10 97: Difference between revisions
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coloured_jones_2 = <math>q^{26}-4 q^{25}+q^{24}+15 q^{23}-21 q^{22}-12 q^{21}+57 q^{20}-37 q^{19}-57 q^{18}+112 q^{17}-30 q^{16}-119 q^{15}+148 q^{14}+q^{13}-165 q^{12}+148 q^{11}+36 q^{10}-172 q^9+113 q^8+53 q^7-130 q^6+60 q^5+43 q^4-65 q^3+20 q^2+18 q-19+5 q^{-1} +3 q^{-2} -3 q^{-3} + q^{-4} </math> | |
coloured_jones_2 = <math>q^{26}-4 q^{25}+q^{24}+15 q^{23}-21 q^{22}-12 q^{21}+57 q^{20}-37 q^{19}-57 q^{18}+112 q^{17}-30 q^{16}-119 q^{15}+148 q^{14}+q^{13}-165 q^{12}+148 q^{11}+36 q^{10}-172 q^9+113 q^8+53 q^7-130 q^6+60 q^5+43 q^4-65 q^3+20 q^2+18 q-19+5 q^{-1} +3 q^{-2} -3 q^{-3} + q^{-4} </math> | |
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computer_talk = |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
</tr> |
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<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15: |
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr> |
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</table> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 97]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[4, 2, 5, 1], X[12, 6, 13, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Knot[10, 97]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[4, 2, 5, 1], X[12, 6, 13, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], |
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X[16, 12, 17, 11], X[10, 18, 11, 17], X[18, 8, 19, 7], |
X[16, 12, 17, 11], X[10, 18, 11, 17], X[18, 8, 19, 7], |
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X[20, 14, 1, 13], X[14, 20, 15, 19], X[6, 16, 7, 15]]</nowiki></ |
X[20, 14, 1, 13], X[14, 20, 15, 19], X[6, 16, 7, 15]]</nowiki></code></td></tr> |
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</table> |
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<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, 97]]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[10, 97]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, |
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-7, 9, -8]</nowiki></ |
-7, 9, -8]</nowiki></code></td></tr> |
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</table> |
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<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, 97]]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 8, 12, 18, 2, 16, 20, 6, 10, 14]</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[10, 97]]</nowiki></code></td></tr> |
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<tr align=left> |
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<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> |
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< |
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 8, 12, 18, 2, 16, 20, 6, 10, 14]</nowiki></code></td></tr> |
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</table> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 97]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_97_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>br = BR[Knot[10, 97]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BR[5, {1, 1, 2, -1, 2, 1, -3, 2, -1, 2, 3, -4, 3, -4}]</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td> |
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<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> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{5, 14}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>BraidIndex[Knot[10, 97]]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>5</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Knot[10, 97]]]</nowiki></code></td></tr> |
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<tr align=left><td></td><td>[[Image:10_97_ML.gif]]</td></tr><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> (#[Knot[10, 97]]&) /@ { |
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SymmetryType, UnknottingNumber, ThreeGenus, |
SymmetryType, UnknottingNumber, ThreeGenus, |
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BridgeIndex, SuperBridgeIndex, NakanishiIndex |
BridgeIndex, SuperBridgeIndex, NakanishiIndex |
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}</nowiki></ |
}</nowiki></code></td></tr> |
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<tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Reversible, 2, 2, 3, NotAvailable, 1}</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 2, 2, 3, NotAvailable, 1}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>alex = Alexander[Knot[10, 97]][t]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 5 22 2 |
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-33 - -- + -- + 22 t - 5 t |
-33 - -- + -- + 22 t - 5 t |
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2 t |
2 t |
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t</nowiki></ |
t</nowiki></code></td></tr> |
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</table> |
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<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, 97]][z]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td> |
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1 + 2 z - 5 z</nowiki></pre></td></tr> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[10, 97]][z]</nowiki></code></td></tr> |
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<tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 97]}</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 |
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1 + 2 z - 5 z</nowiki></code></td></tr> |
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<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, 97]][q]</nowiki></pre></td></tr> |
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</table> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 1 2 3 4 5 6 7 8 9 |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td> |
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<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> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 97]}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[13]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{KnotDet[Knot[10, 97]], KnotSignature[Knot[10, 97]]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[13]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{87, 2}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[14]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Jones[Knot[10, 97]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[14]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 1 2 3 4 5 6 7 8 9 |
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-3 + - + 7 q - 11 q + 14 q - 14 q + 14 q - 11 q + 7 q - 4 q + q |
-3 + - + 7 q - 11 q + 14 q - 14 q + 14 q - 11 q + 7 q - 4 q + q |
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q</nowiki></ |
q</nowiki></code></td></tr> |
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</table> |
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<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> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 97]}</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td> |
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<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> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[15]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[10, 97]}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[16]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Knot[10, 97]][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[16]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -4 -2 2 4 6 8 10 12 14 |
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-1 + q - q + 4 q - 2 q + q + 2 q - 2 q + 2 q - 2 q + |
-1 + q - q + 4 q - 2 q + q + 2 q - 2 q + 2 q - 2 q + |
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16 18 20 22 24 26 28 |
16 18 20 22 24 26 28 |
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2 q + q - 2 q + 3 q - 2 q - 2 q + q</nowiki></ |
2 q + q - 2 q + 3 q - 2 q - 2 q + q</nowiki></code></td></tr> |
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</table> |
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<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, 97]][a, z]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2 2 2 4 4 4 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[17]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Knot[10, 97]][a, z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[17]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 2 2 2 4 4 4 |
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-8 2 2 2 2 z 2 z 4 z 2 z z 3 z z |
-8 2 2 2 2 z 2 z 4 z 2 z z 3 z z |
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-a + -- - -- + -- + z + -- + ---- - ---- + ---- - -- - ---- - -- |
-a + -- - -- + -- + z + -- + ---- - ---- + ---- - -- - ---- - -- |
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6 4 2 8 6 4 2 6 4 2 |
6 4 2 8 6 4 2 6 4 2 |
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a a a a a a a a a a</nowiki></ |
a a a a a a a a a a</nowiki></code></td></tr> |
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</table> |
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<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, 97]][a, z]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2 2 2 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[10, 97]][a, z]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[18]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 2 2 2 |
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-8 2 2 2 4 z 6 z 2 z 2 z z 3 z 10 z |
-8 2 2 2 4 z 6 z 2 z 2 z z 3 z 10 z |
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-a - -- - -- - -- - --- - --- - --- - z + --- + -- + ---- + ----- + |
-a - -- - -- - -- - --- - --- - --- - z + --- + -- + ---- + ----- + |
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Line 138: | Line 224: | ||
----- + ---- + ---- + ---- |
----- + ---- + ---- + ---- |
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6 4 7 5 |
6 4 7 5 |
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a a a a</nowiki></ |
a a a a</nowiki></code></td></tr> |
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</table> |
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<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, 97]], Vassiliev[3][Knot[10, 97]]}</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{2, 4}</nowiki></pre></td></tr> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[10, 97]], Vassiliev[3][Knot[10, 97]]}</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[19]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{2, 4}</nowiki></code></td></tr> |
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</table> |
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<table><tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[20]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Knot[10, 97]][q, t]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[20]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 3 1 2 q 3 5 5 2 7 2 |
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5 q + 3 q + ----- + --- + - + 7 q t + 4 q t + 7 q t + 7 q t + |
5 q + 3 q + ----- + --- + - + 7 q t + 4 q t + 7 q t + 7 q t + |
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3 2 q t t |
3 2 q t t |
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Line 151: | Line 247: | ||
13 6 15 6 15 7 17 7 19 8 |
13 6 15 6 15 7 17 7 19 8 |
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3 q t + 4 q t + q t + 3 q t + q t</nowiki></ |
3 q t + 4 q t + q t + 3 q t + q t</nowiki></code></td></tr> |
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</table> |
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<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, 97], 2][q]</nowiki></pre></td></tr> |
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<table><tr align=left> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -4 3 3 5 2 3 4 5 |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[21]:=</code></td> |
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<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>ColouredJones[Knot[10, 97], 2][q]</nowiki></code></td></tr> |
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<tr align=left> |
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<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[21]:=</code></td> |
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<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -4 3 3 5 2 3 4 5 |
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-19 + q - -- + -- + - + 18 q + 20 q - 65 q + 43 q + 60 q - |
-19 + q - -- + -- + - + 18 q + 20 q - 65 q + 43 q + 60 q - |
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3 2 q |
3 2 q |
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Line 165: | Line 266: | ||
21 22 23 24 25 26 |
21 22 23 24 25 26 |
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12 q - 21 q + 15 q + q - 4 q + q</nowiki></ |
12 q - 21 q + 15 q + q - 4 q + q</nowiki></code></td></tr> |
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</table> }} |
Latest revision as of 17:58, 1 September 2005
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(KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 97's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X4251 X12,6,13,5 X8394 X2,9,3,10 X16,12,17,11 X10,18,11,17 X18,8,19,7 X20,14,1,13 X14,20,15,19 X6,16,7,15 |
Gauss code | 1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, -7, 9, -8 |
Dowker-Thistlethwaite code | 4 8 12 18 2 16 20 6 10 14 |
Conway Notation | [.2.210.2] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 14, width is 5, Braid index is 5 |
[{2, 12}, {1, 7}, {8, 3}, {4, 2}, {3, 6}, {7, 5}, {6, 9}, {11, 8}, {12, 10}, {9, 4}, {5, 11}, {10, 1}] |
[edit Notes on presentations of 10 97]
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 97"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X12,6,13,5 X8394 X2,9,3,10 X16,12,17,11 X10,18,11,17 X18,8,19,7 X20,14,1,13 X14,20,15,19 X6,16,7,15 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -4, 3, -1, 2, -10, 7, -3, 4, -6, 5, -2, 8, -9, 10, -5, 6, -7, 9, -8 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 8 12 18 2 16 20 6 10 14 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[.2.210.2] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 5, 14, 5 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{2, 12}, {1, 7}, {8, 3}, {4, 2}, {3, 6}, {7, 5}, {6, 9}, {11, 8}, {12, 10}, {9, 4}, {5, 11}, {10, 1}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -5 t^2+22 t-33+22 t^{-1} -5 t^{-2} } |
Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -5 z^4+2 z^2+1} |
2nd Alexander ideal (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
Determinant and Signature | { 87, 2 } |
Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9-4 q^8+7 q^7-11 q^6+14 q^5-14 q^4+14 q^3-11 q^2+7 q-3+ q^{-1} } |
HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4 a^{-2} -3 z^4 a^{-4} -z^4 a^{-6} +2 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} +z^2+2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} } |
Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^9 a^{-5} +2 z^9 a^{-7} +6 z^8 a^{-4} +11 z^8 a^{-6} +5 z^8 a^{-8} +8 z^7 a^{-3} +11 z^7 a^{-5} +7 z^7 a^{-7} +4 z^7 a^{-9} +6 z^6 a^{-2} -3 z^6 a^{-4} -21 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} +3 z^5 a^{-1} -12 z^5 a^{-3} -32 z^5 a^{-5} -28 z^5 a^{-7} -11 z^5 a^{-9} -7 z^4 a^{-2} -9 z^4 a^{-4} +5 z^4 a^{-6} +4 z^4 a^{-8} -2 z^4 a^{-10} +z^4-2 z^3 a^{-1} +10 z^3 a^{-3} +24 z^3 a^{-5} +20 z^3 a^{-7} +8 z^3 a^{-9} +6 z^2 a^{-2} +10 z^2 a^{-4} +3 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -z^2-2 z a^{-3} -6 z a^{-5} -4 z a^{-7} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} } |
The A2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-q^2-1+4 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -2 q^{-20} +3 q^{-22} -2 q^{-24} -2 q^{-26} + q^{-28} } |
The G2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+6 q^{10}-5 q^8+10 q^4-19 q^2+31-39 q^{-2} +37 q^{-4} -22 q^{-6} -8 q^{-8} +53 q^{-10} -94 q^{-12} +123 q^{-14} -126 q^{-16} +82 q^{-18} - q^{-20} -105 q^{-22} +205 q^{-24} -241 q^{-26} +205 q^{-28} -91 q^{-30} -61 q^{-32} +196 q^{-34} -257 q^{-36} +214 q^{-38} -82 q^{-40} -83 q^{-42} +199 q^{-44} -208 q^{-46} +106 q^{-48} +71 q^{-50} -229 q^{-52} +296 q^{-54} -241 q^{-56} +68 q^{-58} +148 q^{-60} -329 q^{-62} +404 q^{-64} -336 q^{-66} +158 q^{-68} +73 q^{-70} -266 q^{-72} +359 q^{-74} -328 q^{-76} +186 q^{-78} +3 q^{-80} -172 q^{-82} +252 q^{-84} -210 q^{-86} +77 q^{-88} +98 q^{-90} -221 q^{-92} +231 q^{-94} -136 q^{-96} -38 q^{-98} +207 q^{-100} -299 q^{-102} +277 q^{-104} -150 q^{-106} -24 q^{-108} +177 q^{-110} -251 q^{-112} +231 q^{-114} -142 q^{-116} +28 q^{-118} +61 q^{-120} -111 q^{-122} +108 q^{-124} -71 q^{-126} +33 q^{-128} +2 q^{-130} -19 q^{-132} +20 q^{-134} -16 q^{-136} +8 q^{-138} -3 q^{-140} + q^{-142} } |
A1 Invariants.
Weight | Invariant |
---|---|
1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-2 q+4 q^{-1} -4 q^{-3} +3 q^{-5} +3 q^{-11} -4 q^{-13} +3 q^{-15} -3 q^{-17} + q^{-19} } |
2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-2 q^8+q^6+5 q^4-11 q^2+4+19 q^{-2} -27 q^{-4} -2 q^{-6} +38 q^{-8} -27 q^{-10} -17 q^{-12} +36 q^{-14} -6 q^{-16} -23 q^{-18} +12 q^{-20} +19 q^{-22} -16 q^{-24} -16 q^{-26} +30 q^{-28} - q^{-30} -37 q^{-32} +25 q^{-34} +18 q^{-36} -37 q^{-38} +8 q^{-40} +24 q^{-42} -18 q^{-44} -5 q^{-46} +12 q^{-48} -2 q^{-50} -3 q^{-52} + q^{-54} } |
3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{21}-2 q^{19}+q^{17}+2 q^{15}-2 q^{13}-5 q^{11}+7 q^9+13 q^7-20 q^5-29 q^3+39 q+61 q^{-1} -51 q^{-3} -119 q^{-5} +54 q^{-7} +189 q^{-9} -27 q^{-11} -248 q^{-13} -37 q^{-15} +284 q^{-17} +116 q^{-19} -271 q^{-21} -184 q^{-23} +206 q^{-25} +236 q^{-27} -116 q^{-29} -245 q^{-31} +13 q^{-33} +227 q^{-35} +78 q^{-37} -186 q^{-39} -153 q^{-41} +137 q^{-43} +215 q^{-45} -89 q^{-47} -257 q^{-49} +27 q^{-51} +285 q^{-53} +37 q^{-55} -287 q^{-57} -119 q^{-59} +266 q^{-61} +183 q^{-63} -202 q^{-65} -235 q^{-67} +117 q^{-69} +250 q^{-71} -26 q^{-73} -215 q^{-75} -48 q^{-77} +155 q^{-79} +86 q^{-81} -85 q^{-83} -86 q^{-85} +28 q^{-87} +60 q^{-89} +4 q^{-91} -34 q^{-93} -9 q^{-95} +11 q^{-97} +7 q^{-99} -2 q^{-101} -3 q^{-103} + q^{-105} } |
A2 Invariants.
Weight | Invariant |
---|---|
1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-q^2-1+4 q^{-2} -2 q^{-4} + q^{-6} +2 q^{-8} -2 q^{-10} +2 q^{-12} -2 q^{-14} +2 q^{-16} + q^{-18} -2 q^{-20} +3 q^{-22} -2 q^{-24} -2 q^{-26} + q^{-28} } |
2,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{12}-q^{10}-2 q^8+3 q^6+5 q^4-4 q^2-11+7 q^{-2} +20 q^{-4} -11 q^{-6} -22 q^{-8} +12 q^{-10} +24 q^{-12} -10 q^{-14} -22 q^{-16} +11 q^{-18} +16 q^{-20} -8 q^{-22} -8 q^{-24} +8 q^{-26} +2 q^{-30} +10 q^{-32} -6 q^{-34} -10 q^{-36} +6 q^{-38} +14 q^{-40} -17 q^{-42} -19 q^{-44} +14 q^{-46} +18 q^{-48} -13 q^{-50} -17 q^{-52} +12 q^{-54} +15 q^{-56} -6 q^{-58} -15 q^{-60} +3 q^{-62} +9 q^{-64} +2 q^{-66} -3 q^{-68} -2 q^{-70} + q^{-72} } |
A3 Invariants.
Weight | Invariant |
---|---|
0,1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^8-2 q^6+6 q^2-7-4 q^{-2} +19 q^{-4} -11 q^{-6} -13 q^{-8} +30 q^{-10} -11 q^{-12} -18 q^{-14} +29 q^{-16} -7 q^{-18} -15 q^{-20} +13 q^{-22} +3 q^{-24} -6 q^{-26} -7 q^{-28} +11 q^{-30} +9 q^{-32} -23 q^{-34} +9 q^{-36} +18 q^{-38} -29 q^{-40} +5 q^{-42} +18 q^{-44} -22 q^{-46} +7 q^{-48} +11 q^{-50} -12 q^{-52} +4 q^{-54} +2 q^{-56} -3 q^{-58} + q^{-60} } |
1,0,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^5-q^3- q^{-1} +4 q^{-3} -2 q^{-5} +3 q^{-7} +2 q^{-11} -2 q^{-13} -2 q^{-19} +2 q^{-21} +3 q^{-25} -2 q^{-27} +3 q^{-29} -2 q^{-31} - q^{-33} -2 q^{-35} + q^{-37} } |
B2 Invariants.
Weight | Invariant |
---|---|
0,1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^8-2 q^6+4 q^4-8 q^2+13-18 q^{-2} +27 q^{-4} -31 q^{-6} +35 q^{-8} -34 q^{-10} +29 q^{-12} -18 q^{-14} +3 q^{-16} +13 q^{-18} -31 q^{-20} +47 q^{-22} -61 q^{-24} +68 q^{-26} -67 q^{-28} +63 q^{-30} -49 q^{-32} +35 q^{-34} -15 q^{-36} -2 q^{-38} +17 q^{-40} -29 q^{-42} +34 q^{-44} -36 q^{-46} +33 q^{-48} -29 q^{-50} +20 q^{-52} -14 q^{-54} +8 q^{-56} -3 q^{-58} + q^{-60} } |
1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{14}-2 q^{10}-2 q^8+2 q^6+7 q^4+2 q^2-10-11 q^{-2} +5 q^{-4} +23 q^{-6} +8 q^{-8} -23 q^{-10} -24 q^{-12} +12 q^{-14} +36 q^{-16} +6 q^{-18} -34 q^{-20} -20 q^{-22} +25 q^{-24} +30 q^{-26} -12 q^{-28} -31 q^{-30} + q^{-32} +29 q^{-34} +7 q^{-36} -25 q^{-38} -12 q^{-40} +20 q^{-42} +16 q^{-44} -15 q^{-46} -19 q^{-48} +14 q^{-50} +24 q^{-52} -8 q^{-54} -32 q^{-56} -2 q^{-58} +34 q^{-60} +17 q^{-62} -30 q^{-64} -34 q^{-66} +14 q^{-68} +37 q^{-70} +4 q^{-72} -31 q^{-74} -18 q^{-76} +19 q^{-78} +23 q^{-80} -5 q^{-82} -16 q^{-84} -4 q^{-86} +9 q^{-88} +5 q^{-90} -3 q^{-92} -3 q^{-94} + q^{-98} } |
G2 Invariants.
Weight | Invariant |
---|---|
1,0 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-2 q^{16}+4 q^{14}-6 q^{12}+6 q^{10}-5 q^8+10 q^4-19 q^2+31-39 q^{-2} +37 q^{-4} -22 q^{-6} -8 q^{-8} +53 q^{-10} -94 q^{-12} +123 q^{-14} -126 q^{-16} +82 q^{-18} - q^{-20} -105 q^{-22} +205 q^{-24} -241 q^{-26} +205 q^{-28} -91 q^{-30} -61 q^{-32} +196 q^{-34} -257 q^{-36} +214 q^{-38} -82 q^{-40} -83 q^{-42} +199 q^{-44} -208 q^{-46} +106 q^{-48} +71 q^{-50} -229 q^{-52} +296 q^{-54} -241 q^{-56} +68 q^{-58} +148 q^{-60} -329 q^{-62} +404 q^{-64} -336 q^{-66} +158 q^{-68} +73 q^{-70} -266 q^{-72} +359 q^{-74} -328 q^{-76} +186 q^{-78} +3 q^{-80} -172 q^{-82} +252 q^{-84} -210 q^{-86} +77 q^{-88} +98 q^{-90} -221 q^{-92} +231 q^{-94} -136 q^{-96} -38 q^{-98} +207 q^{-100} -299 q^{-102} +277 q^{-104} -150 q^{-106} -24 q^{-108} +177 q^{-110} -251 q^{-112} +231 q^{-114} -142 q^{-116} +28 q^{-118} +61 q^{-120} -111 q^{-122} +108 q^{-124} -71 q^{-126} +33 q^{-128} +2 q^{-130} -19 q^{-132} +20 q^{-134} -16 q^{-136} +8 q^{-138} -3 q^{-140} + q^{-142} } |
.
KnotTheory`
, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
|
In[3]:=
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K = Knot["10 97"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -5 t^2+22 t-33+22 t^{-1} -5 t^{-2} } |
In[5]:=
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Conway[K][z]
|
Out[5]=
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -5 z^4+2 z^2+1} |
In[6]:=
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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]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}} |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 87, 2 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9-4 q^8+7 q^7-11 q^6+14 q^5-14 q^4+14 q^3-11 q^2+7 q-3+ q^{-1} } |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^4 a^{-2} -3 z^4 a^{-4} -z^4 a^{-6} +2 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} +z^2 a^{-8} +z^2+2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} } |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^9 a^{-5} +2 z^9 a^{-7} +6 z^8 a^{-4} +11 z^8 a^{-6} +5 z^8 a^{-8} +8 z^7 a^{-3} +11 z^7 a^{-5} +7 z^7 a^{-7} +4 z^7 a^{-9} +6 z^6 a^{-2} -3 z^6 a^{-4} -21 z^6 a^{-6} -11 z^6 a^{-8} +z^6 a^{-10} +3 z^5 a^{-1} -12 z^5 a^{-3} -32 z^5 a^{-5} -28 z^5 a^{-7} -11 z^5 a^{-9} -7 z^4 a^{-2} -9 z^4 a^{-4} +5 z^4 a^{-6} +4 z^4 a^{-8} -2 z^4 a^{-10} +z^4-2 z^3 a^{-1} +10 z^3 a^{-3} +24 z^3 a^{-5} +20 z^3 a^{-7} +8 z^3 a^{-9} +6 z^2 a^{-2} +10 z^2 a^{-4} +3 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -z^2-2 z a^{-3} -6 z a^{-5} -4 z a^{-7} -2 a^{-2} -2 a^{-4} -2 a^{-6} - a^{-8} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}
KnotTheory`
. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 97"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -5 t^2+22 t-33+22 t^{-1} -5 t^{-2} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^9-4 q^8+7 q^7-11 q^6+14 q^5-14 q^4+14 q^3-11 q^2+7 q-3+ q^{-1} } } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{} |
Vassiliev invariants
V2 and V3: | (2, 4) |
V2,1 through V6,9: |
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V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.
Khovanov Homology
The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 2 is the signature of 10 97. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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Integral Khovanov Homology
(db, data source) |
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The Coloured Jones Polynomials
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n} | |
2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}-4 q^{25}+q^{24}+15 q^{23}-21 q^{22}-12 q^{21}+57 q^{20}-37 q^{19}-57 q^{18}+112 q^{17}-30 q^{16}-119 q^{15}+148 q^{14}+q^{13}-165 q^{12}+148 q^{11}+36 q^{10}-172 q^9+113 q^8+53 q^7-130 q^6+60 q^5+43 q^4-65 q^3+20 q^2+18 q-19+5 q^{-1} +3 q^{-2} -3 q^{-3} + q^{-4} } |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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