The PD
notation
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 $X_{ijkl}$ where $i$, $j$, $k$ and $l$ 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:
$X_{1928}X_{3,10,4,11}X_{5362}X_{7,1,8,12}X_{9,4,10,5}X_{11,7,12,6}.$
(This of course is the Miller Institute knot, the mirror image of the knot 6_2)
(For In[1] see Setup)
In[2]:=

?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 a named knot (or link) returns the PD presentation of that knot.


In[3]:=

PD::about

The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.


In[4]:=

?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}.


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

Some further details
In[7]:=

?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.


In[8]:=

?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.


In[9]:=

?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.


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

Loop[i] represents a crossingsless loop labeled i.


Hence we can verify that the A2 invariant of the unknot is $q^{2}+1+q^{2}$:
In[13]:=

A2Invariant[Loop[1]][q]

Out[13]=

2 2
1 + q + q
