Planar Diagrams

2008-07-13T20:34:20Z

DelboCacda: racdeldron

drondelolo
{{Manual TOC Sidebar}}

[[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}}



{{HelpAndAbout|
n = 1 |
n1 = 2 |
in = <nowiki>PD</nowiki> |
out= <nowiki>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.</nowiki> |
about= <nowiki>The PD to GaussCode and to MorseLink conversions were written by Siddarth Sankaran at the University of Toronto in the summer of 2005.</nowiki>}}




{{HelpLine|
n = 3 |
in = <nowiki>X</nowiki> |
out= <nowiki>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}.</nowiki>}}


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



{{In|
n = 4 |
in = <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>}}




{{InOut|
n = 5 |
in = <nowiki>Alexander[K][-1]</nowiki> |
out= <nowiki>-11</nowiki>}}


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



{{HelpLine|
n = 6 |
in = <nowiki>Xp</nowiki> |
out= <nowiki>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.</nowiki>}}




{{HelpLine|
n = 7 |
in = <nowiki>Xm</nowiki> |
out= <nowiki>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.</nowiki>}}




{{HelpLine|
n = 8 |
in = <nowiki>P</nowiki> |
out= <nowiki>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.</nowiki>}}


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



{{In|
n = 9 |
in = <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>}}


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



{{InOut|
n = 10 |
in = <nowiki>Jones[K][q] == Jones[K1][q]</nowiki> |
out= <nowiki>True</nowiki>}}




{{HelpLine|
n = 11 |
in = <nowiki>Loop</nowiki> |
out= <nowiki>Loop[i] represents a crossingsless loop labeled i.</nowiki>}}


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



{{InOut|
n = 12 |
in = <nowiki>A2Invariant[Loop[1]][q]</nowiki> |
out= <nowiki> -2 2
1 + q + q</nowiki>}}

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Planar Diagrams