10 20: Difference between revisions

From Knot Atlas
Jump to navigationJump to search
No edit summary
No edit summary
 
Line 1: Line 1:
<!-- WARNING! WARNING! WARNING!
<!-- WARNING! WARNING! WARNING!
<!-- This page was generated from the splice template [[Rolfsen_Splice_Base]]. Please do not edit!
<!-- This page was generated from the splice base [[Rolfsen_Splice_Base]]. Please do not edit!
<!-- You probably want to edit the template referred to immediately below. (See [[Category:Knot Page Template]].)
<!-- You probably want to edit the template referred to immediately below. (See [[Category:Knot Page Template]].)
<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Rolfsen_Splice_Base]]. -->
<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Rolfsen_Splice_Base]]. -->
<!-- <math>\text{Null}</math> -->
<!-- -->
<!-- <math>\text{Null}</math> -->
<!-- -->
{{Rolfsen Knot Page|
{{Rolfsen Knot Page|
n = 10 |
n = 10 |
Line 45: Line 45:
coloured_jones_3 = <math>q^9-q^8+3 q^5-3 q^4-q^3-q^2+7 q-3-4 q^{-1} -4 q^{-2} +11 q^{-3} -4 q^{-5} -9 q^{-6} +8 q^{-7} +6 q^{-8} + q^{-9} -9 q^{-10} -5 q^{-11} +6 q^{-12} +11 q^{-13} -2 q^{-14} -15 q^{-15} -2 q^{-16} +15 q^{-17} +8 q^{-18} -16 q^{-19} -7 q^{-20} +9 q^{-21} +12 q^{-22} -8 q^{-23} -11 q^{-24} +13 q^{-26} +5 q^{-27} -12 q^{-28} -12 q^{-29} +12 q^{-30} +18 q^{-31} -10 q^{-32} -22 q^{-33} +6 q^{-34} +24 q^{-35} - q^{-36} -23 q^{-37} -4 q^{-38} +20 q^{-39} +6 q^{-40} -13 q^{-41} -9 q^{-42} +9 q^{-43} +7 q^{-44} -4 q^{-45} -5 q^{-46} +2 q^{-47} +2 q^{-48} -2 q^{-50} + q^{-51} </math> |
coloured_jones_3 = <math>q^9-q^8+3 q^5-3 q^4-q^3-q^2+7 q-3-4 q^{-1} -4 q^{-2} +11 q^{-3} -4 q^{-5} -9 q^{-6} +8 q^{-7} +6 q^{-8} + q^{-9} -9 q^{-10} -5 q^{-11} +6 q^{-12} +11 q^{-13} -2 q^{-14} -15 q^{-15} -2 q^{-16} +15 q^{-17} +8 q^{-18} -16 q^{-19} -7 q^{-20} +9 q^{-21} +12 q^{-22} -8 q^{-23} -11 q^{-24} +13 q^{-26} +5 q^{-27} -12 q^{-28} -12 q^{-29} +12 q^{-30} +18 q^{-31} -10 q^{-32} -22 q^{-33} +6 q^{-34} +24 q^{-35} - q^{-36} -23 q^{-37} -4 q^{-38} +20 q^{-39} +6 q^{-40} -13 q^{-41} -9 q^{-42} +9 q^{-43} +7 q^{-44} -4 q^{-45} -5 q^{-46} +2 q^{-47} +2 q^{-48} -2 q^{-50} + q^{-51} </math> |
coloured_jones_4 = <math>q^{16}-q^{15}+3 q^{11}-4 q^{10}+9 q^6-9 q^5-2 q^4-3 q^3+q^2+22 q-13-6 q^{-1} -14 q^{-2} +44 q^{-4} -11 q^{-5} -9 q^{-6} -37 q^{-7} -13 q^{-8} +73 q^{-9} +8 q^{-10} - q^{-11} -73 q^{-12} -50 q^{-13} +96 q^{-14} +45 q^{-15} +33 q^{-16} -106 q^{-17} -109 q^{-18} +94 q^{-19} +81 q^{-20} +85 q^{-21} -114 q^{-22} -159 q^{-23} +73 q^{-24} +89 q^{-25} +125 q^{-26} -99 q^{-27} -180 q^{-28} +57 q^{-29} +77 q^{-30} +137 q^{-31} -83 q^{-32} -174 q^{-33} +53 q^{-34} +56 q^{-35} +129 q^{-36} -63 q^{-37} -153 q^{-38} +47 q^{-39} +29 q^{-40} +110 q^{-41} -38 q^{-42} -119 q^{-43} +42 q^{-44} -5 q^{-45} +80 q^{-46} -13 q^{-47} -74 q^{-48} +45 q^{-49} -34 q^{-50} +40 q^{-51} -5 q^{-52} -33 q^{-53} +59 q^{-54} -41 q^{-55} +6 q^{-56} -16 q^{-57} -16 q^{-58} +69 q^{-59} -22 q^{-60} -4 q^{-61} -29 q^{-62} -22 q^{-63} +57 q^{-64} -2 q^{-65} +6 q^{-66} -23 q^{-67} -28 q^{-68} +29 q^{-69} +3 q^{-70} +13 q^{-71} -8 q^{-72} -19 q^{-73} +10 q^{-74} - q^{-75} +7 q^{-76} -7 q^{-78} +3 q^{-79} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} </math> |
coloured_jones_4 = <math>q^{16}-q^{15}+3 q^{11}-4 q^{10}+9 q^6-9 q^5-2 q^4-3 q^3+q^2+22 q-13-6 q^{-1} -14 q^{-2} +44 q^{-4} -11 q^{-5} -9 q^{-6} -37 q^{-7} -13 q^{-8} +73 q^{-9} +8 q^{-10} - q^{-11} -73 q^{-12} -50 q^{-13} +96 q^{-14} +45 q^{-15} +33 q^{-16} -106 q^{-17} -109 q^{-18} +94 q^{-19} +81 q^{-20} +85 q^{-21} -114 q^{-22} -159 q^{-23} +73 q^{-24} +89 q^{-25} +125 q^{-26} -99 q^{-27} -180 q^{-28} +57 q^{-29} +77 q^{-30} +137 q^{-31} -83 q^{-32} -174 q^{-33} +53 q^{-34} +56 q^{-35} +129 q^{-36} -63 q^{-37} -153 q^{-38} +47 q^{-39} +29 q^{-40} +110 q^{-41} -38 q^{-42} -119 q^{-43} +42 q^{-44} -5 q^{-45} +80 q^{-46} -13 q^{-47} -74 q^{-48} +45 q^{-49} -34 q^{-50} +40 q^{-51} -5 q^{-52} -33 q^{-53} +59 q^{-54} -41 q^{-55} +6 q^{-56} -16 q^{-57} -16 q^{-58} +69 q^{-59} -22 q^{-60} -4 q^{-61} -29 q^{-62} -22 q^{-63} +57 q^{-64} -2 q^{-65} +6 q^{-66} -23 q^{-67} -28 q^{-68} +29 q^{-69} +3 q^{-70} +13 q^{-71} -8 q^{-72} -19 q^{-73} +10 q^{-74} - q^{-75} +7 q^{-76} -7 q^{-78} +3 q^{-79} - q^{-80} +2 q^{-81} -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>
Line 54: Line 54:
<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, 20]]</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[11, 14, 12, 15], X[3, 13, 4, 12], X[13, 3, 14, 2],
<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, 20]]</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[11, 14, 12, 15], X[3, 13, 4, 12], X[13, 3, 14, 2],
X[5, 18, 6, 19], X[7, 20, 8, 1], X[19, 6, 20, 7], X[9, 16, 10, 17],
X[5, 18, 6, 19], X[7, 20, 8, 1], X[19, 6, 20, 7], X[9, 16, 10, 17],
X[15, 10, 16, 11], X[17, 8, 18, 9]]</nowiki></pre></td></tr>
X[15, 10, 16, 11], X[17, 8, 18, 9]]</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, 20]]</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, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10,
<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, 20]]</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, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10,
5, -7, 6]</nowiki></pre></td></tr>
5, -7, 6]</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, 20]]</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, 12, 18, 20, 16, 14, 2, 10, 8, 6]</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, 20]]</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[5, {-1, -1, -1, -1, -2, 1, -2, -3, 2, 4, -3, 4}]</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, 20]]</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>{5, 12}</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, 20]]</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, 12, 18, 20, 16, 14, 2, 10, 8, 6]</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>5</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, 20]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_20_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, 20]]&) /@ {
<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, 20]]</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[5, {-1, -1, -1, -1, -2, 1, -2, -3, 2, 4, -3, 4}]</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>{5, 12}</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, 20]]</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>5</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, 20]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:10_20_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, 20]]&) /@ {
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, 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, 20]][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> 3 9 2
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 2, 2, 2, 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, 20]][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> 3 9 2
-11 - -- + - + 9 t - 3 t
-11 - -- + - + 9 t - 3 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, 20]][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> 2 4
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
1 - 3 z - 3 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, 20]][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[10, 20], Knot[10, 162], Knot[11, NonAlternating, 117]}</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, 20]], KnotSignature[Knot[10, 20]]}</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>{35, -2}</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4
1 - 3 z - 3 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, 20]][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 4 5 6 5 4 3
<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[10, 20], Knot[10, 162], Knot[11, NonAlternating, 117]}</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, 20]], KnotSignature[Knot[10, 20]]}</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>{35, -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, 20]][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 4 5 6 5 4 3
-1 + q - -- + -- - -- + -- - -- + -- - -- + - + q
-1 + 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, 20]}</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, 20]][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 -20 -14 -10 -4 2 2 4
<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, 20]}</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, 20]][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 -20 -14 -10 -4 2 2 4
1 + q + q - q - q - q - q + -- + q + q
1 + q + q - q - q - q - q + -- + q + q
2
2
q</nowiki></pre></td></tr>
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, 20]][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 2 4 2 6 2 8 2 2 4
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 20]][a, z]</nowiki></code></td></tr>
<tr align=left>
<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 2 4 2 6 2 8 2 2 4
2 - a - a + a + z - 2 a z - a z - 2 a z + a z - a z -
2 - a - a + a + z - 2 a z - a z - 2 a z + a z - a z -
4 4 6 4
4 4 6 4
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[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 20]][a, z]</nowiki></pre></td></tr>
<table><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
<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, 20]][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 + a + a + a - a z - a z + 3 a z + 2 a z - a z - 3 z -
2 + a + a + a - a z - a z + 3 a z + 2 a z - a z - 3 z -
Line 126: Line 212:
4 8 6 8 8 8 5 9 7 9
4 8 6 8 8 8 5 9 7 9
a z + 3 a z + 2 a z + a z + a z</nowiki></pre></td></tr>
a z + 3 a z + 2 a z + 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, 20]], Vassiliev[3][Knot[10, 20]]}</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>{-3, 6}</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, 20]][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 3 1 1 1 2 1 2
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 20]], Vassiliev[3][Knot[10, 20]]}</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>{-3, 6}</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, 20]][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 3 1 1 1 2 1 2
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 143: Line 239:
---- + - + q t
---- + - + q t
3 q
3 q
q t</nowiki></pre></td></tr>
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, 20], 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 5 6 3 12 7 9 17 5 15
<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, 20], 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 5 6 3 12 7 9 17 5 15
-3 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- +
-3 + q - --- + --- - --- - --- + --- - --- - --- + --- - --- - --- +
25 23 22 21 20 19 18 17 16 15
25 23 22 21 20 19 18 17 16 15
Line 156: Line 257:
3 4
3 4
3 q - q + q</nowiki></pre></td></tr>
3 q - q + q</nowiki></code></td></tr>
</table> }}
</table> }}

Latest revision as of 17:58, 1 September 2005

10 19.gif

10_19

10 21.gif

10_21

10 20.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 20 at Knotilus!

[edit 10 20 Quick Notes]
[edit 10 20 Further Notes and Views]


Knot presentations

Planar diagram presentation X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,18,6,19 X7,20,8,1 X19,6,20,7 X9,16,10,17 X15,10,16,11 X17,8,18,9
Gauss code -1, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10, 5, -7, 6
Dowker-Thistlethwaite code 4 12 18 20 16 14 2 10 8 6
Conway Notation [352]


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

Length is 12, width is 5,

Braid index is 5

10 20 ML.gif 10 20 AP.gif
[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 9}, {8, 10}, {9, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 1}]

[edit Notes on presentations of 10 20]


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 20"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,18,6,19 X7,20,8,1 X19,6,20,7 X9,16,10,17 X15,10,16,11 X17,8,18,9
In[5]:= GaussCode[K]
Out[5]= -1, 4, -3, 1, -5, 7, -6, 10, -8, 9, -2, 3, -4, 2, -9, 8, -10, 5, -7, 6
In[6]:= DTCode[K]
Out[6]= 4 12 18 20 16 14 2 10 8 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [352]
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,-1,-2,1,-2,-3,2,4,-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.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.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 20 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{12, 3}, {4, 2}, {3, 11}, {1, 4}, {10, 12}, {11, 9}, {8, 10}, {9, 5}, {2, 6}, {5, 7}, {6, 8}, {7, 1}]
In[14]:= Draw[ap]
10 20 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 [-11][-1]
Hyperbolic Volume 8.31738
A-Polynomial See Data:10 20/A-polynomial

[edit Notes for 10 20'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 20's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -3 t^2+9 t-11+9 t^{-1} -3 t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ -3 z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 35, -2 }
Jones polynomial [math]\displaystyle{ q-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 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-a^6-z^4 a^4-z^2 a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^6 a^{10}-4 z^4 a^{10}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+7 z^3 a^9-z a^9+2 z^8 a^8-8 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+3 z^5 a^7-4 z^3 a^7+2 z a^7+3 z^8 a^6-12 z^6 a^6+17 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+9 z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+z^7 a^3-z^5 a^3+2 z^3 a^3-z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2 }[/math]
The A2 invariant [math]\displaystyle{ q^{28}+q^{22}-q^{20}-q^{14}-q^{10}-q^4+2 q^2+1+ q^{-2} + q^{-4} }[/math]
The G2 invariant [math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-2 q^{132}-q^{130}+6 q^{128}-9 q^{126}+10 q^{124}-9 q^{122}+3 q^{120}+4 q^{118}-11 q^{116}+16 q^{114}-13 q^{112}+9 q^{110}+q^{108}-8 q^{106}+12 q^{104}-9 q^{102}+3 q^{100}+3 q^{98}-7 q^{96}+7 q^{94}-2 q^{92}-4 q^{90}+11 q^{88}-12 q^{86}+9 q^{84}-4 q^{82}-5 q^{80}+10 q^{78}-15 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+5 q^{68}-12 q^{66}+13 q^{64}-10 q^{62}+4 q^{60}+q^{58}-7 q^{56}+7 q^{54}-3 q^{52}-2 q^{50}+5 q^{48}-6 q^{46}+q^{44}+2 q^{42}-7 q^{40}+7 q^{38}-5 q^{36}+3 q^{34}-q^{32}-3 q^{30}+4 q^{28}-5 q^{26}+5 q^{24}-5 q^{22}+4 q^{20}-3 q^{18}-q^{16}+4 q^{14}-6 q^{12}+8 q^{10}-4 q^8+3 q^6+q^4-q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 10_20.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^{19}-q^{17}+q^{15}-q^{13}+q^{11}-q^9-q^7+q^5-q^3+2 q+ q^{-3} }[/math]
2 [math]\displaystyle{ q^{54}-q^{52}-q^{50}+3 q^{48}-q^{46}-4 q^{44}+3 q^{42}+2 q^{40}-4 q^{38}+q^{36}+3 q^{34}-3 q^{32}-q^{30}+3 q^{28}-q^{26}-q^{24}+2 q^{22}+2 q^{20}-2 q^{18}-2 q^{16}+4 q^{14}-q^{12}-3 q^{10}+3 q^8-q^6-2 q^4+2 q^2-1+2 q^{-4} + q^{-10} }[/math]
3 [math]\displaystyle{ q^{105}-q^{103}-q^{101}+q^{99}+2 q^{97}-q^{95}-5 q^{93}+7 q^{89}+3 q^{87}-6 q^{85}-7 q^{83}+4 q^{81}+9 q^{79}-q^{77}-8 q^{75}-4 q^{73}+6 q^{71}+7 q^{69}-2 q^{67}-8 q^{65}-2 q^{63}+8 q^{61}+6 q^{59}-7 q^{57}-6 q^{55}+6 q^{53}+7 q^{51}-6 q^{49}-7 q^{47}+2 q^{45}+6 q^{43}-2 q^{41}-6 q^{39}+5 q^{35}+6 q^{33}-4 q^{31}-8 q^{29}+10 q^{25}+3 q^{23}-7 q^{21}-7 q^{19}+6 q^{17}+6 q^{15}+q^{13}-5 q^{11}-2 q^9+3 q^7+3 q^5-4 q- q^{-1} +2 q^{-3} +2 q^{-5} -2 q^{-7} - q^{-9} +2 q^{-13} + q^{-21} }[/math]
4 [math]\displaystyle{ q^{172}-q^{170}-q^{168}+q^{166}+2 q^{162}-3 q^{160}-3 q^{158}+2 q^{156}+2 q^{154}+9 q^{152}-3 q^{150}-11 q^{148}-5 q^{146}-q^{144}+18 q^{142}+9 q^{140}-6 q^{138}-13 q^{136}-18 q^{134}+10 q^{132}+16 q^{130}+10 q^{128}-20 q^{124}-8 q^{122}-2 q^{120}+11 q^{118}+21 q^{116}+2 q^{114}-8 q^{112}-25 q^{110}-14 q^{108}+20 q^{106}+27 q^{104}+13 q^{102}-28 q^{100}-36 q^{98}+4 q^{96}+33 q^{94}+30 q^{92}-15 q^{90}-40 q^{88}-10 q^{86}+24 q^{84}+29 q^{82}-5 q^{80}-30 q^{78}-11 q^{76}+16 q^{74}+22 q^{72}+q^{70}-19 q^{68}-11 q^{66}+10 q^{64}+14 q^{62}+8 q^{60}-8 q^{58}-20 q^{56}-8 q^{54}+8 q^{52}+29 q^{50}+14 q^{48}-26 q^{46}-34 q^{44}-13 q^{42}+37 q^{40}+45 q^{38}-7 q^{36}-43 q^{34}-41 q^{32}+18 q^{30}+51 q^{28}+17 q^{26}-20 q^{24}-43 q^{22}-6 q^{20}+30 q^{18}+22 q^{16}+3 q^{14}-26 q^{12}-13 q^{10}+10 q^8+13 q^6+11 q^4-11 q^2-10+ q^{-2} +5 q^{-4} +9 q^{-6} -4 q^{-8} -5 q^{-10} -2 q^{-12} +5 q^{-16} - q^{-18} - q^{-20} - q^{-22} - q^{-24} +2 q^{-26} + q^{-36} }[/math]
5 [math]\displaystyle{ q^{255}-q^{253}-q^{251}+q^{249}-q^{241}-2 q^{239}+2 q^{237}+5 q^{235}+2 q^{233}-q^{231}-6 q^{229}-9 q^{227}-5 q^{225}+9 q^{223}+17 q^{221}+11 q^{219}-q^{217}-19 q^{215}-25 q^{213}-13 q^{211}+14 q^{209}+31 q^{207}+26 q^{205}+5 q^{203}-24 q^{201}-36 q^{199}-23 q^{197}+8 q^{195}+29 q^{193}+29 q^{191}+16 q^{189}-6 q^{187}-22 q^{185}-28 q^{183}-21 q^{181}-6 q^{179}+18 q^{177}+43 q^{175}+44 q^{173}+10 q^{171}-45 q^{169}-75 q^{167}-54 q^{165}+19 q^{163}+95 q^{161}+99 q^{159}+19 q^{157}-88 q^{155}-130 q^{153}-66 q^{151}+60 q^{149}+144 q^{147}+108 q^{145}-23 q^{143}-139 q^{141}-133 q^{139}-15 q^{137}+115 q^{135}+146 q^{133}+48 q^{131}-91 q^{129}-141 q^{127}-62 q^{125}+62 q^{123}+123 q^{121}+72 q^{119}-42 q^{117}-103 q^{115}-63 q^{113}+28 q^{111}+79 q^{109}+53 q^{107}-20 q^{105}-65 q^{103}-42 q^{101}+16 q^{99}+48 q^{97}+35 q^{95}-6 q^{93}-42 q^{91}-42 q^{89}-8 q^{87}+38 q^{85}+54 q^{83}+36 q^{81}-20 q^{79}-78 q^{77}-75 q^{75}-3 q^{73}+89 q^{71}+121 q^{69}+48 q^{67}-83 q^{65}-158 q^{63}-99 q^{61}+55 q^{59}+181 q^{57}+146 q^{55}-18 q^{53}-164 q^{51}-174 q^{49}-33 q^{47}+134 q^{45}+174 q^{43}+58 q^{41}-87 q^{39}-144 q^{37}-81 q^{35}+46 q^{33}+110 q^{31}+71 q^{29}-17 q^{27}-72 q^{25}-55 q^{23}+45 q^{19}+41 q^{17}+4 q^{15}-28 q^{13}-25 q^{11}-3 q^9+18 q^7+20 q^5+3 q^3-13 q-13 q^{-1} -4 q^{-3} +10 q^{-5} +13 q^{-7} +3 q^{-9} -6 q^{-11} -7 q^{-13} -6 q^{-15} +2 q^{-17} +9 q^{-19} +3 q^{-21} - q^{-23} -2 q^{-25} -4 q^{-27} -2 q^{-29} +3 q^{-31} + q^{-33} - q^{-39} - q^{-41} + q^{-43} + q^{-55} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{28}+q^{22}-q^{20}-q^{14}-q^{10}-q^4+2 q^2+1+ q^{-2} + q^{-4} }[/math]
1,1 [math]\displaystyle{ q^{76}-2 q^{74}+4 q^{72}-8 q^{70}+15 q^{68}-22 q^{66}+30 q^{64}-38 q^{62}+43 q^{60}-44 q^{58}+38 q^{56}-28 q^{54}+11 q^{52}+8 q^{50}-26 q^{48}+44 q^{46}-59 q^{44}+64 q^{42}-68 q^{40}+64 q^{38}-55 q^{36}+46 q^{34}-28 q^{32}+18 q^{30}-3 q^{28}-4 q^{26}+12 q^{24}-12 q^{22}+13 q^{20}-12 q^{18}+10 q^{16}-14 q^{14}+11 q^{12}-18 q^{10}+14 q^8-18 q^6+16 q^4-12 q^2+12-6 q^{-2} +8 q^{-4} -2 q^{-6} +4 q^{-8} + q^{-12} }[/math]
2,0 [math]\displaystyle{ q^{72}+q^{64}-3 q^{60}-q^{58}+q^{56}+q^{54}-q^{52}-q^{50}+2 q^{48}+q^{46}-2 q^{44}-2 q^{42}+q^{40}+q^{38}-q^{36}+2 q^{32}+2 q^{30}+3 q^{26}+q^{24}-q^{22}+2 q^{20}+q^{18}-3 q^{16}-3 q^{14}+q^{12}-5 q^8-3 q^6+2 q^4+2 q^{-2} +3 q^{-4} + q^{-6} + q^{-8} + q^{-10} + q^{-12} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{60}-q^{58}+q^{54}-2 q^{52}+q^{50}+q^{48}-2 q^{46}+2 q^{44}-3 q^{40}+2 q^{38}+2 q^{36}-2 q^{34}+2 q^{32}+3 q^{30}-q^{28}+q^{22}-3 q^{20}-q^{18}+2 q^{16}-4 q^{14}-3 q^{12}+2 q^{10}-2 q^8-3 q^6+4 q^4+2 q^2+1+3 q^{-2} +2 q^{-4} + q^{-8} }[/math]
1,0,0 [math]\displaystyle{ q^{37}+q^{33}+q^{29}-q^{27}-q^{23}-q^{19}-q^{13}-q^9-q^5+2 q^3+q+2 q^{-1} + q^{-3} + q^{-5} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{60}-q^{58}+2 q^{56}-3 q^{54}+4 q^{52}-5 q^{50}+5 q^{48}-4 q^{46}+4 q^{44}-2 q^{42}+q^{40}+2 q^{38}-4 q^{36}+6 q^{34}-8 q^{32}+7 q^{30}-9 q^{28}+8 q^{26}-8 q^{24}+5 q^{22}-3 q^{20}+q^{18}-2 q^{14}+3 q^{12}-4 q^{10}+4 q^8-3 q^6+4 q^4-2 q^2+3- q^{-2} +2 q^{-4} + q^{-8} }[/math]
1,0 [math]\displaystyle{ q^{98}-q^{94}-q^{92}+q^{90}+2 q^{88}-q^{86}-3 q^{84}+4 q^{80}+2 q^{78}-4 q^{76}-4 q^{74}+2 q^{72}+5 q^{70}-5 q^{66}-3 q^{64}+3 q^{62}+4 q^{60}-3 q^{56}+3 q^{52}+q^{50}-q^{48}-q^{46}+3 q^{44}+2 q^{42}-2 q^{40}-3 q^{38}+2 q^{36}+3 q^{34}-q^{32}-4 q^{30}-q^{28}+3 q^{26}+q^{24}-4 q^{22}-4 q^{20}+3 q^{16}+q^{14}-3 q^{12}-3 q^{10}+4 q^6+2 q^4-1+ q^{-2} +2 q^{-4} +2 q^{-6} + q^{-14} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{142}-q^{140}+2 q^{138}-3 q^{136}+2 q^{134}-2 q^{132}-q^{130}+6 q^{128}-9 q^{126}+10 q^{124}-9 q^{122}+3 q^{120}+4 q^{118}-11 q^{116}+16 q^{114}-13 q^{112}+9 q^{110}+q^{108}-8 q^{106}+12 q^{104}-9 q^{102}+3 q^{100}+3 q^{98}-7 q^{96}+7 q^{94}-2 q^{92}-4 q^{90}+11 q^{88}-12 q^{86}+9 q^{84}-4 q^{82}-5 q^{80}+10 q^{78}-15 q^{76}+15 q^{74}-9 q^{72}+3 q^{70}+5 q^{68}-12 q^{66}+13 q^{64}-10 q^{62}+4 q^{60}+q^{58}-7 q^{56}+7 q^{54}-3 q^{52}-2 q^{50}+5 q^{48}-6 q^{46}+q^{44}+2 q^{42}-7 q^{40}+7 q^{38}-5 q^{36}+3 q^{34}-q^{32}-3 q^{30}+4 q^{28}-5 q^{26}+5 q^{24}-5 q^{22}+4 q^{20}-3 q^{18}-q^{16}+4 q^{14}-6 q^{12}+8 q^{10}-4 q^8+3 q^6+q^4-q^2+4-3 q^{-2} +4 q^{-4} - q^{-6} + q^{-8} + q^{-10} - q^{-12} +2 q^{-14} + q^{-18} }[/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 20"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -3 t^2+9 t-11+9 t^{-1} -3 t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -3 z^4-3 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]= { 35, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 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-a^6-z^4 a^4-z^2 a^4-z^4 a^2-2 z^2 a^2-a^2+z^2+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}+3 z^2 a^{10}+2 z^7 a^9-8 z^5 a^9+7 z^3 a^9-z a^9+2 z^8 a^8-8 z^6 a^8+9 z^4 a^8-5 z^2 a^8+a^8+z^9 a^7-3 z^7 a^7+3 z^5 a^7-4 z^3 a^7+2 z a^7+3 z^8 a^6-12 z^6 a^6+17 z^4 a^6-9 z^2 a^6+a^6+z^9 a^5-4 z^7 a^5+9 z^5 a^5-8 z^3 a^5+3 z a^5+z^8 a^4-2 z^6 a^4+3 z^4 a^4+z^7 a^3-z^5 a^3+2 z^3 a^3-z a^3+z^6 a^2-2 z^2 a^2+a^2+z^5 a-z^3 a-z a+z^4-3 z^2+2 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_162, K11n117,}

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 20"];
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+9 t-11+9 t^{-1} -3 t^{-2} }[/math], [math]\displaystyle{ q-1+3 q^{-1} -4 q^{-2} +5 q^{-3} -6 q^{-4} +5 q^{-5} -4 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]= {10_162, K11n117,}
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: (-3, 6)
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{ -12 }[/math] [math]\displaystyle{ 48 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 18 }[/math] [math]\displaystyle{ 62 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -896 }[/math] [math]\displaystyle{ -256 }[/math] [math]\displaystyle{ -272 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ -216 }[/math] [math]\displaystyle{ -744 }[/math] [math]\displaystyle{ \frac{30689}{10} }[/math] [math]\displaystyle{ \frac{6826}{15} }[/math] [math]\displaystyle{ \frac{7858}{15} }[/math] [math]\displaystyle{ \frac{2527}{6} }[/math] [math]\displaystyle{ -\frac{1951}{10} }[/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 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -8-7-6-5-4-3-2-1012χ
3          11
1           0
-1        31 2
-3       21  -1
-5      32   1
-7     32    -1
-9    23     -1
-11   23      1
-13  12       -1
-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}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=2 }[/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^4-q^3+3 q-3- q^{-1} +6 q^{-2} -7 q^{-3} +10 q^{-5} -13 q^{-6} +2 q^{-7} +15 q^{-8} -19 q^{-9} +2 q^{-10} +19 q^{-11} -19 q^{-12} - q^{-13} +19 q^{-14} -15 q^{-15} -5 q^{-16} +17 q^{-17} -9 q^{-18} -7 q^{-19} +12 q^{-20} -3 q^{-21} -6 q^{-22} +5 q^{-23} -2 q^{-25} + q^{-26} }[/math]
3 [math]\displaystyle{ q^9-q^8+3 q^5-3 q^4-q^3-q^2+7 q-3-4 q^{-1} -4 q^{-2} +11 q^{-3} -4 q^{-5} -9 q^{-6} +8 q^{-7} +6 q^{-8} + q^{-9} -9 q^{-10} -5 q^{-11} +6 q^{-12} +11 q^{-13} -2 q^{-14} -15 q^{-15} -2 q^{-16} +15 q^{-17} +8 q^{-18} -16 q^{-19} -7 q^{-20} +9 q^{-21} +12 q^{-22} -8 q^{-23} -11 q^{-24} +13 q^{-26} +5 q^{-27} -12 q^{-28} -12 q^{-29} +12 q^{-30} +18 q^{-31} -10 q^{-32} -22 q^{-33} +6 q^{-34} +24 q^{-35} - q^{-36} -23 q^{-37} -4 q^{-38} +20 q^{-39} +6 q^{-40} -13 q^{-41} -9 q^{-42} +9 q^{-43} +7 q^{-44} -4 q^{-45} -5 q^{-46} +2 q^{-47} +2 q^{-48} -2 q^{-50} + q^{-51} }[/math]
4 [math]\displaystyle{ q^{16}-q^{15}+3 q^{11}-4 q^{10}+9 q^6-9 q^5-2 q^4-3 q^3+q^2+22 q-13-6 q^{-1} -14 q^{-2} +44 q^{-4} -11 q^{-5} -9 q^{-6} -37 q^{-7} -13 q^{-8} +73 q^{-9} +8 q^{-10} - q^{-11} -73 q^{-12} -50 q^{-13} +96 q^{-14} +45 q^{-15} +33 q^{-16} -106 q^{-17} -109 q^{-18} +94 q^{-19} +81 q^{-20} +85 q^{-21} -114 q^{-22} -159 q^{-23} +73 q^{-24} +89 q^{-25} +125 q^{-26} -99 q^{-27} -180 q^{-28} +57 q^{-29} +77 q^{-30} +137 q^{-31} -83 q^{-32} -174 q^{-33} +53 q^{-34} +56 q^{-35} +129 q^{-36} -63 q^{-37} -153 q^{-38} +47 q^{-39} +29 q^{-40} +110 q^{-41} -38 q^{-42} -119 q^{-43} +42 q^{-44} -5 q^{-45} +80 q^{-46} -13 q^{-47} -74 q^{-48} +45 q^{-49} -34 q^{-50} +40 q^{-51} -5 q^{-52} -33 q^{-53} +59 q^{-54} -41 q^{-55} +6 q^{-56} -16 q^{-57} -16 q^{-58} +69 q^{-59} -22 q^{-60} -4 q^{-61} -29 q^{-62} -22 q^{-63} +57 q^{-64} -2 q^{-65} +6 q^{-66} -23 q^{-67} -28 q^{-68} +29 q^{-69} +3 q^{-70} +13 q^{-71} -8 q^{-72} -19 q^{-73} +10 q^{-74} - q^{-75} +7 q^{-76} -7 q^{-78} +3 q^{-79} - q^{-80} +2 q^{-81} -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 19.gif

10_19

10 21.gif

10_21

Retrieved from "https://katlas.org/index.php?title=10_20&oldid=61146"