View source for Planar Diagrams

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[[Image:PDNotation.gif|frame|center|The <code>PD</code> notation]]
In the "Planar Diagrams" (<code>PD</code>) presentation we present every knot or link diagram by labeling its edges (with natural numbers, 1,...,n, and with increasing labels as we go around each component) and by a list crossings presented as symbols <math>X_{ijkl}</math> where <math>i</math>, <math>j</math>, <math>k</math> and <math>l</math> are the labels of the edges around that crossing, starting from the incoming lower edge and proceeding counterclockwise. Thus for example, the <code>PD</code> presentation of the knot above is:

<center><math>
  X_{1928} X_{3,10,4,11} X_{5362} X_{7,1,8,12} X_{9,4,10,5} X_{11,7,12,6}.
</math></center>

(This of course is the Miller Institute knot, the mirror image of the knot [[6_2]])

{{Startup Note}}

<!--$$?PD$$-->
<!--Robot Land, no human edits to "END"-->
{{HelpAndAbout1|n=1|s=PD}}
PD[v1, v2, ...] represents a planar diagram whose vertices are v1, v2, .... PD also acts as a "type caster", so for example, PD[K] where K is is a named knot (or link) returns the PD presentation of that knot.
{{HelpAndAbout2|n=2|s=PD}}
The PD to GaussCode and to MorseLink conversions were written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
{{HelpAndAbout3}}
<!--END-->

<!--$$?X$$-->
<!--Robot Land, no human edits to "END"-->
{{Help1|n=3|s=X}}
X[i,j,k,l] represents a crossing between the edges labeled i, j, k and l starting from the incoming lower strand i and going counterclockwise through j, k and l. The (sometimes ambiguous) orientation of the upper strand is determined by the ordering of {j,l}.
{{Help2}}
<!--END-->

Thus, for example, let us compute the determinant of the above knot:

<!--$$K = PD[X[1,9,2,8], X[3,10,4,11], X[5,3,6,2], X[7,1,8,12], X[9,4,10,5], X[11,7,12,6]];$$-->
<!--Robot Land, no human edits to "END"-->
{{In1|n=4}}
<pre style="color: red; border: 0px; padding: 0em"><nowiki>K = PD[X[1,9,2,8], X[3,10,4,11], X[5,3,6,2], X[7,1,8,12], X[9,4,10,5], X[11,7,12,6]];</nowiki></pre>
{{In2}}
<!--END-->

<!--$$Alexander[K][-1]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut1|n=5}}
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Alexander[K][-1]</nowiki></pre>
{{InOut2|n=5}}<pre style="border: 0px; padding: 0em"><nowiki>-11</nowiki></pre>
{{InOut3}}
<!--END-->

<div id="Some further details">
== Some further details ==
</div>

<!--$$?Xp$$-->
<!--Robot Land, no human edits to "END"-->
{{Help1|n=6|s=Xp}}
Xp[i,j,k,l] represents a positive (right handed) crossing between the edges labeled i, j, k and l starting from the incoming lower strand i and going counter clockwise through j, k and l. The upper strand is therefore oriented from l to j regardless of the ordering of {j,l}. Presently Xp is only lightly supported.
{{Help2}}
<!--END-->

<!--$$?Xm$$-->
<!--Robot Land, no human edits to "END"-->
{{Help1|n=7|s=Xm}}
Xm[i,j,k,l] represents a negative (left handed) crossing between the edges labeled i, j, k and l starting from the incoming lower strand i and going counter clockwise through j, k and l. The upper strand is therefore oriented from j to l regardless of the ordering of {j,l}. Presently Xm is only lightly supported.
{{Help2}}
<!--END-->

<!--$$?P$$-->
<!--Robot Land, no human edits to "END"-->
{{Help1|n=8|s=P}}
P[i,j] represents a bivalent vertex whose adjacent edges are i and j (i.e., a "Point" between the segment i and the segment j). Presently P is only lightly supported.
{{Help2}}
<!--END-->

For example, we could add an extra "point" on the Miller Institute knot, splitting edge 12 into two pieces, labeled 12 and 13:

<!--$$K1 = PD[X[1,9,2,8], X[3,10,4,11], X[5,3,6,2], X[7,1,8,13], X[9,4,10,5], X[11,7,12,6], P[12,13]];$$-->
<!--Robot Land, no human edits to "END"-->
{{In1|n=9}}
<pre style="color: red; border: 0px; padding: 0em"><nowiki>K1 = PD[X[1,9,2,8], X[3,10,4,11], X[5,3,6,2], X[7,1,8,13], X[9,4,10,5], X[11,7,12,6], P[12,13]];</nowiki></pre>
{{In2}}
<!--END-->

At the moment, many of our routines do not know to ignore such "extra points". But some do:

<!--$$Jones[K][q] == Jones[K1][q]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut1|n=10}}
<pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[K][q] == Jones[K1][q]</nowiki></pre>
{{InOut2|n=10}}<pre style="border: 0px; padding: 0em"><nowiki>     1            2      3      4    5
-1 + - + 2 q - 2 q  + 2 q  - 2 q  + q  == 
     q
 
                                2       3        5      6
    2         1        2 q     q       q      2 q      q
  ----- + --------- + ----- + ----- - ----- + ----- + ----- + 
  1 + q   q (1 + q)   1 + q   1 + q   1 + q   1 + q   1 + q
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 8]) / (1 + q) + 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 8]) / (1 + q) + 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10]) / (1 + q) + 
 
     9/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10]) / (1 + q) + 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10]) / (1 + q) + 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10]) / (1 + q) - 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[7, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[7, 13]) / (1 + q) - 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[7, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[7, 13]) / (1 + q) - 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 6]) / (1 + q) - 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 6]) / (1 + q) - 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 10]) / (1 + q) - 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 10]) / (1 + q) + 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4]) / (1 + q) + 
 
     9/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4]) / (1 + q) - 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 8]) / (1 + q) - 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 8]) / (1 + q) + 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[7, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[7, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 13]) / (1 + q) + 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
       5/2
   (2 q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
   (Sqrt[q] {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
     9/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     7/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
     3
   (q  {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[9, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 2] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     2
   (q  {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[3, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 6] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
   (Sqrt[q] {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
       3/2
   (2 q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) + 
 
     5/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 10] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[4, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q) - 
 
     3/2
   (q    {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[1, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 4] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[10, 13] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 8] 
 
      {6, 10, 16, 12, 4, 18, 20, 22, 2, 8, 14}[12, 13]) / (1 + q)</nowiki></pre>
{{InOut3}}
<!--END-->

<!--$$?Loop$$-->
<!--Robot Land, no human edits to "END"-->
{{Help1|n=11|s=Loop}}
Loop[i] represents a crossingsless loop labeled i.
{{Help2}}
<!--END-->

Hence we can verify that the A2 invariant of the unknot is <math>q^{-2}+1+q^2</math>:

<!--$$A2Invariant[Loop[1]][q]$$-->
<!--Robot Land, no human edits to "END"-->
{{InOut1|n=12}}
<pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Loop[1]][q]</nowiki></pre>
{{InOut2|n=12}}<pre style="border: 0px; padding: 0em"><nowiki>     -2    2
1 + q   + q</nowiki></pre>
{{InOut3}}
<!--END-->