Notes for AKT-090910-1/0:33:25

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 $\mathbb {R} ^{3}$), which is coloured obeying the local rule of tricolourability, has at least 2 colours $\Leftrightarrow$ 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.)