Planar Diagrams: Difference between revisions
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[[Image:PDNotation.gif|frame|The <code>PD</code> notation|right]] |
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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 on the right is: |
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 on the right is: |
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\begin{displaymath} |
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\end{displaymath} |
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--> |
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<center><math> |
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<div align="CENTER">[[Image:img16.gif]] </div> |
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</math></center> |
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(This of course is the Miller Institute knot, the mirror image of the knot [ |
(This of course is the Miller Institute knot, the mirror image of the knot [[6_2]]) |
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<!--$Startup Note$--> |
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<div align="CENTER"> |
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<!--$$?PD$$--> |
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{| |
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<!--END--> |
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|+ align="BOTTOM" |'''Figure 2:''' Our notation for planar diagrams |
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|- |
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<!--$$?X$$--> |
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| align="center" | |
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<!--END--> |
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[[Image:img18.gif]] |
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|} |
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Thus, for example, let us compute the determinant of the above knot: |
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<!--$$K = PD[X[1,9,2,8], X[3,10,4,11], X[5,3,6,2], X[7,1,8,12], X[9,4,10,5], X[11,7,12,6]];$$--> |
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<!--END--> |
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Alexander[K][-1]$$--> |
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<!--END--> |
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Revision as of 15:58, 23 August 2005
The
PD notationIn 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 on the right is:
(This of course is the Miller Institute knot, the mirror image of the knot 6_2)
Thus, for example, let us compute the determinant of the above knot:
