T(5,4): Difference between revisions
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<!-- a ledmath . uwo . c$Failedv$ildg[h$Failedn.matuwo . ca |
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-------- |
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c$Faidml K$FaidaFaedsKiledaidtort$edaidts$Faileds$Failedg-l$Failed$Fail$Failedd$Failede=$Faile$Failed1, 2, -3, 1} |
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<span id="top"></span> |
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'$Failed (Dowk-Thistlewaite) Codes|DowrhtFaileda$FaediaInvariants|name=T(3,2$Failed=$Failedinite $Failediev)nvaanFl=$iled''$Faileddding-left: 1em;"$Failed) |
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{{Knot Navigation Links|prev=T(7,3).jpg|next=T(15,2).jpg}} |
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Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/14,15,-3,-5,-7,10,11,12,-15,-2,-4,7,8,9,-12,-14,-1,4,5,6,-9,-11,-13,1,2,3,-6,-8,-10,13/goTop.html T(5,4)'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]! |
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Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/5.4.html T(5,4)'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]! |
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===Knot presentations=== |
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{| |
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|'''[[Planar Diagrams|Planar diagram presentation]]''' |
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|style="padding-left: 1em;" | X<sub>17,25,18,24</sub> X<sub>10,26,11,25</sub> X<sub>3,27,4,26</sub> X<sub>11,19,12,18</sub> X<sub>4,20,5,19</sub> X<sub>27,21,28,20</sub> X<sub>5,13,6,12</sub> X<sub>28,14,29,13</sub> X<sub>21,15,22,14</sub> X<sub>29,7,30,6</sub> X<sub>22,8,23,7</sub> X<sub>15,9,16,8</sub> X<sub>23,1,24,30</sub> X<sub>16,2,17,1</sub> X<sub>9,3,10,2</sub> |
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|- |
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|'''[[Gauss Codes|Gauss code]]''' |
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|style="padding-left: 1em;" | {14, 15, -3, -5, -7, 10, 11, 12, -15, -2, -4, 7, 8, 9, -12, -14, -1, 4, 5, 6, -9, -11, -13, 1, 2, 3, -6, -8, -10, 13} |
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|- |
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|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]''' |
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|style="padding-left: 1em;" | 16 -26 -12 22 -2 -18 28 -8 -24 4 -14 -30 10 -20 -6 |
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|} |
|} |
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[[KhovHomolo$Failedeffi oven$Failed> are shoFaile</math>, over ternation < math</math>). e sques with <f$FailedYe2</math>, where <math>s=</th > 2 the signHLRed$Failedrder=1> |
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===Polynomial invariants=== |
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<tr cter> |
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<td wid$Failedled$Fai$Failed$Failed/tr> |
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{{Polynomial Invariants|name=T(5,4)}} |
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===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]=== |
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{| style="margin-left: 1em;" |
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|- |
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|'''V<sub>2</sub> and V<sub>3</sub>''' |
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|style="padding-left: 1em;" | {0, 50}) |
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|} |
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[[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>8 is the signature of T(5,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>. |
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<center><table border=1> |
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<tr align=center> |
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<td width=14.2857%><table cellpadding=0 cellspacing=0> |
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<tr><td>\</td><td> </td><td>r</td></tr> |
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<tr><td> </td><td> \ </td><td> </td></tr> |
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<tr><td>j</td><td> </td><td>\</td></tr> |
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</table></td> |
</table></td> |
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<td wi$F$$Failed;</td><td> $Failedtd bgcolor=yellow>1</$Fl$Faileddd$Failed>$Failed>$Fa$Failed style="color: red; borpding:0d;$Failed=$Failed $Failedn$Failedi$Failedn$Failedp $Failedd$Fae$Failediled > -----$Failed------ |
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<td width=7.14286%>0</td ><td width=7.14286%>1</td ><td width=7.14286%>2</td ><td width=7.14286%>3</td ><td width=7.14286%>4</td ><td width=7.14286%>5</td ><td width=7.14286%>6</td ><td width=7.14286%>7</td ><td width=7.14286%>8</td ><td width=7.14286%>9</td ><td width=14.2857%>χ</td></tr> |
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td$Failed; padding: 0em"><nowiki>PD[TorusKnot[3, 2]]</now$Failededp$Failed $Failedo$F$Failedd,$Faileds$Failedi$Failed<$Failedo$ilede$Failedk$Failedi$Failedp$Failde[-2, 3, -1, 2, -3, 1]</nowiki></pre></td></$Failedolor: bl$$Failed/$Failedrd$Failedo$Faed > $Failea$Failed<$Failedd$Failed $Failedn0rpadding:0<$Failed3$Failedr$Failed: $Failed </now$Failed=borde $Failed - t |
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<tr align=center><td>27</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>-1</td></tr> |
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-($Failediled < $Fai |
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<tr align=center><td>25</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>-1</td></tr> |
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$Failed$Failedailedd$Failed>$Failede$Failed $Failede$Failed 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><t$Failedadding: 0$Faedlede$Failedm$Faid&$Failedr$Failed<$Faed > $Fail<$FailedK$Failedt$Failede$Failed $Failednowiki>Out[9]= </nowiki></pre></td><td><pre style="color: $Fa$Failedrpaddin $Failed/$Failed:$Failede$Failedt$Failedd$Failed>$Failedd$Failedt$Failedcpadding0$Failed"8ding: 0em"><nowiki>Select[AllK$Failed1/q) === Jones[#]$Failedr$Failedp$Failed:$Failedo$Failed>$Failedl$Failedn$Failedl$Failede$Failede$Failedr$Failedd$Failedl$Failed1$Failedi></pre></td><td><pre style="color: black; border: 0px; paddin$Failed2 14 |
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<tr align=center><td>23</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td>-1</td></tr> |
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q + q +$Failed<$Failedl$Failed:d$Failedr$Failed $Failed,$Failed>$Failedo$Failedl$Failed -8 z $Failed- 2 |
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<tr align=center><td>21</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </td><td>0</td></tr> |
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a $Failed></tr> |
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<tr align=center><td>19</td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>1</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td>1</td></tr> |
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<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> |
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<tr align=center><td>17</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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<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> |
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<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> |
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<tr align=center><td>13</td><td bgcolor=red>1</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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<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 |
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<tr align=center><td>11</td><td bgcolor=red>1</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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q + q + q t + q t</nowiki></pre></td></tr> |
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</table></center> |
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</table></math> |
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{{Computer Talk Header}} |
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<table> |
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<tr valign=top> |
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<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:= </pre></td> |
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<td align=left><pre style="color: red; border: 0px; padding: 0em"><< KnotTheory`</pre></td> |
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</tr> |
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<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> |
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<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[5, 4]]</nowiki></pre></td></tr> |
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<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]= </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>15</nowiki></pre></td></tr> |
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<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[5, 4]]</nowiki></pre></td></tr> |
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<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[17, 25, 18, 24], X[10, 26, 11, 25], X[3, 27, 4, 26], |
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X[11, 19, 12, 18], X[4, 20, 5, 19], X[27, 21, 28, 20], |
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X[5, 13, 6, 12], X[28, 14, 29, 13], X[21, 15, 22, 14], |
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X[29, 7, 30, 6], X[22, 8, 23, 7], X[15, 9, 16, 8], X[23, 1, 24, 30], |
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X[16, 2, 17, 1], X[9, 3, 10, 2]]</nowiki></pre></td></tr> |
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<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[5, 4]]</nowiki></pre></td></tr> |
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<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[14, 15, -3, -5, -7, 10, 11, 12, -15, -2, -4, 7, 8, 9, -12, |
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-14, -1, 4, 5, 6, -9, -11, -13, 1, 2, 3, -6, -8, -10, 13]</nowiki></pre></td></tr> |
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<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[5, 4]]</nowiki></pre></td></tr> |
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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[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]</nowiki></pre></td></tr> |
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<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[5, 4]][t]</nowiki></pre></td></tr> |
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<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 -2 2 5 6 |
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-1 + t - t + t + t - t + t</nowiki></pre></td></tr> |
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<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[5, 4]][z]</nowiki></pre></td></tr> |
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<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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1 + 15 z + 56 z + 77 z + 44 z + 11 z + z</nowiki></pre></td></tr> |
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<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> |
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<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>{}</nowiki></pre></td></tr> |
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<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[5, 4]], KnotSignature[TorusKnot[5, 4]]}</nowiki></pre></td></tr> |
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<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>{5, 8}</nowiki></pre></td></tr> |
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<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[5, 4]][q]</nowiki></pre></td></tr> |
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<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> 6 8 10 11 13 |
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q + q + q - q - q</nowiki></pre></td></tr> |
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<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> |
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<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>{}</nowiki></pre></td></tr> |
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Include[ColouredJonesM.mhtml] |
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<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[5, 4]][q]</nowiki></pre></td></tr> |
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<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> 22 24 26 28 30 32 34 36 38 40 |
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q + q + 2 q + 2 q + 3 q + 2 q + q - q - 2 q - 3 q - |
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42 44 46 48 50 52 |
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3 q - 2 q - q + q + q + q</nowiki></pre></td></tr> |
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<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[5, 4]][a, z]</nowiki></pre></td></tr> |
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<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 |
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-18 9 21 14 z 8 z 28 z 21 z z 22 z |
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a + --- + --- + --- - --- - --- - ---- - ---- - --- - ----- - |
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16 14 12 19 17 15 13 18 16 |
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a a a a a a a a a |
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2 2 3 3 3 4 4 4 |
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91 z 70 z 14 z 84 z 70 z 21 z 154 z 133 z |
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----- - ----- + ----- + ----- + ----- + ----- + ------ + ------ - |
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14 12 17 15 13 16 14 12 |
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a a a a a a a a |
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5 5 5 6 6 6 7 7 7 |
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7 z 91 z 84 z 8 z 129 z 121 z z 46 z 45 z |
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---- - ----- - ----- - ---- - ------ - ------ + --- + ----- + ----- + |
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17 15 13 16 14 12 17 15 13 |
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a a a a a a a a a |
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8 8 8 9 9 10 10 11 11 |
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z 56 z 55 z 11 z 11 z 12 z 12 z z z |
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--- + ----- + ----- - ----- - ----- - ------ - ------ + --- + --- + |
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16 14 12 15 13 14 12 15 13 |
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a a a a a a a a a |
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12 12 |
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z z |
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--- + --- |
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14 12 |
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a a</nowiki></pre></td></tr> |
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<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[5, 4]], Vassiliev[3][TorusKnot[5, 4]]}</nowiki></pre></td></tr> |
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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, 50}</nowiki></pre></td></tr> |
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<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[5, 4]][q, t]</nowiki></pre></td></tr> |
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<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 17 4 19 4 21 5 23 5 |
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q + q + q t + q t + q t + q t + q t + q t + |
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19 6 21 6 23 7 25 7 23 8 27 9 |
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q t + q t + q t + q t + q t + q t</nowiki></pre></td></tr> |
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</table> |
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Revision as of 19:28, 26 August 2005
