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{{Knot Navigation Links|prev=T(11,3).jpg|next=T(6,5).jpg}}
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Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-16,17,-18,19,-20,21,-22,23,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15/goTop.html T(23,2)'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
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{{Torus Knot Page|
Visit [http://www.math.toronto.edu/~drorbn/KAtlas/TorusKnots/23.2.html T(23,2)'s page] at the original [http://www.math.toronto.edu/~drorbn/KAtlas/index.html Knot Atlas]!
m = 23 |

n = 2 |
===Knot presentations===
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-16,17,-18,19,-20,21,-22,23,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15/goTop.html |

braid_table = <table cellspacing=0 cellpadding=0 border=0>
{|
<tr><td>[[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]]</td></tr>
|'''[[Planar Diagrams|Planar diagram presentation]]'''
<tr><td>[[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]]</td></tr>
|style="padding-left: 1em;" | X<sub>9,33,10,32</sub> X<sub>33,11,34,10</sub> X<sub>11,35,12,34</sub> X<sub>35,13,36,12</sub> X<sub>13,37,14,36</sub> X<sub>37,15,38,14</sub> X<sub>15,39,16,38</sub> X<sub>39,17,40,16</sub> X<sub>17,41,18,40</sub> X<sub>41,19,42,18</sub> X<sub>19,43,20,42</sub> X<sub>43,21,44,20</sub> X<sub>21,45,22,44</sub> X<sub>45,23,46,22</sub> X<sub>23,1,24,46</sub> X<sub>1,25,2,24</sub> X<sub>25,3,26,2</sub> X<sub>3,27,4,26</sub> X<sub>27,5,28,4</sub> X<sub>5,29,6,28</sub> X<sub>29,7,30,6</sub> X<sub>7,31,8,30</sub> X<sub>31,9,32,8</sub>
</table> |
|-
same_alexander = |
|'''[[Gauss Codes|Gauss code]]'''
same_jones = |
|style="padding-left: 1em;" | {-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15}
khovanov_table = <table border=1>
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 24 26 28 30 32 34 36 38 40 42 44 46 2 4 6 8 10 12 14 16 18 20 22
|}

===Polynomial invariants===

{{Polynomial Invariants|name=T(23,2)}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V<sub>2</sub> and V<sub>3</sub>'''
|style="padding-left: 1em;" | {0, 506})
|}

[[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 <font class=HLYellow>yellow</font> 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>22 is the signature of T(23,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.

<center><table border=1>
<tr align=center>
<tr align=center>
<td width=7.14286%><table cellpadding=0 cellspacing=0>
<td width=7.14286%><table cellpadding=0 cellspacing=0>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
</table></td>
<td width=3.57143%>0</td ><td width=3.57143%>1</td ><td width=3.57143%>2</td ><td width=3.57143%>3</td ><td width=3.57143%>4</td ><td width=3.57143%>5</td ><td width=3.57143%>6</td ><td width=3.57143%>7</td ><td width=3.57143%>8</td ><td width=3.57143%>9</td ><td width=3.57143%>10</td ><td width=3.57143%>11</td ><td width=3.57143%>12</td ><td width=3.57143%>13</td ><td width=3.57143%>14</td ><td width=3.57143%>15</td ><td width=3.57143%>16</td ><td width=3.57143%>17</td ><td width=3.57143%>18</td ><td width=3.57143%>19</td ><td width=3.57143%>20</td ><td width=3.57143%>21</td ><td width=3.57143%>22</td ><td width=3.57143%>23</td ><td width=7.14286%>&chi;</td></tr>
<td width=3.57143%>0</td ><td width=3.57143%>1</td ><td width=3.57143%>2</td ><td width=3.57143%>3</td ><td width=3.57143%>4</td ><td width=3.57143%>5</td ><td width=3.57143%>6</td ><td width=3.57143%>7</td ><td width=3.57143%>8</td ><td width=3.57143%>9</td ><td width=3.57143%>10</td ><td width=3.57143%>11</td ><td width=3.57143%>12</td ><td width=3.57143%>13</td ><td width=3.57143%>14</td ><td width=3.57143%>15</td ><td width=3.57143%>16</td ><td width=3.57143%>17</td ><td width=3.57143%>18</td ><td width=3.57143%>19</td ><td width=3.57143%>20</td ><td width=3.57143%>21</td ><td width=3.57143%>22</td ><td width=3.57143%>23</td ><td width=7.14286%>&chi;</td></tr>
<tr align=center><td>69</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>-1</td></tr>
<tr align=center><td>69</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>-1</td></tr>
<tr align=center><td>67</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>0</td></tr>
<tr align=center><td>67</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>&nbsp;</td><td>0</td></tr>
Line 70: Line 54:
<tr align=center><td>23</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>23</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>21</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
<tr align=center><td>21</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
</table></center>
</table> |
coloured_jones_2 = |

coloured_jones_3 = |
{{Computer Talk Header}}
coloured_jones_4 = |

coloured_jones_5 = |
<table>
coloured_jones_6 = |
<tr valign=top>
coloured_jones_7 = |
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
computer_talk =
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<table>
</tr>
<tr valign=top>
<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[23, 2]]</nowiki></pre></td></tr>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>23</nowiki></pre></td></tr>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</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[23, 2]]</nowiki></pre></td></tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[9, 33, 10, 32], X[33, 11, 34, 10], X[11, 35, 12, 34],
<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[23, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>23</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>TubePlot[TorusKnot[23, 2]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:T(23,2).jpg]]</td></tr><tr valign=top><td><tt><font color=blue>Out[3]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></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>PD[TorusKnot[23, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[9, 33, 10, 32], X[33, 11, 34, 10], X[11, 35, 12, 34],
X[35, 13, 36, 12], X[13, 37, 14, 36], X[37, 15, 38, 14],
X[35, 13, 36, 12], X[13, 37, 14, 36], X[37, 15, 38, 14],
Line 96: Line 85:
X[5, 29, 6, 28], X[29, 7, 30, 6], X[7, 31, 8, 30], X[31, 9, 32, 8]]</nowiki></pre></td></tr>
X[5, 29, 6, 28], X[29, 7, 30, 6], X[7, 31, 8, 30], X[31, 9, 32, 8]]</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[23, 2]]</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>GaussCode[TorusKnot[23, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7,
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7,
8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23,
8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23,
1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]</nowiki></pre></td></tr>
1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15]</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[23, 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>BR[TorusKnot[23, 2]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[2, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1}]</nowiki></pre></td></tr>
1, 1}]</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[23, 2]][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>alex = Alexander[TorusKnot[23, 2]][t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 1
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 1
-1 + t - t + t - t + t - t + t - t + t - t + - +
-1 + t - t + t - t + t - t + t - t + t - t + - +
t
t
Line 113: Line 102:
2 3 4 5 6 7 8 9 10 11
2 3 4 5 6 7 8 9 10 11
t - t + t - t + t - t + t - t + t - t + t</nowiki></pre></td></tr>
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[23, 2]][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>Conway[TorusKnot[23, 2]][z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 8 10 12
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 4 6 8 10 12
1 + 66 z + 715 z + 3003 z + 6435 z + 8008 z + 6188 z +
1 + 66 z + 715 z + 3003 z + 6435 z + 8008 z + 6188 z +
14 16 18 20 22
14 16 18 20 22
3060 z + 969 z + 190 z + 21 z + z</nowiki></pre></td></tr>
3060 z + 969 z + 190 z + 21 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>In[9]:=</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]=&nbsp;&nbsp;&nbsp;</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>Out[9]=&nbsp;&nbsp;</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[23, 2]], KnotSignature[TorusKnot[23, 2]]}</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>{KnotDet[TorusKnot[23, 2]], KnotSignature[TorusKnot[23, 2]]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{23, 22}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{23, 22}</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[23, 2]][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>J=Jones[TorusKnot[23, 2]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 11 13 14 15 16 17 18 19 20 21 22 23
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 11 13 14 15 16 17 18 19 20 21 22 23
q + q - q + q - q + q - q + q - q + q - q + q -
q + q - q + q - q + q - q + q - q + q - q + q -
24 25 26 27 28 29 30 31 32 33 34
24 25 26 27 28 29 30 31 32 33 34
q + q - q + q - q + q - q + q - q + q - q</nowiki></pre></td></tr>
q + q - 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>In[12]:=</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]=&nbsp;&nbsp;&nbsp;</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>Out[12]=&nbsp;&nbsp;</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[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[TorusKnot[23, 2]][q]</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[23, 2]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</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>Out[12]=&nbsp;&nbsp;&nbsp;</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>Kauffman[TorusKnot[23, 2]][a, z]</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[23, 2]][a, z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</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>Out[13]=&nbsp;&nbsp;&nbsp;</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[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][TorusKnot[23, 2]], Vassiliev[3][TorusKnot[23, 2]]}</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[23, 2]], Vassiliev[3][TorusKnot[23, 2]]}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{66, 506}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 506}</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[TorusKnot[23, 2]][q, t]</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[23, 2]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 21 23 25 2 29 3 29 4 33 5 33 6 37 7
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 21 23 25 2 29 3 29 4 33 5 33 6 37 7
q + q + q t + q t + q t + q t + q t + q t +
q + q + q t + q t + q t + q t + q t + q t +
Line 150: Line 138:
65 22 69 23
65 22 69 23
q t + q t</nowiki></pre></td></tr>
q t + q t</nowiki></pre></td></tr>
</table>
</table> }}

Latest revision as of 10:37, 31 August 2005

T(11,3).jpg

T(11,3)

T(6,5).jpg

T(6,5)

T(23,2).jpg See other torus knots

Visit T(23,2) at Knotilus!

Edit T(23,2) Quick Notes


Edit T(23,2) Further Notes and Views


Knot presentations

Planar diagram presentation X9,33,10,32 X33,11,34,10 X11,35,12,34 X35,13,36,12 X13,37,14,36 X37,15,38,14 X15,39,16,38 X39,17,40,16 X17,41,18,40 X41,19,42,18 X19,43,20,42 X43,21,44,20 X21,45,22,44 X45,23,46,22 X23,1,24,46 X1,25,2,24 X25,3,26,2 X3,27,4,26 X27,5,28,4 X5,29,6,28 X29,7,30,6 X7,31,8,30 X31,9,32,8
Gauss code -16, 17, -18, 19, -20, 21, -22, 23, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15
Dowker-Thistlethwaite code 24 26 28 30 32 34 36 38 40 42 44 46 2 4 6 8 10 12 14 16 18 20 22
Braid presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 23, 22 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant Data:T(23,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(23,2)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ): {}

Vassiliev invariants

V2 and V3: (66, 506)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
Data:T(23,2)/V 2,1 Data:T(23,2)/V 3,1 Data:T(23,2)/V 4,1 Data:T(23,2)/V 4,2 Data:T(23,2)/V 4,3 Data:T(23,2)/V 5,1 Data:T(23,2)/V 5,2 Data:T(23,2)/V 5,3 Data:T(23,2)/V 5,4 Data:T(23,2)/V 6,1 Data:T(23,2)/V 6,2 Data:T(23,2)/V 6,3 Data:T(23,2)/V 6,4 Data:T(23,2)/V 6,5 Data:T(23,2)/V 6,6 Data:T(23,2)/V 6,7 Data:T(23,2)/V 6,8 Data:T(23,2)/V 6,9

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 22 is the signature of T(23,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011121314151617181920212223χ
69                       1-1
67                        0
65                     11 0
63                        0
61                   11   0
59                        0
57                 11     0
55                        0
53               11       0
51                        0
49             11         0
47                        0
45           11           0
43                        0
41         11             0
39                        0
37       11               0
35                        0
33     11                 0
31                        0
29   11                   0
27                        0
25  1                     1
231                       1
211                       1
Integral Khovanov Homology

(db, data source)

  

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Torus Knot Page master template (intermediate).

See/edit the Torus Knot_Splice_Base (expert).

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T(11,3).jpg

T(11,3)

T(6,5).jpg

T(6,5)