T(13,2) - Revision history

ScottKnotPageRobot at 15:37, 31 August 2005

2005-08-31T15:37:41Z

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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 = 13 |

ScottKnotPageRobot at 16:15, 30 August 2005

2005-08-30T16:15:25Z

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ScottTestRobot at 00:45, 29 August 2005

2005-08-29T00:45:08Z

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

	<!-- -->		<!-- -->
			<!-- -->

	<span id="top"></span>		<span id="top"></span>
			<!-- -->

	{{Knot Navigation Links\|ext=jpg}}		{{Knot Navigation Links\|ext=jpg}}

		⚫	{{Torus Knot Page Header\|m=13\|n=2\|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-10,11,-12,13,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,1,-2,3/goTop.html}}
	{\| align=left
	\|- valign=top
	\|[[Image:{{PAGENAME}}.jpg]]
⚫	\|{{Torus Knot ~~Site~~ ~~Links~~\|m=13\|n=2\|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-10,11,-12,13,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,1,-2,3/goTop.html}}

	{{:{{PAGENAME}} Quick Notes}}
	\|}

	<br style="clear:both" />		<br style="clear:both" />
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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=11.1111%><table cellpadding=0 cellspacing=0>		<td width=11.1111%><table cellpadding=0 cellspacing=0>
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	<tr align=center><td>13</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> </td><td> </td><td> </td><td> </td><td>1</td></tr>		<tr align=center><td>13</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> </td><td> </td><td> </td><td> </td><td>1</td></tr>
	<tr align=center><td>11</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> </td><td> </td><td> </td><td> </td><td>1</td></tr>		<tr align=center><td>11</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> </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, 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, 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[13, 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[13, 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> -6 -5 -4 -3 -2 1 2 3 4 5 6		<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> -6 -5 -4 -3
	1 + t - t + t - t + ~~t - - - t + t - t + t - t + t~~		1 + Alternating - Alternating + Alternating - Alternating +

⚫	~~t</nowiki></pre></td></tr>~~
			-2 1 2
			Alternating - ----------- - Alternating + Alternating -
			Alternating

		⚫	3 4 5 6
			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[13, 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[13, 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 10 12		<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 10 12
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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, 91}</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, 91}</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[13, 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[13, 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> 11 13 15 2 19 3 19 4 23 5 23 6 27 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> 11 13 2 15 3 19 4 19
	q + q + q ~~t +~~ q t + q ~~t +~~ q t + q ~~t +~~ q t +		q + q + Alternating q + Alternating q + Alternating q +

			5 23 6 23 7 27
			Alternating q + Alternating q + Alternating q +

			8 27 9 31 10 31
			Alternating q + Alternating q + Alternating q +

	27 8 31 9 31 10 35 11 35 12 39 13		11 35 12 35 13 39
	q ~~t +~~ q t + q ~~t +~~ q t + q ~~t +~~ q t</nowiki></pre></td></tr>		Alternating q + Alternating q + Alternating q</nowiki></pre></td></tr>
	</table>		</table>

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

ScottTestRobot at 23:42, 28 August 2005

2005-08-28T23:42:39Z

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 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-10,11,-12,13,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,1,-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=13|n=2|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-8,9,-10,11,-12,13,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,1,-2,3/goTop.html}}
 {{:{{PAGENAME}} Quick Notes}}
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 {{Knot Presentations}}
 {{Polynomial Invariants}}
 {{Vassiliev Invariants}}
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   q   t  + q   t  + q   t   + q   t   + q   t   + q   t</nowiki></pre></td></tr>
 </table>

ScottTestRobot at 01:24, 28 August 2005

2005-08-28T01:24:23Z

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DrorsRobot at 02:46, 27 August 2005

2005-08-27T02:46:04Z

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DrorsRobot at 23:35, 26 August 2005

2005-08-26T23:35:25Z

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DrorsRobot at 23:33, 26 August 2005

2005-08-26T23:33:17Z

← Older revision		Revision as of 19:33, 26 August 2005
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	<!-- $Failedmath . ~~uwo~~ . c$Failedv$~~ildge~~] at [h$Failedn.matuwo . ca		<!-- ailed $Failedn$F $Failedmath . u . c$Failedv$ileddge] at [h$Failedn.matuwo . ca
	--------		--------
	~~c$Faidml~~ K$~~FaidaFaedsKt~~$~~iledpaaidtoront~~$~~edailedtres~~$Failedgrams$~~Faileding~~-left: 1em;" \| X<sub>3146<u b 152/s$Failed62</sub>		caidml K$FaidailedsKt$idpa$Faidront$iledailedtres$Failedgrams \| dir en Planartio$Failede="padding-left: 1em;" \| X<sub>3146<u b 152/s$Failed62</sub>
	\|-		\|-
	\|'''[[GausCeGss cod$Failede=$Faile$Failed1, 2, -3, 1}		\|'''[[GausCeGss cod$Failede=$Faile$Failed1, 2, -3, 1}
	'$Failed (Dowk-~~Thistlewaite) Codes~~\|~~DowrhtFaileda~~$~~FaediaInvariants~~\|name=T(3,2)}}		'$Failed (Dowk-ThistlaeCes\|Dowr - Thistlailedepa$Faedial Invariants\|name=T(3,2)}}

	===[[Finite Type (Vassiliev)nvaanFailed===$Failed'''		===[[Finite Type (Vassiliev)nvaanFailed===$Failed'''
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	[[KhovHomolo$Failedeffi oven$Failed> are shoFaile</math>, over ~~ternation < math</math~~>). e ~~sques~~ with <f$FailedYe2</math>, where <math>s=</math>2 is the ~~signHLRed$Failedrder~~=1>		[[KhovHomolo$Failedeffi oven$Failed> are shoFaile</math>, over ternationmathmath>). The squares with <f$FailedYe2</math>, where <math>s=</math>22 signHLRed$Fail the<center><table border=1>
	<tr align=center>		<tr align=center>
	<td wid$Failedled$Failed>j</td><td> </td$Failed/tr>		<td wid$Failedled$Failed>j</td><td> </td$Failed/tr>
		⚫	</tab$Failed/$Failedlednter><td>9</td><td> </td><td>&nbsp$Failedtd bgcyello1</$Faileded<td> </td><t$Failedo$Failedo$Fail$Failediled>$Failed&$Failedd$Failed>$Failed>$Fa$Failed style="color: red; borpadding:0"><< KnotTheory$Failed
	</table></td>
		⚫	</tr>$$Failed9$Failedd$Failed<$Failed;$Failed=$Failed $Failedn$Failedi$Failedn$Failedp $Failedd$Faile$Failed
⚫	<~~td wi~~$F$~~Failedtd~~>9</td><td> </td><td>&nbsp$Failedtd ~~bgcolor=yellow>1~~</$~~Fleded~~ < ~~tdnbsp~~;</td><t$Failedo$Failedo$Fail$Failediled>$Failed&$Failedd$Failed>$Failed>$Fa$Failed style="color: red; borpadding:0"><< KnotTheory$Failed
⚫	</tr>$~~Failed="border: 0~~$Failed$Failed;$Failed=$Failed $Failedn$Failedi$Failedn$Failedp $Failedd$Faile$Failed
	ailed > -----$Failed------		ailed > -----$Failed------
	tdtd><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[TorusKnot[3, 2]]</now$Failededp$Failed $Failedo$F$Failedd,$Faileds$Failedi$Failed<$Failedo$Failede$Failedk$Failedi$Failedp$Failedde[-2, 3, -1, 2, -3, 1]</nowiki></pre></td></$Failedolor: bl$~~$Failed~~/$~~Failedrd~~$~~Failedo~~$~~Faed~~ > $~~Failea~~$Failed<$Failedd$Failed $Failedn0rpadding:0<$Failed3$Failedr$Failed: $Failed  </now$~~Failed=borde~~ ~~$Failed~~ - t		tdtd><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[TorusKnot[3, 2]]</now$Failededp$Failed $Failedo$F$Failedd,$Faileds$Failedi$Failed<$Failedo$Failede$Failedk$Failedi$Failedp$Failedde[-2, 3, -1, 2, -3, 1]</nowiki></pre></td><Failedolor:bl$Faidn[5]:=</nowiki></$Faedrd$Faido$Failed>$Failedea$Failed<$Failedd$Failed $Failedn0rpadding:0<$Failed3$Failedr$Failed: $Failed  </now$Faile t
	-(-------) < ~~$Failedo~~$Failed < $~~Fai~~		-1 - ------- + borde $Failed < $Fa
	$~~Failedde~~$Failed $Failedd$Failed>$Failede$Failed $Failede$Failed 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><t$Failedadding: 0em"><now$Failed=$Failed/$Faileda$Failede$Failedm$Failed&$Failedr$Failed<$Failed>$Failed<$FailedK$Failedt$Failede$Failed $Failednowiki>Out[9]=   </nowiki></pre></td><td><~~pre~~ ~~style=~~"~~color: $Failedi~~>~~{3, 2}~~</~~nowiki~~>$~~Failedo$Failedrpaddin~~ $Failed/$Failed:$~~Failede~~$~~Failedt~~$~~Failedd~~$Failed>$Failedd~~$Failedt$Failedcpadding0~~$Failed~~"8ding~~: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>		$Failededo$Failed<$Failede$Failed $Failedd$Failed>$Failede$Failed $Failede$Failed 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><t$Failedadding: 0em"><now$Failed=$Failed/$Faileda$Failede$Failedm$Failed&$Failedr$Failed<$Failed>$Failed<$FailedK$Failedt$Failede$Failed $Failednowiki>Out[9]=   </nowiki></pre></td><td><p$Failedding: 0em"><nowik$Failed/$Failed>$Failedepadding:0<$Failed<$Failed:$Failedi$Failed"pai$Failed;$Failedi$Failed>$Failedr$Failedd$Failed 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>		<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]		Include[ColouredJonesM.mhtml]

DrorsRobot at 23:28, 26 August 2005

2005-08-26T23:28:40Z

New page

<!-- $Failedmath . uwo . c$Failedv$ildge] at [h$Failedn.matuwo . ca
--------
c$Faidml K$FaidaFaedsKt$iledpaaidtoront$edailedtres$Failedgrams$Faileding-left: 1em;" | X<sub>3146<u b 152/s$Failed62</sub>
|-
|'''[[GausCeGss cod$Failede=$Faile$Failed1, 2, -3, 1}
'$Failed (Dowk-Thistlewaite) Codes|DowrhtFaileda$FaediaInvariants|name=T(3,2)}}

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

[[KhovHomolo$Failedeffi oven$Failed> are shoFaile</math>, over ternation < math</math>). e sques with <f$FailedYe2</math>, where <math>s=</math>2 is the signHLRed$Failedrder=1>
<tr align=center>
<td wid$Failedled$Failed>j</td><td> </td$Failed/tr>
</table></td>
<td wi$F$Failedtd>9</td><td> </td><td>&nbsp$Failedtd bgcolor=yellow>1</$Fleded < tdnbsp;</td><t$Failedo$Failedo$Fail$Failediled>$Failed&$Failedd$Failed>$Failed>$Fa$Failed style="color: red; borpadding:0"><< KnotTheory$Failed
</tr>$Failed="border: 0$Failed$Failed;$Failed=$Failed $Failedn$Failedi$Failedn$Failedp $Failedd$Faile$Failed
ailed > -----$Failed------
tdtd><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[TorusKnot[3, 2]]</now$Failededp$Failed $Failedo$F$Failedd,$Faileds$Failedi$Failed<$Failedo$Failede$Failedk$Failedi$Failedp$Failedde[-2, 3, -1, 2, -3, 1]</nowiki></pre></td></$Failedolor: bl$$Failed/$Failedrd$Failedo$Faed > $Failea$Failed<$Failedd$Failed $Failedn0rpadding:0<$Failed3$Failedr$Failed: $Failed  </now$Failed=borde $Failed - t
-(-------) < $Failedo$Failed < $Fai
$Failedde$Failed $Failedd$Failed>$Failede$Failed $Failede$Failed 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><t$Failedadding: 0em"><now$Failed=$Failed/$Faileda$Failede$Failedm$Failed&$Failedr$Failed<$Failed>$Failed<$FailedK$Failedt$Failede$Failed $Failednowiki>Out[9]=   </nowiki></pre></td><td><pre style="color: $Failedi>{3, 2}</nowiki>$Failedo$Failedrpaddin $Failed/$Failed:$Failede$Failedt$Failedd$Failed>$Failedd$Failedt$Failedcpadding0$Failed"8ding: 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>