ScottKnotPageRobot at 15:37, 31 August 2005

2005-08-31T15:37:25Z

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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 = 14 |

ScottKnotPageRobot at 16:15, 30 August 2005

2005-08-30T16:15:10Z

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ScottTestRobot at 00:44, 29 August 2005

2005-08-29T00:44:51Z

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ScottTestRobot at 23:42, 28 August 2005

2005-08-28T23:42:22Z

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 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/26,-28,-1,3,4,-6,-7,9,10,-12,-13,15,16,-18,-19,21,22,-24,-25,27,28,-2,-3,5,6,-8,-9,11,12,-14,-15,17,18,-20,-21,23,24,-26,-27,1,2,-4,-5,7,8,-10,-11,13,14,-16,-17,19,20,-22,-23,25/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=14|n=3|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/26,-28,-1,3,4,-6,-7,9,10,-12,-13,15,16,-18,-19,21,22,-24,-25,27,28,-2,-3,5,6,-8,-9,11,12,-14,-15,17,18,-20,-21,23,24,-26,-27,1,2,-4,-5,7,8,-10,-11,13,14,-16,-17,19,20,-22,-23,25/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 01:25, 28 August 2005

2005-08-28T01:25:17Z

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DrorsRobot at 02:46, 27 August 2005

2005-08-27T02:46:58Z

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DrorsRobot at 02:19, 27 August 2005

2005-08-27T02:19:34Z

New page

{{Knot Navigation Links|prev=T(7,5).jpg|next=T(29,2).jpg}}

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/26,-28,-1,3,4,-6,-7,9,10,-12,-13,15,16,-18,-19,21,22,-24,-25,27,28,-2,-3,5,6,-8,-9,11,12,-14,-15,17,18,-20,-21,23,24,-26,-27,1,2,-4,-5,7,8,-10,-11,13,14,-16,-17,19,20,-22,-23,25/goTop.html T(14,3)'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/14.3.html T(14,3)'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;" | X3,41,4,40 X22,42,23,41 X23,5,24,4 X42,6,43,5 X43,25,44,24 X6,26,7,25 X7,45,8,44 X26,46,27,45 X27,9,28,8 X46,10,47,9 X47,29,48,28 X10,30,11,29 X11,49,12,48 X30,50,31,49 X31,13,32,12 X50,14,51,13 X51,33,52,32 X14,34,15,33 X15,53,16,52 X34,54,35,53 X35,17,36,16 X54,18,55,17 X55,37,56,36 X18,38,19,37 X19,1,20,56 X38,2,39,1 X39,21,40,20 X2,22,3,21
|-
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | {26, -28, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19, 21, 22, -24, -25, 27, 28, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17, 18, -20, -21, 23, 24, -26, -27, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -17, 19, 20, -22, -23, 25}
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 38 -40 42 -44 46 -48 50 -52 54 -56 2 -4 6 -8 10 -12 14 -16 18 -20 22 -24 26 -28 30 -32 34 -36
|}

===Polynomial invariants===

{{Polynomial Invariants|name=T(14,3)}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V2 and V3'''
|style="padding-left: 1em;" | {0, 455})
|}

[[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>18 is the signature of T(14,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<center><table border=1>
<tr align=center>
<td width=8.33333%><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=4.16667%>0</td ><td width=4.16667%>1</td ><td width=4.16667%>2</td ><td width=4.16667%>3</td ><td width=4.16667%>4</td ><td width=4.16667%>5</td ><td width=4.16667%>6</td ><td width=4.16667%>7</td ><td width=4.16667%>8</td ><td width=4.16667%>9</td ><td width=4.16667%>10</td ><td width=4.16667%>11</td ><td width=4.16667%>12</td ><td width=4.16667%>13</td ><td width=4.16667%>14</td ><td width=4.16667%>15</td ><td width=4.16667%>16</td ><td width=4.16667%>17</td ><td width=4.16667%>18</td ><td width=4.16667%>19</td ><td width=8.33333%>χ</td></tr>
<tr align=center><td>57</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> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>-1</td></tr>
<tr align=center><td>55</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> </td><td bgcolor=red>1</td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td>-1</td></tr>
<tr align=center><td>53</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> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td>0</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 bgcolor=red>1</td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td> </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 bgcolor=red>1</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </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 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>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 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>0</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 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>41</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>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>39</td><td> </td><td> </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>37</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=red>1</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>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 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>0</td></tr>
<tr align=center><td>33</td><td> </td><td> </td><td> </td><td bgcolor=red>1</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>0</td></tr>
<tr align=center><td>31</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> </td><td> </td><td>1</td></tr>
<tr align=center><td>29</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> </td><td> </td><td>1</td></tr>
<tr align=center><td>27</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> </td><td> </td><td>1</td></tr>
<tr align=center><td>25</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> </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[14, 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>28</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[14, 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[3, 41, 4, 40], X[22, 42, 23, 41], X[23, 5, 24, 4],

X[42, 6, 43, 5], X[43, 25, 44, 24], X[6, 26, 7, 25], X[7, 45, 8, 44],

X[26, 46, 27, 45], X[27, 9, 28, 8], X[46, 10, 47, 9],

X[47, 29, 48, 28], X[10, 30, 11, 29], X[11, 49, 12, 48],

X[30, 50, 31, 49], X[31, 13, 32, 12], X[50, 14, 51, 13],

X[51, 33, 52, 32], X[14, 34, 15, 33], X[15, 53, 16, 52],

X[34, 54, 35, 53], X[35, 17, 36, 16], X[54, 18, 55, 17],

X[55, 37, 56, 36], X[18, 38, 19, 37], X[19, 1, 20, 56],

X[38, 2, 39, 1], X[39, 21, 40, 20], X[2, 22, 3, 21]]</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[14, 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[26, -28, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -18, -19,

21, 22, -24, -25, 27, 28, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 17,

18, -20, -21, 23, 24, -26, -27, 1, 2, -4, -5, 7, 8, -10, -11, 13, 14,

-16, -17, 19, 20, -22, -23, 25]</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[14, 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, 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[14, 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> -13 -12 -10 -9 -7 -6 -4 -3 1 3
-1 + t - t + t - t + t - t + t - t + - + t - t +
t

4 6 7 9 10 12 13
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[14, 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
1 + 65 z + 1040 z + 6448 z + 20540 z + 38532 z + 45942 z +

14 16 18 20 22 24 26
36366 z + 19532 z + 7144 z + 1751 z + 275 z + 25 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[14, 3]], KnotSignature[TorusKnot[14, 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, 18}</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[14, 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> 13 15 28
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[14, 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[14, 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[14, 3]], Vassiliev[3][TorusKnot[14, 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, 455}</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[14, 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> 25 27 29 2 33 3 31 4 33 4 35 5 37 5
q + q + q t + q t + q t + q t + q t + q t +

35 6 39 7 37 8 39 8 41 9 43 9 41 10
q t + q t + q t + q t + q t + q t + q t +

45 11 43 12 45 12 47 13 49 13 47 14 51 15
q t + q t + q t + q t + q t + q t + q t +

49 16 51 16 53 17 55 17 53 18 57 19
q t + q t + q t + q t + q t + q t</nowiki></pre></td></tr>
</table>

T(14,3) - Revision history

ScottKnotPageRobot at 15:37, 31 August 2005

ScottKnotPageRobot at 16:15, 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