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For KnotTheory`, we present every knot or link diagram (every
Planar Diagram or just PD) 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
where
,
,
and
are the labels of the edges around that
crossing, starting from the incoming lower edge
and proceeding counterclockwise. Thus for example, the PD presentation
of the knot in Figure 2 is:
In[2]:= ?PD
In[3]:= PD::about
In[4]:= ?X
|
Thus, for example, let us compute the determinant of the above knot:
In[5]:= | 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] ]; |
In[6]:= | Alexander[K][-1] |
Out[6]= | -11 |
In[7]:= ?Xp
In[8]:= ?Xm
In[9]:= ?P
|
For example, we could add an extra ``point'' on the Miller Institute knot, splitting edge 12 into two pieces, labeled 12 and 13:
In[10]:= | 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] ]; |
At the moment, many of our routines do not know to ignore such ``extra points''. But some do:
In[11]:= | Jones[K][q] == Jones[K1][q] |
Out[11]= | True |
In[12]:= ?Loop
|
Hence we can verify that the A2 invariant of the unknot is
:
In[13]:= | A2Invariant[Loop[1]][q] |
Out[13]= | -2 2 1 + q + q |