Planar Diagrams: Difference between revisions
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In the "Planar Diagrams" (PD
) 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 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 above is:
(This of course is the Miller Institute knot, the mirror image of the knot 6_2)
(For In[1] see Setup)
|
|
|
Thus, for example, let us compute the determinant of the above knot:
In[4]:=
|
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[5]:=
|
Alexander[K][-1]
|
Out[5]=
|
-11
|
Some further details
|
|
|
For example, we could add an extra "point" on the Miller Institute knot, splitting edge 12 into two pieces, labeled 12 and 13:
In[9]:=
|
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[10]:=
|
Jones[K][q] == Jones[K1][q]
|
Out[10]=
|
True
|
|
Hence we can verify that the A2 invariant of the unknot is :
In[12]:=
|
A2Invariant[Loop[1]][q]
|
Out[12]=
|
-2 2
1 + q + q
|