T(9,2) - Revision history

ScottKnotPageRobot at 15:38, 31 August 2005

2005-08-31T15:38:05Z

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-<!-- This page was  generated from the splice template "Torus_Knot_Splice_Template". Please do not edit! -->
+<!-- This page was generated from the splice template [[Torus_Knot_Splice_Base]]. Please do not edit!
 {{Torus Knot Page|
 m = 9 |

ScottKnotPageRobot at 16:15, 30 August 2005

2005-08-30T16:15:51Z

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ScottTestRobot at 00:45, 29 August 2005

2005-08-29T00:45:48Z

← Older revision		Revision as of 20:45, 28 August 2005
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			<!-- This knot page was produced from [[Torus Knots Splice Template]] -->

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	<span id="top"></span>		<span id="top"></span>
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	{{Knot Navigation Links\|ext=jpg}}		{{Knot Navigation Links\|ext=jpg}}

		⚫	{{Torus Knot Page Header\|m=9\|n=2\|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-1,2,-3,4,-5,6,-7,8,-9,1,-2,3/goTop.html}}
	{\| align=left
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	\|[[Image:{{PAGENAME}}.jpg]]
⚫	\|{{Torus Knot ~~Site~~ ~~Links~~\|m=9\|n=2\|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-1,2,-3,4,-5,6,-7,8,-9,1,-2,3/goTop.html}}

	{{:{{PAGENAME}} Quick Notes}}
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	{{Vassiliev Invariants}}		{{Vassiliev Invariants}}

	~~===[[~~Khovanov Homology~~]]=~~==		{{Khovanov Homology\|table=<table border=1>

	The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.

	<center><table border=1>
	<tr align=center>		<tr align=center>
	<td width=14.2857%><table cellpadding=0 cellspacing=0>		<td width=14.2857%><table cellpadding=0 cellspacing=0>
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	<tr align=center><td>9</td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>		<tr align=center><td>9</td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>
	<tr align=center><td>7</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>		<tr align=center><td>7</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr>
	</table>~~</center>~~		</table>}}

	{{Computer Talk Header}}		{{Computer Talk Header}}
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	<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=  </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]</nowiki></pre></td></tr>		<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=  </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1}]</nowiki></pre></td></tr>
	<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>alex = Alexander[TorusKnot[9, 2]][t]</nowiki></pre></td></tr>		<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>alex = Alexander[TorusKnot[9, 2]][t]</nowiki></pre></td></tr>
	<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=  </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -4 -3 -2 1 2 3 4		<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=  </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -4 -3 -2 1
	1 + t - t + t - - - ~~t + t~~ - ~~t + t~~		1 + Alternating - Alternating + Alternating - ----------- -
	~~t</nowiki></pre></td></tr>~~		Alternating

			2 3 4
			Alternating + Alternating - Alternating + Alternating</nowiki></pre></td></tr>
	<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>Conway[TorusKnot[9, 2]][z]</nowiki></pre></td></tr>		<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>Conway[TorusKnot[9, 2]][z]</nowiki></pre></td></tr>
	<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> 2 4 6 8		<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> 2 4 6 8
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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>{0, 30}</nowiki></pre></td></tr>		<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>{0, 30}</nowiki></pre></td></tr>
	<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>Kh[TorusKnot[9, 2]][q, t]</nowiki></pre></td></tr>		<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>Kh[TorusKnot[9, 2]][q, t]</nowiki></pre></td></tr>
	<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> 7 9 11 2 15 3 15 4 19 5 19 6 23 7		<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> 7 9 2 11 3 15 4 15
	q + q + q ~~t +~~ q t + q ~~t +~~ q t + q ~~t +~~ q t +		q + q + Alternating q + Alternating q + Alternating q +

			5 19 6 19 7 23
			Alternating q + Alternating q + Alternating q +

	23 8 27 9		8 23 9 27
	q t + q t</nowiki></pre></td></tr>		Alternating q + Alternating q</nowiki></pre></td></tr>
	</table>		</table>

	{{Category:Knot Page}}		[[Category:Knot Page]]

ScottTestRobot at 23:43, 28 August 2005

2005-08-28T23:43:13Z

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-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-1,2,-3,4,-5,6,-7,8,-9,1,-2,3/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=9|n=2|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-1,2,-3,4,-5,6,-7,8,-9,1,-2,3/goTop.html}}
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 {{Knot Presentations}}
 {{Polynomial Invariants}}
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   q   t  + q   t</nowiki></pre></td></tr>
 </table>

ScottTestRobot at 01:24, 28 August 2005

2005-08-28T01:24:16Z

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DrorsRobot at 02:45, 27 August 2005

2005-08-27T02:45:55Z

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DrorsRobot at 23:35, 26 August 2005

2005-08-26T23:35:14Z

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DrorsRobot at 23:28, 26 August 2005

2005-08-26T23:28:34Z

New page

<!-- ailed $Failedmath . uwo . c$Failedtrv$Faileddge] at [h$Failedn.matuwo . ca
--------
c$Faidml K$Failedp://www.maFailedsKts/3.2.hl(2s pa$Failedtoronto.edu/~drorbn/KAtlas/inxtot Atl$Failedt presentations===

{|
|'''[[Planar Diagrams|Planar dir entatio$Failede="padding-left: 1em;" | X<sub>3146<u b 152/s$Failed62</sub>
|-
|'''[[Gauss Codes|Gauss code]]'''
|style=$Faile$Failed1, 2, -3, 1}
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistl$Failedepa$Faedial Invariants|name=T(3,2)}}

===[[Finite Type (Vassiliev)nvaanFailed===$Failed'''
|style="padding-left: 1em;" | {0, 1})
|}

[[KhovHomolo$Failedeffi oven$Failed> are show$Failedj</math>, alternation over <math>r</math>). The squares with <f$FailedYel$Faile$Failedh>j-2r=s+1</math>, where <math>s=</math>2 is the signature of T$Failedro entries off the $Failed<font class=HLRed$Failed

<center><table border=1>
<tr align=center>
<td width=25.%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbs$Fa$Failed$Failed>j</td><td> </td$Failed/tr>
</table></td>
<td width=12.5%>0</td ><td width=12.5%>1</td ><td width=12.5%>2</td ><td width=12.5%>3</td ><td wid$Failednter><td>9</td><td> </td><td>&nbsp$Failedtd bgcolor=yellow>1</$Faileded<td> </td><t$Failedo$Failedo$Failed>$Failed>$Failedl$Failedl$Failedd$Failed>$Failed&$Failedd$Failed>$Failed>$Fa$Failed style="color: red; borpadding:0"><< KnotTheory$Failed
</tr>$Failed="border: 0px; pa$Failedy$Failed9$Failedd$Failed<$Failed;$Failed=$Failed $Failedn$Failedi$Failedn$Failedp $Failedd$Failedo$Failed
ailed > -----------------------
tdtd><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[TorusKnot[3, 2]]</nowiki></pre></td$Failedlor: blue; border$FailedO$Failedp$Failed $Failedo$Failed,$Failed,$Faileds$Failedi$Failed<$Failedo$Failede$Failedk$Failedi$Failedp$Failedde[-2, 3, -1, 2, -3, 1]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: bl$Failedn[5]:=</nowiki></$Failedrd$Failedo$Failed>$Failedea$Failed<$Failedd$Failed $Failedn0rpadding:0<$Failed3$Failedr$Failed: 0px; padding: 0em"><nowiki>Out[6]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 1
-1 + - + t
t</nowiki></pre></td></tr>
<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>Conway[TorusKnot[3, 2]][z]</nowiki></pre></td></tr>
<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> 2
1 + z</nowiki></pre></td></tr>
<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>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[3, 1]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[TorusKnot[3, 2]], KnotSignature[TorusKnot[3, 2]]}</nowiki></pre></td></tr>
<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>{3, 2}</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>J=Jones[TorusKnot[3, 2]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 4
q + q - q</nowiki></pre></td></tr>
<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>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
<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>{Knot[3, 1]}</nowiki></pre></td></tr>
Include[ColouredJonesM.mhtml]
<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>A2Invariant[TorusKnot[3, 2]][q]</nowiki></pre></td></tr>
<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> 2 4 6 8 12 14
q + q + 2 q + q - q - q</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>Kauffman[TorusKnot[3, 2]][a, z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2
-4 2 z z z z
-a - -- + -- + -- + -- + --
2 5 3 4 2
a a a a a</nowiki></pre></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>{Vassiliev[2][TorusKnot[3, 2]], Vassiliev[3][TorusKnot[3, 2]]}</nowiki></pre></td></tr>
<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>{0, 1}</nowiki></pre></td></tr>
<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>Kh[TorusKnot[3, 2]][q, t]</nowiki></pre></td></tr>
<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> 3 5 2 9 3
q + q + q t + q t</nowiki></pre></td></tr>
</table>