

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["10 131"];
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{ -2 t^2+8 t-11+8 t^{-1} -2 t^{-2} }[/math], [math]\displaystyle{ 2 q^{-1} -3 q^{-2} +5 q^{-3} -5 q^{-4} +5 q^{-5} -5 q^{-6} +3 q^{-7} -2 q^{-8} + q^{-9} }[/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]= {8_14, 9_8,}
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: (0, 2)
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{ 0 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -\frac{128}{3} }[/math] [math]\displaystyle{ -80 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 456 }[/math] [math]\displaystyle{ -\frac{464}{3} }[/math] [math]\displaystyle{ \frac{1952}{3} }[/math] [math]\displaystyle{ \frac{200}{3} }[/math] [math]\displaystyle{ 56 }[/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 10 131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-10χ
-1        22
-3       21-1
-5      31 2
-7     22  0
-9    33   0
-11   22    0
-13  13     -2
-15 12      1
-17 1       -1
-191        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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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^{-1} + q^{-2} -4 q^{-3} +5 q^{-4} +3 q^{-5} -13 q^{-6} +11 q^{-7} +7 q^{-8} -23 q^{-9} +14 q^{-10} +12 q^{-11} -26 q^{-12} +10 q^{-13} +17 q^{-14} -23 q^{-15} +3 q^{-16} +18 q^{-17} -16 q^{-18} -2 q^{-19} +13 q^{-20} -7 q^{-21} -4 q^{-22} +6 q^{-23} - q^{-24} -2 q^{-25} + q^{-26} }[/math]
3 [math]\displaystyle{ 2 q^{-1} -2 q^{-2} + q^{-3} -2 q^{-4} +7 q^{-5} -5 q^{-6} -6 q^{-7} +3 q^{-8} +18 q^{-9} -10 q^{-10} -25 q^{-11} +6 q^{-12} +43 q^{-13} -9 q^{-14} -50 q^{-15} - q^{-16} +63 q^{-17} +4 q^{-18} -63 q^{-19} -15 q^{-20} +63 q^{-21} +22 q^{-22} -57 q^{-23} -27 q^{-24} +46 q^{-25} +35 q^{-26} -38 q^{-27} -37 q^{-28} +24 q^{-29} +43 q^{-30} -16 q^{-31} -40 q^{-32} + q^{-33} +41 q^{-34} +6 q^{-35} -33 q^{-36} -15 q^{-37} +25 q^{-38} +19 q^{-39} -15 q^{-40} -18 q^{-41} +5 q^{-42} +15 q^{-43} -9 q^{-45} -3 q^{-46} +5 q^{-47} +2 q^{-48} - q^{-49} -2 q^{-50} + q^{-51} }[/math]
4 [math]\displaystyle{ 1+ q^{-1} -2 q^{-2} - q^{-3} +4 q^{-4} -2 q^{-5} +2 q^{-6} -7 q^{-7} -4 q^{-8} +22 q^{-9} -5 q^{-11} -37 q^{-12} -17 q^{-13} +73 q^{-14} +34 q^{-15} -11 q^{-16} -108 q^{-17} -72 q^{-18} +132 q^{-19} +115 q^{-20} +21 q^{-21} -184 q^{-22} -170 q^{-23} +150 q^{-24} +192 q^{-25} +92 q^{-26} -207 q^{-27} -256 q^{-28} +119 q^{-29} +211 q^{-30} +154 q^{-31} -174 q^{-32} -282 q^{-33} +76 q^{-34} +172 q^{-35} +181 q^{-36} -117 q^{-37} -262 q^{-38} +39 q^{-39} +112 q^{-40} +186 q^{-41} -57 q^{-42} -222 q^{-43} + q^{-44} +44 q^{-45} +180 q^{-46} +12 q^{-47} -168 q^{-48} -38 q^{-49} -29 q^{-50} +151 q^{-51} +74 q^{-52} -89 q^{-53} -50 q^{-54} -94 q^{-55} +85 q^{-56} +95 q^{-57} -5 q^{-58} -17 q^{-59} -112 q^{-60} +8 q^{-61} +58 q^{-62} +36 q^{-63} +32 q^{-64} -73 q^{-65} -27 q^{-66} +5 q^{-67} +21 q^{-68} +46 q^{-69} -21 q^{-70} -16 q^{-71} -15 q^{-72} -3 q^{-73} +25 q^{-74} -7 q^{-77} -7 q^{-78} +6 q^{-79} + q^{-80} +2 q^{-81} - q^{-82} -2 q^{-83} + q^{-84} }[/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.

10 130.gif

10_130

10 132.gif

10_132

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