ScottKnotPageRobot at 15:37, 31 August 2005

2005-08-31T15:37:19Z

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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 = 7 |

ScottKnotPageRobot at 16:15, 30 August 2005

2005-08-30T16:15:03Z

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ScottTestRobot at 00:44, 29 August 2005

2005-08-29T00:44:40Z

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ScottTestRobot at 23:42, 28 August 2005

2005-08-28T23:42:13Z

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 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-25,1,2,3,4,-8,-11,-14,-17,21,22,23,24,-28,-3,-6,-9,13,14,15,16,-20,-23,-26,-1,5,6,7,8,-12,-15,-18,-21,25,26,27,28,-4,-7,-10,-13,17,18,19,20,-24,-27,-2,-5,9,10,11,12,-16,-19,-22/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=7|n=5|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-25,1,2,3,4,-8,-11,-14,-17,21,22,23,24,-28,-3,-6,-9,13,14,15,16,-20,-23,-26,-1,5,6,7,8,-12,-15,-18,-21,25,26,27,28,-4,-7,-10,-13,17,18,19,20,-24,-27,-2,-5,9,10,11,12,-16,-19,-22/goTop.html}}
 {{:{{PAGENAME}} Quick Notes}}
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 {{Knot Presentations}}
 {{Polynomial Invariants}}
 {{Vassiliev Invariants}}
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   q   t   + q   t</nowiki></pre></td></tr>
 </table>

ScottTestRobot at 01:25, 28 August 2005

2005-08-28T01:25:14Z

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DrorsRobot at 02:46, 27 August 2005

2005-08-27T02:46:55Z

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DrorsRobot at 02:19, 27 August 2005

2005-08-27T02:19:31Z

New page

{{Knot Navigation Links|prev=T(27,2).jpg|next=T(14,3).jpg}}

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-25,1,2,3,4,-8,-11,-14,-17,21,22,23,24,-28,-3,-6,-9,13,14,15,16,-20,-23,-26,-1,5,6,7,8,-12,-15,-18,-21,25,26,27,28,-4,-7,-10,-13,17,18,19,20,-24,-27,-2,-5,9,10,11,12,-16,-19,-22/goTop.html T(7,5)'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/7.5.html T(7,5)'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;" | X25,3,26,2 X48,4,49,3 X15,5,16,4 X38,6,39,5 X49,27,50,26 X16,28,17,27 X39,29,40,28 X6,30,7,29 X17,51,18,50 X40,52,41,51 X7,53,8,52 X30,54,31,53 X41,19,42,18 X8,20,9,19 X31,21,32,20 X54,22,55,21 X9,43,10,42 X32,44,33,43 X55,45,56,44 X22,46,23,45 X33,11,34,10 X56,12,1,11 X23,13,24,12 X46,14,47,13 X1,35,2,34 X24,36,25,35 X47,37,48,36 X14,38,15,37
|-
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | {-25, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26, 27, 28, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -2, -5, 9, 10, 11, 12, -16, -19, -22}
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 34 -48 -38 52 42 -56 -46 4 50 -8 -54 12 2 -16 -6 20 10 -24 -14 28 18 -32 -22 36 26 -40 -30 44
|}

===Polynomial invariants===

{{Polynomial Invariants|name=T(7,5)}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V2 and V3'''
|style="padding-left: 1em;" | {0, 280})
|}

[[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(7,5). 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 bgcolor=red>1</td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</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> </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> </td><td> </td><td bgcolor=yellow>2</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>3</td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </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 bgcolor=red>2</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </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 bgcolor=red>1</td><td> </td><td bgcolor=red>1</td><td bgcolor=yellow>2</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>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>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[7, 5]]</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[7, 5]]</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[25, 3, 26, 2], X[48, 4, 49, 3], X[15, 5, 16, 4], X[38, 6, 39, 5],

X[49, 27, 50, 26], X[16, 28, 17, 27], X[39, 29, 40, 28],

X[6, 30, 7, 29], X[17, 51, 18, 50], X[40, 52, 41, 51],

X[7, 53, 8, 52], X[30, 54, 31, 53], X[41, 19, 42, 18],

X[8, 20, 9, 19], X[31, 21, 32, 20], X[54, 22, 55, 21],

X[9, 43, 10, 42], X[32, 44, 33, 43], X[55, 45, 56, 44],

X[22, 46, 23, 45], X[33, 11, 34, 10], X[56, 12, 1, 11],

X[23, 13, 24, 12], X[46, 14, 47, 13], X[1, 35, 2, 34],

X[24, 36, 25, 35], X[47, 37, 48, 36], X[14, 38, 15, 37]]</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, 5]]</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[-25, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -3,

-6, -9, 13, 14, 15, 16, -20, -23, -26, -1, 5, 6, 7, 8, -12, -15, -18,

-21, 25, 26, 27, 28, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -2,

-5, 9, 10, 11, 12, -16, -19, -22]</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, 5]]</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[5, {1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1,

2, 3, 4, 1, 2, 3, 4}]</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, 5]][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 -7 -6 -5 -4 -2 1 2 4 5
1 + t - t + t - t + t - t + t - - - t + t - t + t -
t

6 7 11 12
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, 5]][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 + 48 z + 628 z + 3498 z + 10032 z + 16511 z + 16757 z +

14 16 18 20 22 24
10949 z + 4692 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[7, 5]], KnotSignature[TorusKnot[7, 5]]}</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>{1, 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[7, 5]][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 20 22
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[7, 5]][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[7, 5]][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[7, 5]], Vassiliev[3][TorusKnot[7, 5]]}</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, 280}</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, 5]][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 33 8 35 8 37 9
q t + q t + q t + q t + q t + 2 q t + q t +

39 9 37 10 41 11 39 12 41 12 45 12
2 q t + 2 q t + 3 q t + q t + 2 q t + q t +

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

51 16 51 17
q t + q t</nowiki></pre></td></tr>
</table>

T(7,5) - 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