Tricolorability
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.
Rules of tricolorability
A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:
- 1. At least two colors must be used, and
- 2. At each crossing, the three incident strands are either all the same color or all different colors.
Examples
Here is an example of how to color a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue.
Example of a tricolorable knot
The granny knot is tricolorable. In this coloring the three strands at every crossing have three different colors. Coloring one but not both of the trefoil knots all red would also give an admissible coloring.
Example of a non-tricolorable knot
The figure eight knot is not tricolorable. Looking at the figure with three of the strands coloured, one can see there is a dilemma: all three crossings involving the black strand need a different color to complete its tricoloring. The top crossing needs a green strand, the middle crossing needs a blue strand and the bottom crossing needs a red strand. There is no way to colour two of the colored strands the same to avoid problems either. Since one cannot color this knot in accordance with the rules of tricolorability, it is considered NOT tricolorable.
Isotopy invariant
Tricolorability is an isotopy invariant, which is a property of a knot or link that remains constant regardless of any ambient isotopy. This can be proven by examining Reidemeister moves. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant.
| Reidemeister Move I is tricolorable. | Reidemeister Move II is tricolorable. | Reidemeister Move III is tricolorable. |
|---|---|---|
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 |
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Properties
Tricolorability is a weak knot invariant. The composition of a tricolorable knot with another knot is always tricolorable. Any separable link with a tricolorable separable component is also tricolorable.
In Torus Knots
If the torus knot/link denoted by (m,n) is tricolorable, then so are (j*m,i*n) and (i*n,j*m) for any natural numbers i and j.
References
- Weisstein, Eric W. "Three-Colorable Knot." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Three-ColorableKnot.html