ScottKnotPageRobot at 15:37, 31 August 2005

2005-08-31T15:37:15Z

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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 16:14, 30 August 2005

2005-08-30T16:14:58Z

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ScottTestRobot at 00:44, 29 August 2005

2005-08-29T00:44:36Z

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ScottTestRobot at 23:42, 28 August 2005

2005-08-28T23:42:06Z

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 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/8,9,-12,-14,-16,19,20,21,-24,-26,-1,4,5,6,-9,-11,-13,16,17,18,-21,-23,-25,1,2,3,-6,-8,-10,13,14,15,-18,-20,-22,25,26,27,-3,-5,-7,10,11,12,-15,-17,-19,22,23,24,-27,-2,-4,7/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=4|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/8,9,-12,-14,-16,19,20,21,-24,-26,-1,4,5,6,-9,-11,-13,16,17,18,-21,-23,-25,1,2,3,-6,-8,-10,13,14,15,-18,-20,-22,25,26,27,-3,-5,-7,10,11,12,-15,-17,-19,22,23,24,-27,-2,-4,7/goTop.html}}
 {{:{{PAGENAME}} Quick Notes}}
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 {{Knot Presentations}}
 {{Polynomial Invariants}}
 {{Vassiliev Invariants}}
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   q   t</nowiki></pre></td></tr>
 </table>

ScottTestRobot at 01:25, 28 August 2005

2005-08-28T01:25:07Z

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DrorsRobot at 02:46, 27 August 2005

2005-08-27T02:46:48Z

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DrorsRobot at 02:19, 27 August 2005

2005-08-27T02:19:23Z

New page

{{Knot Navigation Links|prev=T(13,3).jpg|next=T(27,2).jpg}}

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/8,9,-12,-14,-16,19,20,21,-24,-26,-1,4,5,6,-9,-11,-13,16,17,18,-21,-23,-25,1,2,3,-6,-8,-10,13,14,15,-18,-20,-22,25,26,27,-3,-5,-7,10,11,12,-15,-17,-19,22,23,24,-27,-2,-4,7/goTop.html T(9,4)'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/9.4.html T(9,4)'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;" | X11,25,12,24 X52,26,53,25 X39,27,40,26 X53,13,54,12 X40,14,41,13 X27,15,28,14 X41,1,42,54 X28,2,29,1 X15,3,16,2 X29,43,30,42 X16,44,17,43 X3,45,4,44 X17,31,18,30 X4,32,5,31 X45,33,46,32 X5,19,6,18 X46,20,47,19 X33,21,34,20 X47,7,48,6 X34,8,35,7 X21,9,22,8 X35,49,36,48 X22,50,23,49 X9,51,10,50 X23,37,24,36 X10,38,11,37 X51,39,52,38
|-
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | {8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -1, 4, 5, 6, -9, -11, -13, 16, 17, 18, -21, -23, -25, 1, 2, 3, -6, -8, -10, 13, 14, 15, -18, -20, -22, 25, 26, 27, -3, -5, -7, 10, 11, 12, -15, -17, -19, 22, 23, 24, -27, -2, -4, 7}
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 28 -44 -18 34 -50 -24 40 -2 -30 46 -8 -36 52 -14 -42 4 -20 -48 10 -26 -54 16 -32 -6 22 -38 -12
|}

===Polynomial invariants===

{{Polynomial Invariants|name=T(9,4)}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V2 and V3'''
|style="padding-left: 1em;" | {0, 300})
|}

[[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 yellow 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>16 is the signature of T(9,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<center><table border=1>
<tr align=center>
<td width=9.09091%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td> </td><td>r</td></tr>
<tr><td> </td><td> \ </td><td> </td></tr>
<tr><td>j</td><td> </td><td>\</td></tr>
</table></td>
<td width=4.54545%>0</td ><td width=4.54545%>1</td ><td width=4.54545%>2</td ><td width=4.54545%>3</td ><td width=4.54545%>4</td ><td width=4.54545%>5</td ><td width=4.54545%>6</td ><td width=4.54545%>7</td ><td width=4.54545%>8</td ><td width=4.54545%>9</td ><td width=4.54545%>10</td ><td width=4.54545%>11</td ><td width=4.54545%>12</td ><td width=4.54545%>13</td ><td width=4.54545%>14</td ><td width=4.54545%>15</td ><td width=4.54545%>16</td ><td width=4.54545%>17</td ><td width=9.09091%>χ</td></tr>
<tr align=center><td>51</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> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=yellow>1</td><td>0</td></tr>
<tr align=center><td>49</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> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>0</td></tr>
<tr align=center><td>47</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> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td>-1</td></tr>
<tr align=center><td>45</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 bgcolor=red>1</td><td bgcolor=red>2</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td>-1</td></tr>
<tr align=center><td>43</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </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> </td><td> </td><td>-1</td></tr>
<tr align=center><td>41</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>2</td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>39</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>2</td><td bgcolor=red>1</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>37</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>35</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td> </td><td bgcolor=red>1</td><td bgcolor=red>2</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>33</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</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> </td><td> </td><td>0</td></tr>
<tr align=center><td>31</td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td bgcolor=red>1</td><td> </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> </td><td> </td><td>1</td></tr>
<tr align=center><td>29</td><td> </td><td> </td><td> </td><td> </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> </td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>27</td><td> </td><td> </td><td bgcolor=red>1</td><td> </td><td> </td><td bgcolor=yellow> </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>1</td></tr>
<tr align=center><td>25</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td bgcolor=yellow> </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>23</td><td bgcolor=red>1</td><td> </td><td> </td><td bgcolor=yellow> </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> </td><td>1</td></tr>
</table></center>

{{Computer Talk Header}}

<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=    </pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em"><< 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[9, 4]]</nowiki></pre></td></tr>
<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>27</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[9, 4]]</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[11, 25, 12, 24], X[52, 26, 53, 25], X[39, 27, 40, 26],

X[53, 13, 54, 12], X[40, 14, 41, 13], X[27, 15, 28, 14],

X[41, 1, 42, 54], X[28, 2, 29, 1], X[15, 3, 16, 2],

X[29, 43, 30, 42], X[16, 44, 17, 43], X[3, 45, 4, 44],

X[17, 31, 18, 30], X[4, 32, 5, 31], X[45, 33, 46, 32],

X[5, 19, 6, 18], X[46, 20, 47, 19], X[33, 21, 34, 20],

X[47, 7, 48, 6], X[34, 8, 35, 7], X[21, 9, 22, 8], X[35, 49, 36, 48],

X[22, 50, 23, 49], X[9, 51, 10, 50], X[23, 37, 24, 36],

X[10, 38, 11, 37], X[51, 39, 52, 38]]</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[9, 4]]</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[8, 9, -12, -14, -16, 19, 20, 21, -24, -26, -1, 4, 5, 6, -9,

-11, -13, 16, 17, 18, -21, -23, -25, 1, 2, 3, -6, -8, -10, 13, 14,

15, -18, -20, -22, 25, 26, 27, -3, -5, -7, 10, 11, 12, -15, -17, -19,

22, 23, 24, -27, -2, -4, 7]</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[9, 4]]</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[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3,

1, 2, 3, 1, 2, 3}]</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, 4]][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> -12 -11 -8 -7 -4 -2 2 4 7 8 11 12
1 + t - t + t - t + t - t - 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[9, 4]][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
1 + 50 z + 665 z + 3675 z + 10318 z + 16720 z + 16834 z +

14 16 18 20 22 24
10963 z + 4693 z + 1311 z + 230 z + 23 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]=   </nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{}</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[9, 4]], KnotSignature[TorusKnot[9, 4]]}</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>{9, 16}</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[9, 4]][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> 12 14 16 17 18 19 20 21 23
q + q + q - q + q - q + 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>{}</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[9, 4]][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>NotAvailable</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[9, 4]][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>NotAvailable</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[9, 4]], Vassiliev[3][TorusKnot[9, 4]]}</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, 300}</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, 4]][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> 23 25 27 2 31 3 29 4 31 4 33 5 35 5
q + q + q t + q t + q t + q t + q t + q t +

31 6 33 6 35 7 37 7 35 8 39 9 37 10
q t + q t + q t + q t + 2 q t + 2 q t + q t +

39 10 41 11 43 11 39 12 41 12 45 12
q t + 2 q t + q t + q t + 2 q t + q t +

43 13 45 13 43 14 47 15 47 16 51 16
q t + 2 q t + q t + 2 q t + q t + q t +

51 17
q t</nowiki></pre></td></tr>
</table>

T(9,4) - Revision history

ScottKnotPageRobot at 15:37, 31 August 2005

ScottKnotPageRobot at 16:14, 30 August 2005

ScottTestRobot at 00:44, 29 August 2005

ScottTestRobot at 23:42, 28 August 2005

ScottTestRobot at 01:25, 28 August 2005

DrorsRobot at 02:46, 27 August 2005

DrorsRobot at 02:19, 27 August 2005