Lzhang: video annotation

2009-09-22T21:25:59Z

video annotation

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Cont'd: Invariance of tricolourability under R2. In particular, the subtlety (as viewed locally) about the total number of colours used (globally) is discussed.

Problem/Concern:
Sometimes, in a local picture, only 2 colours appear on one side of an isotopy move (e.g. R2) whereas all 3 colours appear on the other. One might worry that this could lead to the violation of the global rule.

Solution:
One can prove that a knot (consisting of only 1 connected piece of material in <math>\mathbb{R}^3</math>), which is coloured obeying the local rule of tricolourability, has at least 2 colours <math>\Leftrightarrow</math> it has all 3 colours. The proof relies on the fact that the same piece of material can change colour (from one colour to a 2nd colour) only by going 'under' a crossing, and whenever a crossing involves 2 colours it must involve a 3rd. (This argument fails, however, for links. Just consider two knots, one red and one blue say, placed side by side.)

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Lzhang: video annotation