T(31,2) - Revision history

ScottKnotPageRobot at 15:37, 31 August 2005

2005-08-31T15:37:35Z

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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 = 31 |

ScottKnotPageRobot at 16:15, 30 August 2005

2005-08-30T16:15:18Z

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ScottTestRobot at 00:45, 29 August 2005

2005-08-29T00:45:00Z

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ScottTestRobot at 23:42, 28 August 2005

2005-08-28T23:42:31Z

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-<!-- Script generated - do not edit! -->
+<!--  -->
 <!--  -->
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 |- valign=top
 |[[Image:{{PAGENAME}}.jpg]]
-|Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-16,17,-18,19,-20,21,-22,23,-24,25,-26,27,-28,29,-30,31,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,24,-25,26,-27,28,-29,30,-31,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15/goTop.html {{PAGENAME}}'s page] at [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html Knotilus]!
+|{{Torus Knot Site Links|m=31|n=2|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-16,17,-18,19,-20,21,-22,23,-24,25,-26,27,-28,29,-30,31,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,24,-25,26,-27,28,-29,30,-31,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15/goTop.html}}
 {{:{{PAGENAME}} Quick Notes}}
@@ Line 22: / Line 20: @@
 {{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

2005-08-28T01:25:27Z

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DrorsRobot at 02:47, 27 August 2005

2005-08-27T02:47:06Z

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DrorsRobot at 02:19, 27 August 2005

2005-08-27T02:19:41Z

New page

{{Knot Navigation Links|prev=T(29,2).jpg|next=T(8,5).jpg}}

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-16,17,-18,19,-20,21,-22,23,-24,25,-26,27,-28,29,-30,31,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,18,-19,20,-21,22,-23,24,-25,26,-27,28,-29,30,-31,1,-2,3,-4,5,-6,7,-8,9,-10,11,-12,13,-14,15/goTop.html T(31,2)'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/31.2.html T(31,2)'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;" | X17,49,18,48 X49,19,50,18 X19,51,20,50 X51,21,52,20 X21,53,22,52 X53,23,54,22 X23,55,24,54 X55,25,56,24 X25,57,26,56 X57,27,58,26 X27,59,28,58 X59,29,60,28 X29,61,30,60 X61,31,62,30 X31,1,32,62 X1,33,2,32 X33,3,34,2 X3,35,4,34 X35,5,36,4 X5,37,6,36 X37,7,38,6 X7,39,8,38 X39,9,40,8 X9,41,10,40 X41,11,42,10 X11,43,12,42 X43,13,44,12 X13,45,14,44 X45,15,46,14 X15,47,16,46 X47,17,48,16
|-
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em;" | {-16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15}
|-
|'''[[DT (Dowker-Thistlethwaite) Codes|Dowker-Thistlethwaite code]]'''
|style="padding-left: 1em;" | 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
|}

===Polynomial invariants===

{{Polynomial Invariants|name=T(31,2)}}

===[[Finite Type (Vassiliev) Invariants|Vassiliev invariants]]===
{| style="margin-left: 1em;"
|-
|'''V2 and V3'''
|style="padding-left: 1em;" | {0, 1240})
|}

[[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>30 is the signature of T(31,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<center><table border=1>
<tr align=center>
<td width=5.55556%><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=2.77778%>0</td ><td width=2.77778%>1</td ><td width=2.77778%>2</td ><td width=2.77778%>3</td ><td width=2.77778%>4</td ><td width=2.77778%>5</td ><td width=2.77778%>6</td ><td width=2.77778%>7</td ><td width=2.77778%>8</td ><td width=2.77778%>9</td ><td width=2.77778%>10</td ><td width=2.77778%>11</td ><td width=2.77778%>12</td ><td width=2.77778%>13</td ><td width=2.77778%>14</td ><td width=2.77778%>15</td ><td width=2.77778%>16</td ><td width=2.77778%>17</td ><td width=2.77778%>18</td ><td width=2.77778%>19</td ><td width=2.77778%>20</td ><td width=2.77778%>21</td ><td width=2.77778%>22</td ><td width=2.77778%>23</td ><td width=2.77778%>24</td ><td width=2.77778%>25</td ><td width=2.77778%>26</td ><td width=2.77778%>27</td ><td width=2.77778%>28</td ><td width=2.77778%>29</td ><td width=2.77778%>30</td ><td width=2.77778%>31</td ><td width=5.55556%>χ</td></tr>
<tr align=center><td>93</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> </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>91</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> </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>89</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> </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>87</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> </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> </td><td> </td><td>0</td></tr>
<tr align=center><td>85</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> </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>83</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> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>81</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> </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> </td><td> </td><td>0</td></tr>
<tr align=center><td>79</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> </td><td> </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>0</td></tr>
<tr align=center><td>77</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> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr>
<tr align=center><td>75</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> </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>0</td></tr>
<tr align=center><td>73</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> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</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>71</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> </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>69</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 bgcolor=yellow>1</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>67</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> </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>0</td></tr>
<tr align=center><td>65</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> </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>63</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> </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>0</td></tr>
<tr align=center><td>61</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> </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>0</td></tr>
<tr align=center><td>59</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> </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>0</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 bgcolor=yellow>1</td><td bgcolor=yellow>1</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>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 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> </td><td> </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 bgcolor=yellow>1</td><td bgcolor=yellow>1</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>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 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> </td><td> </td><td> </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 bgcolor=yellow>1</td><td bgcolor=yellow>1</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> </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 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> </td><td> </td><td> </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 bgcolor=yellow>1</td><td bgcolor=yellow>1</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> </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 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> </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>41</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> </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> </td><td> </td><td>0</td></tr>
<tr align=center><td>39</td><td> </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> </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>37</td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</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> </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 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> </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>0</td></tr>
<tr align=center><td>33</td><td> </td><td bgcolor=yellow> </td><td bgcolor=yellow>1</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> </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 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> </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>1</td></tr>
<tr align=center><td>29</td><td bgcolor=yellow>1</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> </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[31, 2]]</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>31</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[31, 2]]</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[17, 49, 18, 48], X[49, 19, 50, 18], X[19, 51, 20, 50],

X[51, 21, 52, 20], X[21, 53, 22, 52], X[53, 23, 54, 22],

X[23, 55, 24, 54], X[55, 25, 56, 24], X[25, 57, 26, 56],

X[57, 27, 58, 26], X[27, 59, 28, 58], X[59, 29, 60, 28],

X[29, 61, 30, 60], X[61, 31, 62, 30], X[31, 1, 32, 62],

X[1, 33, 2, 32], X[33, 3, 34, 2], X[3, 35, 4, 34], X[35, 5, 36, 4],

X[5, 37, 6, 36], X[37, 7, 38, 6], X[7, 39, 8, 38], X[39, 9, 40, 8],

X[9, 41, 10, 40], X[41, 11, 42, 10], X[11, 43, 12, 42],

X[43, 13, 44, 12], X[13, 45, 14, 44], X[45, 15, 46, 14],

X[15, 47, 16, 46], X[47, 17, 48, 16]]</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[31, 2]]</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[-16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28,

29, -30, 31, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14,

-15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29,

30, -31, 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[31, 2]]</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[2, {1, 1, 1, 1, 1, 1, 1, 1, 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>
<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[31, 2]][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> -15 -14 -13 -12 -11 -10 -9 -8 -7 -6
-1 + t - t + t - t + t - t + t - t + t - t +

-5 -4 -3 -2 1 2 3 4 5 6 7 8
t - t + t - t + - + t - t + t - t + t - t + t - t +
t

9 10 11 12 13 14 15
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[31, 2]][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 + 120 z + 2380 z + 18564 z + 75582 z + 184756 z + 293930 z +

14 16 18 20 22
319770 z + 245157 z + 134596 z + 53130 z + 14950 z +

24 26 28 30
2925 z + 378 z + 29 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[31, 2]], KnotSignature[TorusKnot[31, 2]]}</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>{31, 30}</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[31, 2]][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> 15 17 18 19 20 21 22 23 24 25 26 27
q + q - q + q - q + q - q + q - q + q - q + q -

28 29 30 31 32 33 34 35 36 37 38
q + q - q + q - q + q - q + q - q + q - q +

39 40 41 42 43 44 45 46
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>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[31, 2]][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[31, 2]][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[31, 2]], Vassiliev[3][TorusKnot[31, 2]]}</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, 1240}</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[31, 2]][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> 29 31 33 2 37 3 37 4 41 5 41 6 45 7
q + q + q t + q t + q t + q t + q t + q t +

45 8 49 9 49 10 53 11 53 12 57 13 57 14
q t + q t + q t + q t + q t + q t + q t +

61 15 61 16 65 17 65 18 69 19 69 20 73 21
q t + q t + q t + q t + q t + q t + q t +

73 22 77 23 77 24 81 25 81 26 85 27 85 28
q t + q t + q t + q t + q t + q t + q t +

89 29 89 30 93 31
q t + q t + q t</nowiki></pre></td></tr>
</table>