T(8,5) - Revision history

ScottKnotPageRobot at 15:37, 31 August 2005

2005-08-31T15:37:28Z

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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 = 8 |

ScottKnotPageRobot at 16:15, 30 August 2005

2005-08-30T16:15:11Z

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ScottTestRobot at 00:44, 29 August 2005

2005-08-29T00:44:53Z

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ScottTestRobot at 23:42, 28 August 2005

2005-08-28T23:42:25Z

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 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/27,28,-32,-3,-6,-9,13,14,15,16,-20,-23,-26,-29,1,2,3,4,-8,-11,-14,-17,21,22,23,24,-28,-31,-2,-5,9,10,11,12,-16,-19,-22,-25,29,30,31,32,-4,-7,-10,-13,17,18,19,20,-24,-27,-30,-1,5,6,7,8,-12,-15,-18,-21,25,26/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=8|n=5|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/27,28,-32,-3,-6,-9,13,14,15,16,-20,-23,-26,-29,1,2,3,4,-8,-11,-14,-17,21,22,23,24,-28,-31,-2,-5,9,10,11,12,-16,-19,-22,-25,29,30,31,32,-4,-7,-10,-13,17,18,19,20,-24,-27,-30,-1,5,6,7,8,-12,-15,-18,-21,25,26/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</nowiki></pre></td></tr>
 </table>

ScottTestRobot at 01:25, 28 August 2005

2005-08-28T01:25:31Z

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DrorsRobot at 02:47, 27 August 2005

2005-08-27T02:47:10Z

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DrorsRobot at 02:19, 27 August 2005

2005-08-27T02:19:45Z

New page

{{Knot Navigation Links|prev=T(31,2).jpg|next=T(16,3).jpg}}

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/27,28,-32,-3,-6,-9,13,14,15,16,-20,-23,-26,-29,1,2,3,4,-8,-11,-14,-17,21,22,23,24,-28,-31,-2,-5,9,10,11,12,-16,-19,-22,-25,29,30,31,32,-4,-7,-10,-13,17,18,19,20,-24,-27,-30,-1,5,6,7,8,-12,-15,-18,-21,25,26/goTop.html T(8,5)'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/8.5.html T(8,5)'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;" | X54,16,55,15 X29,17,30,16 X4,18,5,17 X43,19,44,18 X30,56,31,55 X5,57,6,56 X44,58,45,57 X19,59,20,58 X6,32,7,31 X45,33,46,32 X20,34,21,33 X59,35,60,34 X46,8,47,7 X21,9,22,8 X60,10,61,9 X35,11,36,10 X22,48,23,47 X61,49,62,48 X36,50,37,49 X11,51,12,50 X62,24,63,23 X37,25,38,24 X12,26,13,25 X51,27,52,26 X38,64,39,63 X13,1,14,64 X52,2,53,1 X27,3,28,2 X14,40,15,39 X53,41,54,40 X28,42,29,41 X3,43,4,42
|-
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | {27, 28, -32, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29, 1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -2, -5, 9, 10, 11, 12, -16, -19, -22, -25, 29, 30, 31, 32, -4, -7, -10, -13, 17, 18, 19, 20, -24, -27, -30, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26}
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 52 -42 -56 46 60 -50 -64 54 4 -58 -8 62 12 -2 -16 6 20 -10 -24 14 28 -18 -32 22 36 -26 -40 30 44 -34 -48 38
|}

===Polynomial invariants===

{{Polynomial Invariants|name=T(8,5)}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V2 and V3'''
|style="padding-left: 1em;" | {0, 420})
|}

[[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>20 is the signature of T(8,5). 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 bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td>0</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 bgcolor=red>1</td><td bgcolor=yellow>1</td><td bgcolor=yellow> </td><td> </td><td>0</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 bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td bgcolor=red>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 bgcolor=red>1</td><td bgcolor=yellow>3</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td>-1</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 bgcolor=red>1</td><td bgcolor=red>2</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td bgcolor=red>1</td><td> </td><td> </td><td> </td><td>-1</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=yellow>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </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 bgcolor=red>3</td><td bgcolor=yellow>2</td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </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 bgcolor=red>2</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>2</td><td> </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 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> </td><td> </td><td>0</td></tr>
<tr align=center><td>39</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> </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 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> </td><td> </td><td>1</td></tr>
<tr align=center><td>35</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> </td><td> </td><td>1</td></tr>
<tr align=center><td>33</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>31</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>29</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>27</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[8, 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>32</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[8, 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[54, 16, 55, 15], X[29, 17, 30, 16], X[4, 18, 5, 17],

X[43, 19, 44, 18], X[30, 56, 31, 55], X[5, 57, 6, 56],

X[44, 58, 45, 57], X[19, 59, 20, 58], X[6, 32, 7, 31],

X[45, 33, 46, 32], X[20, 34, 21, 33], X[59, 35, 60, 34],

X[46, 8, 47, 7], X[21, 9, 22, 8], X[60, 10, 61, 9],

X[35, 11, 36, 10], X[22, 48, 23, 47], X[61, 49, 62, 48],

X[36, 50, 37, 49], X[11, 51, 12, 50], X[62, 24, 63, 23],

X[37, 25, 38, 24], X[12, 26, 13, 25], X[51, 27, 52, 26],

X[38, 64, 39, 63], X[13, 1, 14, 64], X[52, 2, 53, 1],

X[27, 3, 28, 2], X[14, 40, 15, 39], X[53, 41, 54, 40],

X[28, 42, 29, 41], X[3, 43, 4, 42]]</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[8, 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[27, 28, -32, -3, -6, -9, 13, 14, 15, 16, -20, -23, -26, -29,

1, 2, 3, 4, -8, -11, -14, -17, 21, 22, 23, 24, -28, -31, -2, -5, 9,

10, 11, 12, -16, -19, -22, -25, 29, 30, 31, 32, -4, -7, -10, -13, 17,

18, 19, 20, -24, -27, -30, -1, 5, 6, 7, 8, -12, -15, -18, -21, 25, 26]</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[8, 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, 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[8, 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> -14 -13 -9 -8 -6 -5 -4 -3 1 3
-1 + t - t + t - t + t - t + t - t + - + t - t +
t

4 5 6 8 9 13 14
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[8, 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 + 63 z + 1092 z + 8169 z + 32055 z + 73876 z + 107849 z +

14 16 18 20 22 24
104771 z + 69785 z + 32320 z + 10395 z + 2277 z + 324 z +

26 28
27 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[8, 5]], KnotSignature[TorusKnot[8, 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>{5, 20}</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[8, 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> 14 16 18 23 25
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[8, 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[8, 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[8, 5]], Vassiliev[3][TorusKnot[8, 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, 420}</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[8, 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> 27 29 31 2 35 3 33 4 35 4 37 5 39 5
q + q + q t + q t + q t + q t + q t + q t +

35 6 37 6 39 7 41 7 37 8 39 8 41 9
q t + q t + q t + q t + q t + 2 q t + q t +

43 9 41 10 45 11 43 12 45 12 49 12
2 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t +

47 13 49 13 47 14 51 14 49 15 51 15
3 q t + 2 q t + 2 q t + q t + q t + 3 q t +

49 16 51 16 53 16 55 16 53 17 55 17
q t + q t + q t + q t + 2 q t + q t +

53 18 57 18 57 19
q t + q t + q t</nowiki></pre></td></tr>
</table>