7 1: Difference between revisions

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<tr align=center>
<tr align=center>
<td width=16.6667%><table cellpadding=0 cellspacing=0>
<td width=16.6667%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>\</td><td>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
<td width=8.33333%>-7</td ><td width=8.33333%>-6</td ><td width=8.33333%>-5</td ><td width=8.33333%>-4</td ><td width=8.33333%>-3</td ><td width=8.33333%>-2</td ><td width=8.33333%>-1</td ><td width=8.33333%>0</td ><td width=16.6667%>&chi;</td></tr>
<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>-7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>-9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>-11</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-13</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-15</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-17</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-19</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-21</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
</table> |
coloured_jones_2 = <math> q^{-6} + q^{-9} - q^{-11} + q^{-12} - q^{-14} + q^{-15} - q^{-17} + q^{-18} - q^{-20} - q^{-23} + q^{-24} - q^{-26} + q^{-27} </math> |
coloured_jones_3 = <math> q^{-9} + q^{-13} - q^{-16} + q^{-17} - q^{-20} + q^{-21} - q^{-24} + q^{-25} - q^{-28} + q^{-29} - q^{-31} - q^{-32} + q^{-33} - q^{-35} + q^{-37} - q^{-39} + q^{-41} - q^{-43} + q^{-45} + q^{-46} - q^{-47} + q^{-50} - q^{-51} </math> |
coloured_jones_4 = <math> q^{-12} + q^{-17} - q^{-21} + q^{-22} - q^{-26} + q^{-27} - q^{-31} + q^{-32} - q^{-36} + q^{-37} -2 q^{-41} + q^{-42} -2 q^{-46} + q^{-47} + q^{-48} -2 q^{-51} + q^{-52} + q^{-53} -2 q^{-56} + q^{-57} + q^{-58} -2 q^{-61} + q^{-62} +2 q^{-63} -2 q^{-66} + q^{-67} + q^{-68} -2 q^{-71} + q^{-72} + q^{-73} -2 q^{-76} + q^{-77} - q^{-81} + q^{-82} </math> |
coloured_jones_5 = <math> q^{-15} + q^{-21} - q^{-26} + q^{-27} - q^{-32} + q^{-33} - q^{-38} + q^{-39} - q^{-44} + q^{-45} - q^{-50} - q^{-56} + q^{-60} - q^{-62} + q^{-66} - q^{-68} + q^{-72} - q^{-74} + q^{-78} + q^{-84} - q^{-87} + q^{-90} - q^{-93} + q^{-96} - q^{-99} - q^{-105} + q^{-107} - q^{-111} + q^{-113} + q^{-119} - q^{-120} </math> |
coloured_jones_6 = <math> q^{-18} + q^{-25} - q^{-31} + q^{-32} - q^{-38} + q^{-39} - q^{-45} + q^{-46} - q^{-52} + q^{-53} - q^{-59} + q^{-60} - q^{-61} - q^{-66} + q^{-67} - q^{-68} + q^{-72} - q^{-73} + q^{-74} - q^{-75} + q^{-79} - q^{-80} + q^{-81} - q^{-82} + q^{-86} - q^{-87} + q^{-88} - q^{-89} + q^{-93} - q^{-94} + q^{-95} - q^{-96} + q^{-97} + q^{-100} - q^{-101} + q^{-102} - q^{-103} + q^{-104} - q^{-106} + q^{-107} - q^{-108} + q^{-109} - q^{-110} + q^{-111} - q^{-113} + q^{-114} - q^{-115} + q^{-116} - q^{-117} + q^{-118} - q^{-120} + q^{-121} - q^{-122} + q^{-123} - q^{-124} + q^{-125} - q^{-126} - q^{-127} + q^{-128} - q^{-129} + q^{-130} - q^{-131} + q^{-132} - q^{-134} + q^{-135} - q^{-136} + q^{-137} - q^{-138} + q^{-139} - q^{-141} + q^{-142} - q^{-143} + q^{-144} - q^{-145} + q^{-146} + q^{-149} - q^{-150} + q^{-151} - q^{-152} + q^{-156} - q^{-157} + q^{-158} - q^{-159} - q^{-164} + q^{-165} </math> |
coloured_jones_7 = <math> q^{-21} + q^{-29} - q^{-36} + q^{-37} - q^{-44} + q^{-45} - q^{-52} + q^{-53} - q^{-60} + q^{-61} - q^{-68} + q^{-69} - q^{-71} - q^{-76} + q^{-77} - q^{-79} + q^{-85} - q^{-87} + q^{-93} - q^{-95} + q^{-101} - q^{-103} + q^{-109} - q^{-111} + q^{-114} + q^{-117} - q^{-119} + q^{-122} - q^{-127} + q^{-130} - q^{-135} + q^{-138} - q^{-143} + q^{-146} - q^{-150} - q^{-151} + q^{-154} - q^{-158} + q^{-162} - q^{-166} + q^{-170} - q^{-174} + q^{-178} + q^{-179} - q^{-182} + q^{-187} - q^{-190} + q^{-195} - q^{-198} - q^{-201} + q^{-203} - q^{-209} + q^{-211} + q^{-216} - q^{-217} </math> |
computer_talk =
<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<table><tr align=left>
<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[7, 1]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[1, 8, 2, 9], X[3, 10, 4, 11], X[5, 12, 6, 13], X[7, 14, 8, 1],
X[9, 2, 10, 3], X[11, 4, 12, 5], X[13, 6, 14, 7]]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Knot[7, 1]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[-1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>DTCode[Knot[7, 1]]</nowiki></code></td></tr>
<tr align=left>
<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[8, 10, 12, 14, 2, 4, 6]</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[7, 1]]</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[2, {-1, -1, -1, -1, -1, -1, -1}]</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>{2, 7}</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[7, 1]]</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>2</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[7, 1]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:7_1_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[7, 1]]&) /@ {
SymmetryType, UnknottingNumber, ThreeGenus,
BridgeIndex, SuperBridgeIndex, NakanishiIndex
}</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Reversible, 3, 3, 2, 4, 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[7, 1]][t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -3 -2 1 2 3
-1 + t - t + - + t - t + t
t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Conway[Knot[7, 1]][z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 2 4 6
1 + 6 z + 5 z + z</nowiki></code></td></tr>
</table>
<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[7, 1]}</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[7, 1]], KnotSignature[Knot[7, 1]]}</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>{7, -6}</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[7, 1]][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> -10 -9 -8 -7 -6 -5 -3
-q + q - q + q - q + q + q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[15]:=</code></td>
<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[7, 1]}</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[7, 1]][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> -30 -28 -26 -18 -16 2 -12 -10
-q - q - q + q + q + --- + q + q
14
q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<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[7, 1]][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> 6 8 6 2 8 2 6 4 8 4 6 6
4 a - 3 a + 10 a z - 4 a z + 6 a z - a z + a z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[18]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Knot[7, 1]][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> 6 8 7 9 11 13 6 2 8 2
-4 a - 3 a + 3 a z + a z - a z + a z + 10 a z + 7 a z -
10 2 12 2 7 3 9 3 11 3 6 4 8 4
2 a z + a z - 4 a z - 3 a z + a z - 6 a z - 5 a z +
10 4 7 5 9 5 6 6 8 6
a z + a z + a z + a z + a z</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[19]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>{Vassiliev[2][Knot[7, 1]], Vassiliev[3][Knot[7, 1]]}</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>{6, -14}</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[7, 1]][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> -7 -5 1 1 1 1 1 1
q + q + ------ + ------ + ------ + ------ + ------ + -----
21 7 17 6 17 5 13 4 13 3 9 2
q t q t q t q t q t q t</nowiki></code></td></tr>
</table>
<table><tr align=left>
<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[7, 1], 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> -27 -26 -24 -23 -20 -18 -17 -15 -14 -12
q - q + q - q - q + q - q + q - q + q -
-11 -9 -6
q + q + q</nowiki></code></td></tr>
</table> }}

Revision as of 23:03, 12 June 2007

{{Rolfsen Knot Page| n = 7 | k = 1 | KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,5,-2,6,-3,7,-4,1,-5,2,-6,3,-7,4/goTop.html |

braid_table =

BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif

|

braid_crossings = 7 | braid_width = 2 | braid_index = 2 | same_alexander = | same_jones = |

khovanov_table =

\
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