T(7,2): Difference between revisions

Line 1:

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Visit $Failedmath.uwo.c$Failedtriev$Faileddge] at [h$Failedn.math.uwo.ca/c$Failedml Knotilus]!

⚫

Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/3.2.hl(2s pageat o theginal [http://www.math.toronto.edu/~drorbn/KAtlas/inxtot Atl$Failedt presentations===

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-1,2,-3,4,-5,6,-7,1,-2,3/goTop.html T(7,2)'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!

⚫

Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/7.2.~~html T~~(~~7,2)'s~~ ~~page]~~ at ~~the original~~ [http://www.math.toronto.edu/~drorbn/KAtlas/~~index.html~~ ~~Knot~~ ~~Atlas]!~~

===Knot presentations===

{|

|'''[[Planar Diagrams|Planar diagram presentation]]'''

|style="padding-left: 1em;" | X<sub>~~5,13,6,12~~</sub> X<~~sub>13,7,14,6</sub>~~ ~~X<sub>7,1,8,14</sub~~> ~~X<sub>1928</sub> X<sub>9,3,10,2</sub> X<sub>3,11,4,10<~~/sub> X<sub>~~11,5,12,4~~</sub>

|style="padding-left: 1em;" | X<sub>3146</sub> X<b > 152/sub> X<sub>5362</sub>

|-

|'''[[Gauss Codes|Gauss code]]'''

|style=~~"padding-left: 1em;" | {-4, 5, -6, 7, -1~~, 2, -3~~, 4, -5, 6, -7~~, 1~~, -2, 3~~}

|style=$Faile$Failed1, 2, -3, 1}

|-

|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-~~Thistlethwaite~~ ~~code]]'''~~

|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistl$Failede="pa$Failedial Invariants|name=T(3,2)}}

⚫

|style="padding-left: 1em;" | 8 ~~10 12 14 2 4 6~~

|}

===~~Polynomial~~ invariants===

===[[Finite Type (Vassiliev) Invaants | Vasliev invariants]]===$Failed'''

⚫

|style="padding-left: 1em;" | {0, 1})

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===

{| style="margin-left: 1em;"

|-

|'''V<sub>2</sub> and V<sub>3</sub>'''

|style="padding-left: 1em;" | {0, 14})

|}

[[Khovanov ~~Homology]]. 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>6 is the signature of T(7,2). Nonzero entries off the critical diagonals (if any ~~exist) are highlighted~~ in <font class=HLRed~~>red</font>.~~

[[Khovanov Homolo$Failedefficien$Failed> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYel$Failedse on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>2 is the signature of T(3,2). Nonzero entries off the critical diagonals (if any $Failed in <font class=HLRed$Failed

<tr><td>\</td><td>&nbs$Failed;</td><td> \ </td><td>&n$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$Failednter><td>9</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>$Failed<td> </td><t$Failedo$Failedo$Failed>$Failed>$Failedl$Failedl$Failedd$Failed>$Failed&$Failedd$Failed>$Failed>$Failedo$Failedp$Failed align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td>

⚫

</tr>$Failed="border: 0px; pa$Failedy$Failed9$Failedd$Failed<$Failed;$Failed=$Failed $Failedn$Failedi$Failedn$Failedp$Failed>$Failedd$Failedo$Failed/td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[TorusKnot[3, 2]]</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>GaussCode[TorusKnot[3, 2]]</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-2, 3, -1, 2, -3, 1]</nowiki></pre></td></tr>

⚫

<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[TorusKnot[3, 2]]</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[3, 2]][t]</nowiki></pre></td></tr>

<tr ~~align~~=~~center~~><td>7<~~/td><td~~ ~~bgcolor~~=~~yellow~~>1<~~/td~~>~~<td bgcolor~~=~~yellow>~~ ~~</td><td>~~ ~~</td><td>~~ </td>~~<td> ~~</td>~~<td> ~~</td><td>~~ ~~<~~/td><td>&nbsp~~;~~</td~~><td>1~~</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> 1

-1 + - + t

⚫

t</nowiki></pre></td></tr>

</table></center>

⚫

<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>

<table>

</tr>

<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 19, 2005, 13:11:25)...</pre></td></tr>

<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>Crossings[TorusKnot[7, 2]]</nowiki></pre></td></tr>

⚫

⚫

<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>PD[TorusKnot[7, 2]]</nowiki></pre></td></tr>

⚫

<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>~~PD[X[5,~~ ~~13,~~ 6, ~~12],~~ ~~X[13,~~ 7, ~~14,~~ ~~6],~~ ~~X[7,~~ 1, 8, ~~14],~~ ~~X[1,~~ 9, 2, ~~8],~~

X[9, 3, 10, 2], X[3, 11, 4, 10], X[11, 5, 12, 4]]</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>GaussCode[TorusKnot[7, 2]]</nowiki></pre></td></tr>

<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3]</nowiki></pre></td></tr>

⚫

<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[TorusKnot[7, 2]]</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[7, 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> -3 -2 1 2 3

-1 + t - t + - + t - t + 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[7, 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~~

⚫

1 ~~+ 6 z + 5 z~~ + 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>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[TorusKnot[7, 2]], KnotSignature[TorusKnot[7, 2]]}</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>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[TorusKnot[7, 2]][q]</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 ~~5 6 7 8 9~~ 10

<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 + q - q~~ + q - q</nowiki></pre></td></tr>

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>

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[7, 2]][q]</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>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> 10 12 14 16 18 ~~26 28 30~~

<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~~ - q - q</nowiki></pre></td></tr>

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[7, 2]][a, z]</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 ~~2 2 3~~

<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

-3 4 z z z ~~3 z z 2 z 7 z 10~~ z z

-4 2 z z z z

-- - ~~-- +~~ --- ~~- -~~-- + -- + --- + ~~--- - --~~-- + --~~-- + ----- + --- -~~

-a - -- + -- + -- + -- + --

8 6 13 11 9 7 ~~12 10 8 6 11~~

2 5 3 4 2

a a a a a a ~~a a a a~~ a

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>

⚫

3 3 4 4 4 5 5 6 6

⚫

<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>

3 z 4 z z 5 z 6 z z z z z

⚫

<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

---- - ---- + --- - ---- - ---- + -- + -- + -- + --

9 7 ~~10 8 6 9 7 8 6~~

q + q + q t + q t</nowiki></pre></td></tr>

a a a a 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[7, 2]], Vassiliev[3][TorusKnot[7, 2]]}</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[7, 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> 5 7 9 2 13 3 13 4 17 5 17 6 21 7

q + q + q t + q t + q t + q t + q t + q t</nowiki></pre></td></tr>

</table>

T(7,2): Difference between revisions

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@@ Line 1: / Line 1: @@
 <!-- Script generated - do not edit! -->
+<!-- $Failed.jpg|next=T(5,2).jpg}}
-<!--  -->
+Visit $Failedmath.uwo.c$Failedtriev$Faileddge] at [h$Failedn.math.uwo.ca/c$Failedml Knotilus]!
-<span id="top"></span>
+Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/3.2.hl(2s pageat o theginal [http://www.math.toronto.edu/~drorbn/KAtlas/inxtot Atl$Failedt presentations===
-{{Knot Navigation Links|prev=T(5,2).jpg|next=T(4,3).jpg}}
-Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-4,5,-6,7,-1,2,-3,4,-5,6,-7,1,-2,3/goTop.html T(7,2)'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
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-===Knot presentations===
 {|
 |'''[[Planar Diagrams|Planar diagram presentation]]'''
-|style="padding-left: 1em;" | X<sub>5,13,6,12</sub> X<sub>13,7,14,6</sub> X<sub>7,1,8,14</sub> X<sub>1928</sub> X<sub>9,3,10,2</sub> X<sub>3,11,4,10</sub> X<sub>11,5,12,4</sub>
+|style="padding-left: 1em;" | X<sub>3146</sub> X<b > 152/sub> X<sub>5362</sub>
 |-
 |'''[[Gauss Codes|Gauss code]]'''
-|style="padding-left: 1em;" | {-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3}
+|style=$Faile$Failed1, 2, -3, 1}
 |-
-|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
+|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistl$Failede="pa$Failedial Invariants|name=T(3,2)}}
-|style="padding-left: 1em;" | 8 10 12 14 2 4 6
-|}
-===Polynomial invariants===
+===[[Finite Type (Vassiliev) Invaants | Vasliev invariants]]===$Failed'''
+|style="padding-left: 1em;" | {0, 1})
-{{Polynomial Invariants|name=T(7,2)}}
-===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
-{| style="margin-left: 1em;"
-|-
-|'''V<sub>2</sub> and V<sub>3</sub>'''
-|style="padding-left: 1em;" | {0, 14})
 |}
-[[Khovanov Homology]]. The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>6 is the signature of T(7,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
+[[Khovanov Homolo$Failedefficien$Failed> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYel$Failedse on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>2 is the signature of T(3,2). Nonzero entries off the critical diagonals (if any $Failed in <font class=HLRed$Failed
 <center><table border=1>
 <tr align=center>
-<td width=16.6667%><table cellpadding=0 cellspacing=0>
+<td width=25.%><table cellpadding=0 cellspacing=0>
-<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
+<tr><td>\</td><td>&nbs$Failed;</td><td>&nbsp;\&nbsp;</td><td>&n$Failed>j</td><td>&nbsp;</td$Failed/tr>
-<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
-<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
 </table></td>
-<td width=8.33333%>0</td ><td width=8.33333%>1</td ><td width=8.33333%>2</td ><td width=8.33333%>3</td ><td width=8.33333%>4</td ><td width=8.33333%>5</td ><td width=8.33333%>6</td ><td width=8.33333%>7</td ><td width=16.6667%>&chi;</td></tr>
+<td width=12.5%>0</td ><td width=12.5%>1</td ><td width=12.5%>2</td ><td width=12.5%>3</td ><td wid$Failednter><td>9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>$Failed<td>&nbsp;</td><t$Failedo$Failedo$Failed>$Failed>$Failedl$Failedl$Failedd$Failed>$Failed&$Failedd$Failed>$Failed>$Failedo$Failedp$Failed align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
+</tr>$Failed="border: 0px; pa$Failedy$Failed9$Failedd$Failed<$Failed;$Failed=$Failed $Failedn$Failedi$Failedn$Failedp$Failed>$Failedd$Failedo$Failed/td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[TorusKnot[3, 2]]</nowiki></pre></td></tr>
-<tr align=center><td>21</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>19</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>&nbsp;</td><td>0</td></tr>
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[3, 1, 4, 6], X[1, 5, 2, 4], X[5, 3, 6, 2]]</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>GaussCode[TorusKnot[3, 2]]</nowiki></pre></td></tr>
-<tr align=center><td>17</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>0</td></tr>
-<tr align=center><td>15</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 valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-2, 3, -1, 2, -3, 1]</nowiki></pre></td></tr>
+<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[TorusKnot[3, 2]]</nowiki></pre></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>11</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 valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {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[3, 2]][t]</nowiki></pre></td></tr>
-<tr align=center><td>9</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-<tr align=center><td>7</td><td bgcolor=yellow>1</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>1</td></tr>
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     1
+-1 + - + t
-<tr align=center><td>5</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>
+     t</nowiki></pre></td></tr>
-</table></center>
+<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]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     2
-{{Computer Talk Header}}
++ z</nowiki></pre></td></tr>
-<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><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 19, 2005, 13:11:25)...</pre></td></tr>
-<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>Crossings[TorusKnot[7, 2]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>7</nowiki></pre></td></tr>
-<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>PD[TorusKnot[7, 2]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[5, 13, 6, 12], X[13, 7, 14, 6], X[7, 1, 8, 14], X[1, 9, 2, 8],
-  X[9, 3, 10, 2], X[3, 11, 4, 10], X[11, 5, 12, 4]]</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>GaussCode[TorusKnot[7, 2]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3]</nowiki></pre></td></tr>
-<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[TorusKnot[7, 2]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {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[7, 2]][t]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      -3    -2   1        2    3
--1 + t   - t   + - + t - t  + 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[7, 2]][z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       2      4    6
-+ 6 z  + 5 z  + 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]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[7, 1]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;&nbsp;</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[7, 2]], KnotSignature[TorusKnot[7, 2]]}</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]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{7, 6}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;&nbsp;</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[7, 2]][q]</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]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3    5    6    7    8    9    10
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     3    4
-q  + q  - q  + q  - q  + q  - q</nowiki></pre></td></tr>
+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]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[7, 1]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;&nbsp;</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[7, 2]][q]</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>A2Invariant[TorusKnot[3, 2]][q]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 10    12      14    16    18    26    28    30
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;&nbsp;</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   - q   - q</nowiki></pre></td></tr>
+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[7, 2]][a, z]</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]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                                  2       2      2       2    3
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                       2    2
--3   4     z     z    z    3 z   z     2 z    7 z    10 z    z
+  -4   2    z    z    z    z
--- - -- + --- - --- + -- + --- + --- - ---- + ---- + ----- + --- -
+-a   - -- + -- + -- + -- + --
-6    13    11    9    7     12    10      8      6      11
+5    3    4    2
-a    a    a     a     a    a     a     a       a      a      a
+       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]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 1}</nowiki></pre></td></tr>
-3    4       4      4    5    5    6    6
+<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>
-z    4 z    z     5 z    6 z    z    z    z    z
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>     3    5  2    9  3
-  ---- - ---- + --- - ---- - ---- + -- + -- + -- + --
-7     10     8      6     9    7    8    6
+q + q  + q  t  + q  t</nowiki></pre></td></tr>
-   a      a     a      a      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[7, 2]], Vassiliev[3][TorusKnot[7, 2]]}</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 14}</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[7, 2]][q, t]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 5    7    9  2    13  3    13  4    17  5    17  6    21  7
-q  + q  + q  t  + q   t  + q   t  + q   t  + q   t  + q   t</nowiki></pre></td></tr>
 </table>