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<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
</tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</td></tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[8, 12]]</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[4, 2, 5, 1], X[10, 8, 11, 7], X[8, 3, 9, 4], X[2, 9, 3, 10],
<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[8, 12]]</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[4, 2, 5, 1], X[10, 8, 11, 7], X[8, 3, 9, 4], X[2, 9, 3, 10],
X[14, 6, 15, 5], X[16, 11, 1, 12], X[12, 15, 13, 16], X[6, 14, 7, 13]]</nowiki></pre></td></tr>
X[14, 6, 15, 5], X[16, 11, 1, 12], X[12, 15, 13, 16], X[6, 14, 7, 13]]</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[8, 12]]</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, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6]</nowiki></pre></td></tr>
<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[8, 12]]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[4, 8, 14, 10, 2, 16, 6, 12]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[8, 12]]</nowiki></code></td></tr>
<tr align=left>
<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[8, 12]]</nowiki></pre></td></tr>
<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, 2, -1, -3, 2, 4, -3, 4}]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<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>
<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, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6]</nowiki></code></td></tr>
</table>
<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, 8}</nowiki></pre></td></tr>
<table><tr align=left>
<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[8, 12]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5</nowiki></pre></td></tr>
<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>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[8, 12]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:8_12_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>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[8, 12]]</nowiki></code></td></tr>
<tr align=left>
<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[8, 12]]&) /@ {
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[4, 8, 14, 10, 2, 16, 6, 12]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<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[8, 12]]</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, 2, -1, -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, 8}</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[8, 12]]</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[8, 12]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:8_12_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[8, 12]]&) /@ {
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>{FullyAmphicheiral, 2, 2, 2, {4, 6}, 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[8, 12]][t]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -2 7 2
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{FullyAmphicheiral, 2, 2, 2, {4, 6}, 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[8, 12]][t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -2 7 2
13 + t - - - 7 t + t
13 + t - - - 7 t + 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[8, 12]][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 + 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[8, 12]][z]</nowiki></code></td></tr>
<tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[8, 12]}</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[8, 12]], KnotSignature[Knot[8, 12]]}</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>{29, 0}</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 + 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[8, 12]][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> -4 2 4 5 2 3 4
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Knot[8, 12]}</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[8, 12]], KnotSignature[Knot[8, 12]]}</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>{29, 0}</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[8, 12]][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> -4 2 4 5 2 3 4
5 + q - -- + -- - - - 5 q + 4 q - 2 q + q
5 + q - -- + -- - - - 5 q + 4 q - 2 q + q
3 2 q
3 2 q
q q</nowiki></pre></td></tr>
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[8, 12]}</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[8, 12]][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> -14 -12 -10 -8 -4 -2 2 4 8 10 12
<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[8, 12]}</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[8, 12]][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> -14 -12 -10 -8 -4 -2 2 4 8 10 12
-1 + q + q - q + q - q + q + q - q + q - q + q +
-1 + q + q - q + q - q + q + q - q + q - q + q +
14
14
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[8, 12]][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
<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[8, 12]][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
-4 -2 2 4 2 2 z 2 2 4
-4 -2 2 4 2 2 z 2 2 4
1 + a - a - a + a + z - ---- - 2 a z + z
1 + a - a - a + a + z - ---- - 2 a z + z
2
2
a</nowiki></pre></td></tr>
a</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[8, 12]][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 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[8, 12]][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 2
-4 -2 2 4 z 3 2 z 2 z 2 2 4 2
-4 -2 2 4 z 3 2 z 2 z 2 2 4 2
1 + a + a + a + a + -- + a z - ---- - ---- - 2 a z - 2 a z -
1 + a + a + a + a + -- + a z - ---- - ---- - 2 a z - 2 a z -
Line 120: Line 206:
---- + ---- + 2 a z + 2 a z + 4 z + ---- + 2 a z + -- + a z
---- + ---- + 2 a z + 2 a z + 4 z + ---- + 2 a z + -- + a z
3 a 2 a
3 a 2 a
a a</nowiki></pre></td></tr>
a a</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[8, 12]], Vassiliev[3][Knot[8, 12]]}</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, 0}</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[8, 12]][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 1 1 1 3 1 2 3
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[8, 12]], Vassiliev[3][Knot[8, 12]]}</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, 0}</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[8, 12]][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 1 1 1 3 1 2 3
- + 3 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t +
- + 3 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t +
q 9 4 7 3 5 3 5 2 3 2 3 q t
q 9 4 7 3 5 3 5 2 3 2 3 q t
Line 130: Line 226:
3 3 2 5 2 5 3 7 3 9 4
3 3 2 5 2 5 3 7 3 9 4
2 q t + q t + 3 q t + q t + q t + q t</nowiki></pre></td></tr>
2 q t + q t + 3 q t + q t + q t + q t</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[21]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>ColouredJones[Knot[8, 12], 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> -12 2 6 8 3 18 15 10 30 18 16
<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[8, 12], 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> -12 2 6 8 3 18 15 10 30 18 16
35 + q - --- + -- - -- - -- + -- - -- - -- + -- - -- - -- - 16 q -
35 + q - --- + -- - -- - -- + -- - -- - -- + -- - -- - -- - 16 q -
11 9 8 7 6 5 4 3 2 q
11 9 8 7 6 5 4 3 2 q
Line 141: Line 242:
12
12
q</nowiki></pre></td></tr>
q</nowiki></code></td></tr>
</table> }}
</table> }}

Revision as of 18:02, 1 September 2005

8 11.gif

8_11

8 13.gif

8_13

8 12.gif
(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 8 12's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 12 at Knotilus!

[edit 8 12 Quick Notes]
[edit 8 12 Further Notes and Views]


In symmetric decorative form

Knot presentations

Planar diagram presentation X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13
Gauss code 1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6
Dowker-Thistlethwaite code 4 8 14 10 2 16 6 12
Conway Notation [2222]


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

Length is 8, width is 5,

Braid index is 5

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

[edit Notes on presentations of 8 12]

Knot 8_12.
A graph, knot 8_12.
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["8 12"];
In[4]:= PD[K]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= X4251 X10,8,11,7 X8394 X2,9,3,10 X14,6,15,5 X16,11,1,12 X12,15,13,16 X6,14,7,13
In[5]:= GaussCode[K]
Out[5]= 1, -4, 3, -1, 5, -8, 2, -3, 4, -2, 6, -7, 8, -5, 7, -6
In[6]:= DTCode[K]
Out[6]= 4 8 14 10 2 16 6 12

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];
In[8]:= ConwayNotation[K]
Out[8]= [2222]
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,\{-1,2,-1,-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, 8, 5 }
In[11]:= Show[BraidPlot[br]]
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.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."
8 12 ML.gif
Out[12]= -Graphics-
In[13]:= ap = ArcPresentation[K]
Out[13]= ArcPresentation[{3, 9}, {4, 2}, {1, 3}, {2, 7}, {6, 8}, {7, 5}, {10, 6}, {9, 4}, {5, 10}, {8, 1}]
In[14]:= Draw[ap]
8 12 AP.gif
Out[14]= -Graphics-

Three dimensional invariants

Symmetry type Fully amphicheiral
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-5][-5]
Hyperbolic Volume 8.93586
A-Polynomial See Data:8 12/A-polynomial

[edit Notes for 8 12'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 0

[edit Notes for 8 12's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^2-7 t+13-7 t^{-1} + t^{-2} }[/math]
Conway polynomial [math]\displaystyle{ z^4-3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 29, 0 }
Jones polynomial [math]\displaystyle{ q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ a^4-2 z^2 a^2-a^2+z^4+z^2+1-2 z^2 a^{-2} - a^{-2} + a^{-4} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 a z^5+2 z^5 a^{-1} +2 z^5 a^{-3} +a^4 z^4-a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -4 z^4-3 a^3 z^3-3 a z^3-3 z^3 a^{-1} -3 z^3 a^{-3} -2 a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +a^3 z+z a^{-3} +a^4+a^2+ a^{-2} + a^{-4} +1 }[/math]
The A2 invariant [math]\displaystyle{ q^{14}+q^{12}-q^{10}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
The G2 invariant [math]\displaystyle{ q^{66}-q^{64}+3 q^{62}-3 q^{60}+2 q^{58}-3 q^{54}+9 q^{52}-11 q^{50}+12 q^{48}-8 q^{46}+10 q^{42}-17 q^{40}+23 q^{38}-18 q^{36}+8 q^{34}+4 q^{32}-16 q^{30}+17 q^{28}-13 q^{26}+3 q^{24}+6 q^{22}-12 q^{20}+9 q^{18}-12 q^{14}+21 q^{12}-23 q^{10}+15 q^8-q^6-14 q^4+27 q^2-29+27 q^{-2} -14 q^{-4} - q^{-6} +15 q^{-8} -23 q^{-10} +21 q^{-12} -12 q^{-14} +9 q^{-18} -12 q^{-20} +6 q^{-22} +3 q^{-24} -13 q^{-26} +17 q^{-28} -16 q^{-30} +4 q^{-32} +8 q^{-34} -18 q^{-36} +23 q^{-38} -17 q^{-40} +10 q^{-42} -8 q^{-46} +12 q^{-48} -11 q^{-50} +9 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_12.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q^9-q^7+2 q^5-q^3- q^{-3} +2 q^{-5} - q^{-7} + q^{-9} }[/math]
2 [math]\displaystyle{ q^{26}-q^{24}-q^{22}+4 q^{20}-2 q^{18}-5 q^{16}+7 q^{14}-7 q^{10}+5 q^8+2 q^6-4 q^4+q^2+3+ q^{-2} -4 q^{-4} +2 q^{-6} +5 q^{-8} -7 q^{-10} +7 q^{-14} -5 q^{-16} -2 q^{-18} +4 q^{-20} - q^{-22} - q^{-24} + q^{-26} }[/math]
3 [math]\displaystyle{ q^{51}-q^{49}-q^{47}+q^{45}+3 q^{43}-2 q^{41}-7 q^{39}+2 q^{37}+12 q^{35}+q^{33}-16 q^{31}-7 q^{29}+20 q^{27}+12 q^{25}-20 q^{23}-17 q^{21}+16 q^{19}+22 q^{17}-12 q^{15}-20 q^{13}+7 q^{11}+18 q^9-2 q^7-13 q^5-4 q^3+9 q+9 q^{-1} -4 q^{-3} -13 q^{-5} -2 q^{-7} +18 q^{-9} +7 q^{-11} -20 q^{-13} -12 q^{-15} +22 q^{-17} +16 q^{-19} -17 q^{-21} -20 q^{-23} +12 q^{-25} +20 q^{-27} -7 q^{-29} -16 q^{-31} + q^{-33} +12 q^{-35} +2 q^{-37} -7 q^{-39} -2 q^{-41} +3 q^{-43} + q^{-45} - q^{-47} - q^{-49} + q^{-51} }[/math]
4 [math]\displaystyle{ q^{84}-q^{82}-q^{80}+q^{78}+3 q^{74}-4 q^{72}-5 q^{70}+3 q^{68}+5 q^{66}+14 q^{64}-9 q^{62}-22 q^{60}-7 q^{58}+11 q^{56}+44 q^{54}+4 q^{52}-40 q^{50}-42 q^{48}-7 q^{46}+77 q^{44}+44 q^{42}-32 q^{40}-76 q^{38}-49 q^{36}+76 q^{34}+80 q^{32}+5 q^{30}-78 q^{28}-82 q^{26}+44 q^{24}+81 q^{22}+33 q^{20}-50 q^{18}-75 q^{16}+8 q^{14}+52 q^{12}+42 q^{10}-15 q^8-49 q^6-20 q^4+18 q^2+41+18 q^{-2} -20 q^{-4} -49 q^{-6} -15 q^{-8} +42 q^{-10} +52 q^{-12} +8 q^{-14} -75 q^{-16} -50 q^{-18} +33 q^{-20} +81 q^{-22} +44 q^{-24} -82 q^{-26} -78 q^{-28} +5 q^{-30} +80 q^{-32} +76 q^{-34} -49 q^{-36} -76 q^{-38} -32 q^{-40} +44 q^{-42} +77 q^{-44} -7 q^{-46} -42 q^{-48} -40 q^{-50} +4 q^{-52} +44 q^{-54} +11 q^{-56} -7 q^{-58} -22 q^{-60} -9 q^{-62} +14 q^{-64} +5 q^{-66} +3 q^{-68} -5 q^{-70} -4 q^{-72} +3 q^{-74} + q^{-78} - q^{-80} - q^{-82} + q^{-84} }[/math]
5 [math]\displaystyle{ q^{125}-q^{123}-q^{121}+q^{119}+q^{113}-2 q^{111}-4 q^{109}+3 q^{107}+7 q^{105}+5 q^{103}-13 q^{99}-19 q^{97}-6 q^{95}+23 q^{93}+39 q^{91}+22 q^{89}-23 q^{87}-67 q^{85}-59 q^{83}+8 q^{81}+95 q^{79}+114 q^{77}+28 q^{75}-105 q^{73}-174 q^{71}-96 q^{69}+88 q^{67}+233 q^{65}+180 q^{63}-46 q^{61}-258 q^{59}-266 q^{57}-31 q^{55}+253 q^{53}+337 q^{51}+116 q^{49}-221 q^{47}-367 q^{45}-192 q^{43}+157 q^{41}+366 q^{39}+250 q^{37}-93 q^{35}-333 q^{33}-262 q^{31}+25 q^{29}+273 q^{27}+260 q^{25}+20 q^{23}-207 q^{21}-230 q^{19}-56 q^{17}+140 q^{15}+196 q^{13}+81 q^{11}-80 q^9-157 q^7-104 q^5+27 q^3+127 q+127 q^{-1} +27 q^{-3} -104 q^{-5} -157 q^{-7} -80 q^{-9} +81 q^{-11} +196 q^{-13} +140 q^{-15} -56 q^{-17} -230 q^{-19} -207 q^{-21} +20 q^{-23} +260 q^{-25} +273 q^{-27} +25 q^{-29} -262 q^{-31} -333 q^{-33} -93 q^{-35} +250 q^{-37} +366 q^{-39} +157 q^{-41} -192 q^{-43} -367 q^{-45} -221 q^{-47} +116 q^{-49} +337 q^{-51} +253 q^{-53} -31 q^{-55} -266 q^{-57} -258 q^{-59} -46 q^{-61} +180 q^{-63} +233 q^{-65} +88 q^{-67} -96 q^{-69} -174 q^{-71} -105 q^{-73} +28 q^{-75} +114 q^{-77} +95 q^{-79} +8 q^{-81} -59 q^{-83} -67 q^{-85} -23 q^{-87} +22 q^{-89} +39 q^{-91} +23 q^{-93} -6 q^{-95} -19 q^{-97} -13 q^{-99} +5 q^{-103} +7 q^{-105} +3 q^{-107} -4 q^{-109} -2 q^{-111} + q^{-113} + q^{-119} - q^{-121} - q^{-123} + q^{-125} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{14}+q^{12}-q^{10}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} - q^{-10} + q^{-12} + q^{-14} }[/math]
1,1 [math]\displaystyle{ q^{36}-2 q^{34}+6 q^{32}-10 q^{30}+19 q^{28}-30 q^{26}+42 q^{24}-54 q^{22}+64 q^{20}-70 q^{18}+62 q^{16}-46 q^{14}+23 q^{12}+10 q^{10}-44 q^8+80 q^6-106 q^4+124 q^2-130+124 q^{-2} -106 q^{-4} +80 q^{-6} -44 q^{-8} +10 q^{-10} +23 q^{-12} -46 q^{-14} +62 q^{-16} -70 q^{-18} +64 q^{-20} -54 q^{-22} +42 q^{-24} -30 q^{-26} +19 q^{-28} -10 q^{-30} +6 q^{-32} -2 q^{-34} + q^{-36} }[/math]
2,0 [math]\displaystyle{ q^{36}+q^{34}-2 q^{30}+3 q^{26}-4 q^{22}-q^{20}+4 q^{18}+2 q^{16}-5 q^{14}+4 q^{10}-q^8-2 q^6+q^4+2 q^2+2 q^{-2} + q^{-4} -2 q^{-6} - q^{-8} +4 q^{-10} -5 q^{-14} +2 q^{-16} +4 q^{-18} - q^{-20} -4 q^{-22} +3 q^{-26} -2 q^{-30} + q^{-34} + q^{-36} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{28}-q^{26}+q^{24}+3 q^{22}-3 q^{20}+q^{18}+5 q^{16}-6 q^{14}+4 q^{10}-5 q^8-q^6+3 q^4+q^2+ q^{-2} +3 q^{-4} - q^{-6} -5 q^{-8} +4 q^{-10} -6 q^{-14} +5 q^{-16} + q^{-18} -3 q^{-20} +3 q^{-22} + q^{-24} - q^{-26} + q^{-28} }[/math]
1,0,0 [math]\displaystyle{ q^{19}+q^{17}+q^{15}-q^{13}+q^{11}-q^9-q^5+q^3+ q^{-3} - q^{-5} - q^{-9} + q^{-11} - q^{-13} + q^{-15} + q^{-17} + q^{-19} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ q^{28}-q^{26}+3 q^{24}-3 q^{22}+5 q^{20}-5 q^{18}+5 q^{16}-4 q^{14}+2 q^{12}-3 q^8+5 q^6-7 q^4+9 q^2-10+9 q^{-2} -7 q^{-4} +5 q^{-6} -3 q^{-8} +2 q^{-12} -4 q^{-14} +5 q^{-16} -5 q^{-18} +5 q^{-20} -3 q^{-22} +3 q^{-24} - q^{-26} + q^{-28} }[/math]
1,0 [math]\displaystyle{ q^{46}-q^{42}-q^{40}+2 q^{38}+3 q^{36}-4 q^{32}-2 q^{30}+4 q^{28}+5 q^{26}-2 q^{24}-6 q^{22}-q^{20}+5 q^{18}+2 q^{16}-4 q^{14}-3 q^{12}+2 q^{10}+3 q^8-q^6-3 q^4+q^2+5+ q^{-2} -3 q^{-4} - q^{-6} +3 q^{-8} +2 q^{-10} -3 q^{-12} -4 q^{-14} +2 q^{-16} +5 q^{-18} - q^{-20} -6 q^{-22} -2 q^{-24} +5 q^{-26} +4 q^{-28} -2 q^{-30} -4 q^{-32} +3 q^{-36} +2 q^{-38} - q^{-40} - q^{-42} + q^{-46} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{66}-q^{64}+3 q^{62}-3 q^{60}+2 q^{58}-3 q^{54}+9 q^{52}-11 q^{50}+12 q^{48}-8 q^{46}+10 q^{42}-17 q^{40}+23 q^{38}-18 q^{36}+8 q^{34}+4 q^{32}-16 q^{30}+17 q^{28}-13 q^{26}+3 q^{24}+6 q^{22}-12 q^{20}+9 q^{18}-12 q^{14}+21 q^{12}-23 q^{10}+15 q^8-q^6-14 q^4+27 q^2-29+27 q^{-2} -14 q^{-4} - q^{-6} +15 q^{-8} -23 q^{-10} +21 q^{-12} -12 q^{-14} +9 q^{-18} -12 q^{-20} +6 q^{-22} +3 q^{-24} -13 q^{-26} +17 q^{-28} -16 q^{-30} +4 q^{-32} +8 q^{-34} -18 q^{-36} +23 q^{-38} -17 q^{-40} +10 q^{-42} -8 q^{-46} +12 q^{-48} -11 q^{-50} +9 q^{-52} -3 q^{-54} +2 q^{-58} -3 q^{-60} +3 q^{-62} - q^{-64} + q^{-66} }[/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["8 12"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^2-7 t+13-7 t^{-1} + t^{-2} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 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]= { 29, 0 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ a^4-2 z^2 a^2-a^2+z^4+z^2+1-2 z^2 a^{-2} - a^{-2} + a^{-4} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 a z^5+2 z^5 a^{-1} +2 z^5 a^{-3} +a^4 z^4-a^2 z^4-z^4 a^{-2} +z^4 a^{-4} -4 z^4-3 a^3 z^3-3 a z^3-3 z^3 a^{-1} -3 z^3 a^{-3} -2 a^4 z^2-2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +a^3 z+z a^{-3} +a^4+a^2+ a^{-2} + a^{-4} +1 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk
The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

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

In[3]:= K = Knot["8 12"];
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{ t^2-7 t+13-7 t^{-1} + t^{-2} }[/math], [math]\displaystyle{ q^4-2 q^3+4 q^2-5 q+5-5 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4} }[/math] }
In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
Out[5]= {}
In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
Out[6]= {}

Vassiliev invariants

V2 and V3: (-3, 0)
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{ 0 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 82 }[/math] [math]\displaystyle{ 30 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -288 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -984 }[/math] [math]\displaystyle{ -360 }[/math] [math]\displaystyle{ -\frac{8031}{10} }[/math] [math]\displaystyle{ \frac{1226}{15} }[/math] [math]\displaystyle{ -\frac{9262}{15} }[/math] [math]\displaystyle{ \frac{479}{6} }[/math] [math]\displaystyle{ -\frac{1311}{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]0 is the signature of 8 12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -4-3-2-101234χ
9        11
7       1 -1
5      31 2
3     21  -1
1    33   0
-1   33    0
-3  12     -1
-5 13      2
-7 1       -1
-91        1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math]
[math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=4 }[/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^{12}-2 q^{11}+6 q^9-8 q^8-3 q^7+18 q^6-15 q^5-10 q^4+30 q^3-18 q^2-16 q+35-16 q^{-1} -18 q^{-2} +30 q^{-3} -10 q^{-4} -15 q^{-5} +18 q^{-6} -3 q^{-7} -8 q^{-8} +6 q^{-9} -2 q^{-11} + q^{-12} }[/math]
3 [math]\displaystyle{ q^{24}-2 q^{23}+2 q^{21}+3 q^{20}-7 q^{19}-5 q^{18}+11 q^{17}+13 q^{16}-18 q^{15}-22 q^{14}+20 q^{13}+40 q^{12}-26 q^{11}-54 q^{10}+23 q^9+73 q^8-20 q^7-88 q^6+15 q^5+100 q^4-9 q^3-108 q^2+4 q+109+4 q^{-1} -108 q^{-2} -9 q^{-3} +100 q^{-4} +15 q^{-5} -88 q^{-6} -20 q^{-7} +73 q^{-8} +23 q^{-9} -54 q^{-10} -26 q^{-11} +40 q^{-12} +20 q^{-13} -22 q^{-14} -18 q^{-15} +13 q^{-16} +11 q^{-17} -5 q^{-18} -7 q^{-19} +3 q^{-20} +2 q^{-21} -2 q^{-23} + q^{-24} }[/math]
4 [math]\displaystyle{ q^{40}-2 q^{39}+2 q^{37}-q^{36}+4 q^{35}-9 q^{34}-q^{33}+10 q^{32}+q^{31}+13 q^{30}-32 q^{29}-14 q^{28}+25 q^{27}+19 q^{26}+46 q^{25}-72 q^{24}-58 q^{23}+23 q^{22}+54 q^{21}+130 q^{20}-105 q^{19}-134 q^{18}-21 q^{17}+81 q^{16}+255 q^{15}-101 q^{14}-209 q^{13}-104 q^{12}+77 q^{11}+381 q^{10}-64 q^9-257 q^8-187 q^7+52 q^6+464 q^5-20 q^4-267 q^3-244 q^2+18 q+493+18 q^{-1} -244 q^{-2} -267 q^{-3} -20 q^{-4} +464 q^{-5} +52 q^{-6} -187 q^{-7} -257 q^{-8} -64 q^{-9} +381 q^{-10} +77 q^{-11} -104 q^{-12} -209 q^{-13} -101 q^{-14} +255 q^{-15} +81 q^{-16} -21 q^{-17} -134 q^{-18} -105 q^{-19} +130 q^{-20} +54 q^{-21} +23 q^{-22} -58 q^{-23} -72 q^{-24} +46 q^{-25} +19 q^{-26} +25 q^{-27} -14 q^{-28} -32 q^{-29} +13 q^{-30} + q^{-31} +10 q^{-32} - q^{-33} -9 q^{-34} +4 q^{-35} - q^{-36} +2 q^{-37} -2 q^{-39} + q^{-40} }[/math]
5 [math]\displaystyle{ q^{60}-2 q^{59}+2 q^{57}-q^{56}+2 q^{54}-5 q^{53}-2 q^{52}+9 q^{51}+3 q^{50}-2 q^{49}-3 q^{48}-18 q^{47}-8 q^{46}+22 q^{45}+32 q^{44}+14 q^{43}-20 q^{42}-63 q^{41}-52 q^{40}+30 q^{39}+99 q^{38}+101 q^{37}-q^{36}-149 q^{35}-185 q^{34}-39 q^{33}+177 q^{32}+285 q^{31}+144 q^{30}-202 q^{29}-411 q^{28}-251 q^{27}+169 q^{26}+520 q^{25}+428 q^{24}-118 q^{23}-632 q^{22}-588 q^{21}+23 q^{20}+695 q^{19}+777 q^{18}+91 q^{17}-748 q^{16}-931 q^{15}-217 q^{14}+766 q^{13}+1064 q^{12}+339 q^{11}-761 q^{10}-1171 q^9-444 q^8+743 q^7+1238 q^6+535 q^5-705 q^4-1286 q^3-605 q^2+666 q+1291+666 q^{-1} -605 q^{-2} -1286 q^{-3} -705 q^{-4} +535 q^{-5} +1238 q^{-6} +743 q^{-7} -444 q^{-8} -1171 q^{-9} -761 q^{-10} +339 q^{-11} +1064 q^{-12} +766 q^{-13} -217 q^{-14} -931 q^{-15} -748 q^{-16} +91 q^{-17} +777 q^{-18} +695 q^{-19} +23 q^{-20} -588 q^{-21} -632 q^{-22} -118 q^{-23} +428 q^{-24} +520 q^{-25} +169 q^{-26} -251 q^{-27} -411 q^{-28} -202 q^{-29} +144 q^{-30} +285 q^{-31} +177 q^{-32} -39 q^{-33} -185 q^{-34} -149 q^{-35} - q^{-36} +101 q^{-37} +99 q^{-38} +30 q^{-39} -52 q^{-40} -63 q^{-41} -20 q^{-42} +14 q^{-43} +32 q^{-44} +22 q^{-45} -8 q^{-46} -18 q^{-47} -3 q^{-48} -2 q^{-49} +3 q^{-50} +9 q^{-51} -2 q^{-52} -5 q^{-53} +2 q^{-54} - q^{-56} +2 q^{-57} -2 q^{-59} + q^{-60} }[/math]
6 [math]\displaystyle{ q^{84}-2 q^{83}+2 q^{81}-q^{80}-2 q^{78}+6 q^{77}-6 q^{76}-3 q^{75}+11 q^{74}-q^{73}-2 q^{72}-12 q^{71}+11 q^{70}-16 q^{69}-7 q^{68}+38 q^{67}+16 q^{66}+3 q^{65}-41 q^{64}+q^{63}-68 q^{62}-33 q^{61}+98 q^{60}+93 q^{59}+77 q^{58}-58 q^{57}-37 q^{56}-237 q^{55}-183 q^{54}+124 q^{53}+255 q^{52}+331 q^{51}+93 q^{50}+5 q^{49}-556 q^{48}-604 q^{47}-102 q^{46}+356 q^{45}+782 q^{44}+595 q^{43}+402 q^{42}-827 q^{41}-1294 q^{40}-786 q^{39}+85 q^{38}+1183 q^{37}+1419 q^{36}+1336 q^{35}-712 q^{34}-1957 q^{33}-1851 q^{32}-715 q^{31}+1194 q^{30}+2244 q^{29}+2655 q^{28}-93 q^{27}-2262 q^{26}-2933 q^{25}-1845 q^{24}+746 q^{23}+2752 q^{22}+3943 q^{21}+806 q^{20}-2153 q^{19}-3708 q^{18}-2911 q^{17}+70 q^{16}+2891 q^{15}+4876 q^{14}+1634 q^{13}-1812 q^{12}-4100 q^{11}-3656 q^{10}-556 q^9+2791 q^8+5381 q^7+2212 q^6-1428 q^5-4191 q^4-4055 q^3-1036 q^2+2568 q+5533+2568 q^{-1} -1036 q^{-2} -4055 q^{-3} -4191 q^{-4} -1428 q^{-5} +2212 q^{-6} +5381 q^{-7} +2791 q^{-8} -556 q^{-9} -3656 q^{-10} -4100 q^{-11} -1812 q^{-12} +1634 q^{-13} +4876 q^{-14} +2891 q^{-15} +70 q^{-16} -2911 q^{-17} -3708 q^{-18} -2153 q^{-19} +806 q^{-20} +3943 q^{-21} +2752 q^{-22} +746 q^{-23} -1845 q^{-24} -2933 q^{-25} -2262 q^{-26} -93 q^{-27} +2655 q^{-28} +2244 q^{-29} +1194 q^{-30} -715 q^{-31} -1851 q^{-32} -1957 q^{-33} -712 q^{-34} +1336 q^{-35} +1419 q^{-36} +1183 q^{-37} +85 q^{-38} -786 q^{-39} -1294 q^{-40} -827 q^{-41} +402 q^{-42} +595 q^{-43} +782 q^{-44} +356 q^{-45} -102 q^{-46} -604 q^{-47} -556 q^{-48} +5 q^{-49} +93 q^{-50} +331 q^{-51} +255 q^{-52} +124 q^{-53} -183 q^{-54} -237 q^{-55} -37 q^{-56} -58 q^{-57} +77 q^{-58} +93 q^{-59} +98 q^{-60} -33 q^{-61} -68 q^{-62} + q^{-63} -41 q^{-64} +3 q^{-65} +16 q^{-66} +38 q^{-67} -7 q^{-68} -16 q^{-69} +11 q^{-70} -12 q^{-71} -2 q^{-72} - q^{-73} +11 q^{-74} -3 q^{-75} -6 q^{-76} +6 q^{-77} -2 q^{-78} - q^{-80} +2 q^{-81} -2 q^{-83} + q^{-84} }[/math]
7 [math]\displaystyle{ q^{112}-2 q^{111}+2 q^{109}-q^{108}-2 q^{106}+2 q^{105}+5 q^{104}-7 q^{103}-q^{102}+7 q^{101}-q^{100}-12 q^{98}-2 q^{97}+17 q^{96}-13 q^{95}+3 q^{94}+22 q^{93}+7 q^{92}+7 q^{91}-43 q^{90}-35 q^{89}+10 q^{88}-26 q^{87}+25 q^{86}+81 q^{85}+63 q^{84}+69 q^{83}-77 q^{82}-148 q^{81}-104 q^{80}-156 q^{79}+17 q^{78}+206 q^{77}+282 q^{76}+356 q^{75}+55 q^{74}-268 q^{73}-435 q^{72}-661 q^{71}-340 q^{70}+204 q^{69}+654 q^{68}+1130 q^{67}+788 q^{66}+49 q^{65}-750 q^{64}-1693 q^{63}-1545 q^{62}-598 q^{61}+680 q^{60}+2252 q^{59}+2499 q^{58}+1547 q^{57}-198 q^{56}-2717 q^{55}-3672 q^{54}-2860 q^{53}-655 q^{52}+2844 q^{51}+4768 q^{50}+4537 q^{49}+2104 q^{48}-2562 q^{47}-5828 q^{46}-6373 q^{45}-3891 q^{44}+1784 q^{43}+6441 q^{42}+8212 q^{41}+6132 q^{40}-511 q^{39}-6764 q^{38}-9928 q^{37}-8380 q^{36}-1108 q^{35}+6552 q^{34}+11294 q^{33}+10685 q^{32}+2988 q^{31}-6052 q^{30}-12328 q^{29}-12727 q^{28}-4870 q^{27}+5238 q^{26}+12969 q^{25}+14476 q^{24}+6663 q^{23}-4286 q^{22}-13308 q^{21}-15867 q^{20}-8241 q^{19}+3344 q^{18}+13388 q^{17}+16876 q^{16}+9543 q^{15}-2421 q^{14}-13285 q^{13}-17618 q^{12}-10578 q^{11}+1640 q^{10}+13082 q^9+18052 q^8+11361 q^7-915 q^6-12789 q^5-18322 q^4-11965 q^3+311 q^2+12426 q+18379+12426 q^{-1} +311 q^{-2} -11965 q^{-3} -18322 q^{-4} -12789 q^{-5} -915 q^{-6} +11361 q^{-7} +18052 q^{-8} +13082 q^{-9} +1640 q^{-10} -10578 q^{-11} -17618 q^{-12} -13285 q^{-13} -2421 q^{-14} +9543 q^{-15} +16876 q^{-16} +13388 q^{-17} +3344 q^{-18} -8241 q^{-19} -15867 q^{-20} -13308 q^{-21} -4286 q^{-22} +6663 q^{-23} +14476 q^{-24} +12969 q^{-25} +5238 q^{-26} -4870 q^{-27} -12727 q^{-28} -12328 q^{-29} -6052 q^{-30} +2988 q^{-31} +10685 q^{-32} +11294 q^{-33} +6552 q^{-34} -1108 q^{-35} -8380 q^{-36} -9928 q^{-37} -6764 q^{-38} -511 q^{-39} +6132 q^{-40} +8212 q^{-41} +6441 q^{-42} +1784 q^{-43} -3891 q^{-44} -6373 q^{-45} -5828 q^{-46} -2562 q^{-47} +2104 q^{-48} +4537 q^{-49} +4768 q^{-50} +2844 q^{-51} -655 q^{-52} -2860 q^{-53} -3672 q^{-54} -2717 q^{-55} -198 q^{-56} +1547 q^{-57} +2499 q^{-58} +2252 q^{-59} +680 q^{-60} -598 q^{-61} -1545 q^{-62} -1693 q^{-63} -750 q^{-64} +49 q^{-65} +788 q^{-66} +1130 q^{-67} +654 q^{-68} +204 q^{-69} -340 q^{-70} -661 q^{-71} -435 q^{-72} -268 q^{-73} +55 q^{-74} +356 q^{-75} +282 q^{-76} +206 q^{-77} +17 q^{-78} -156 q^{-79} -104 q^{-80} -148 q^{-81} -77 q^{-82} +69 q^{-83} +63 q^{-84} +81 q^{-85} +25 q^{-86} -26 q^{-87} +10 q^{-88} -35 q^{-89} -43 q^{-90} +7 q^{-91} +7 q^{-92} +22 q^{-93} +3 q^{-94} -13 q^{-95} +17 q^{-96} -2 q^{-97} -12 q^{-98} - q^{-100} +7 q^{-101} - q^{-102} -7 q^{-103} +5 q^{-104} +2 q^{-105} -2 q^{-106} - q^{-108} +2 q^{-109} -2 q^{-111} + q^{-112} }[/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.

8 11.gif

8_11

8 13.gif

8_13

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