ScottKnotPageRobot at 15:38, 31 August 2005

2005-08-31T15:38:03Z

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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 = 10 |

ScottKnotPageRobot at 16:15, 30 August 2005

2005-08-30T16:15:49Z

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ScottTestRobot at 00:45, 29 August 2005

2005-08-29T00:45:45Z

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ScottTestRobot at 23:43, 28 August 2005

2005-08-28T23:43:11Z

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 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-12,-13,15,16,-18,-19,1,2,-4,-5,7,8,-10,-11,13,14,-16,-17,19,20,-2,-3,5,6,-8,-9,11,12,-14,-15,17,18,-20,-1,3,4,-6,-7,9,10/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=10|n=3|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-12,-13,15,16,-18,-19,1,2,-4,-5,7,8,-10,-11,13,14,-16,-17,19,20,-2,-3,5,6,-8,-9,11,12,-14,-15,17,18,-20,-1,3,4,-6,-7,9,10/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</nowiki></pre></td></tr>
 </table>

ScottTestRobot at 01:24, 28 August 2005

2005-08-28T01:24:39Z

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DrorsRobot at 02:46, 27 August 2005

2005-08-27T02:46:24Z

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DrorsRobot at 02:19, 27 August 2005

2005-08-27T02:19:00Z

New page

{{Knot Navigation Links|prev=T(19,2).jpg|next=T(7,4).jpg}}

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-12,-13,15,16,-18,-19,1,2,-4,-5,7,8,-10,-11,13,14,-16,-17,19,20,-2,-3,5,6,-8,-9,11,12,-14,-15,17,18,-20,-1,3,4,-6,-7,9,10/goTop.html T(10,3)'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/10.3.html T(10,3)'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;" | X34,8,35,7 X21,9,22,8 X22,36,23,35 X9,37,10,36 X10,24,11,23 X37,25,38,24 X38,12,39,11 X25,13,26,12 X26,40,27,39 X13,1,14,40 X14,28,15,27 X1,29,2,28 X2,16,3,15 X29,17,30,16 X30,4,31,3 X17,5,18,4 X18,32,19,31 X5,33,6,32 X6,20,7,19 X33,21,34,20
|-
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | {-12, -13, 15, 16, -18, -19, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -17, 19, 20, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -1, 3, 4, -6, -7, 9, 10}
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 28 -30 32 -34 36 -38 40 -2 4 -6 8 -10 12 -14 16 -18 20 -22 24 -26
|}

===Polynomial invariants===

{{Polynomial Invariants|name=T(10,3)}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V2 and V3'''
|style="padding-left: 1em;" | {0, 165})
|}

[[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>14 is the signature of T(10,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<center><table border=1>
<tr align=center>
<td width=11.1111%><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=5.55556%>0</td ><td width=5.55556%>1</td ><td width=5.55556%>2</td ><td width=5.55556%>3</td ><td width=5.55556%>4</td ><td width=5.55556%>5</td ><td width=5.55556%>6</td ><td width=5.55556%>7</td ><td width=5.55556%>8</td ><td width=5.55556%>9</td ><td width=5.55556%>10</td ><td width=5.55556%>11</td ><td width=5.55556%>12</td ><td width=5.55556%>13</td ><td width=11.1111%>χ</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> </td><td> </td><td bgcolor=yellow>1</td><td>-1</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> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td>-1</td></tr>
<tr align=center><td>37</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>1</td><td bgcolor=yellow>1</td><td> </td><td>0</td></tr>
<tr align=center><td>35</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> </td><td bgcolor=yellow> </td><td bgcolor=red>1</td><td> </td><td>0</td></tr>
<tr align=center><td>33</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 bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>31</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 bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>29</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</td><td> </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>27</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> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>25</td><td> </td><td> </td><td> </td><td bgcolor=red>1</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>0</td></tr>
<tr align=center><td>23</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> </td><td> </td><td> </td><td> </td><td>1</td></tr>
<tr align=center><td>21</td><td> </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>19</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>17</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> </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[10, 3]]</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>20</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[10, 3]]</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[34, 8, 35, 7], X[21, 9, 22, 8], X[22, 36, 23, 35],

X[9, 37, 10, 36], X[10, 24, 11, 23], X[37, 25, 38, 24],

X[38, 12, 39, 11], X[25, 13, 26, 12], X[26, 40, 27, 39],

X[13, 1, 14, 40], X[14, 28, 15, 27], X[1, 29, 2, 28],

X[2, 16, 3, 15], X[29, 17, 30, 16], X[30, 4, 31, 3], X[17, 5, 18, 4],

X[18, 32, 19, 31], X[5, 33, 6, 32], X[6, 20, 7, 19],

X[33, 21, 34, 20]]</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[10, 3]]</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[-12, -13, 15, 16, -18, -19, 1, 2, -4, -5, 7, 8, -10, -11, 13,

14, -16, -17, 19, 20, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18,

-20, -1, 3, 4, -6, -7, 9, 10]</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[10, 3]]</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[3, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 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[10, 3]][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> -9 -8 -6 -5 -3 -2 2 3 5 6 8 9
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[10, 3]][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 14
1 + 33 z + 264 z + 792 z + 1166 z + 946 z + 443 z + 119 z +

16 18
17 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[10, 3]], KnotSignature[TorusKnot[10, 3]]}</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, 14}</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[10, 3]][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> 9 11 20
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[10, 3]][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[10, 3]][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[10, 3]], Vassiliev[3][TorusKnot[10, 3]]}</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, 165}</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[10, 3]][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> 17 19 21 2 25 3 23 4 25 4 27 5 29 5
q + q + q t + q t + q t + q t + q t + q t +

27 6 31 7 29 8 31 8 33 9 35 9 33 10
q t + q t + q t + q t + q t + q t + q t +

37 11 35 12 37 12 39 13 41 13
q t + q t + q t + q t + q t</nowiki></pre></td></tr>
</table>

T(10,3) - Revision history

ScottKnotPageRobot at 15:38, 31 August 2005

ScottKnotPageRobot at 16:15, 30 August 2005

ScottTestRobot at 00:45, 29 August 2005

ScottTestRobot at 23:43, 28 August 2005

ScottTestRobot at 01:24, 28 August 2005

DrorsRobot at 02:46, 27 August 2005

DrorsRobot at 02:19, 27 August 2005