Tricolorability

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The Tricolorability of a knot refers to the ability of a knot to be colored with three colors according to two rules. The tricolorability of a knot is another methed used by mathematicians in the realm of Knot theory to distinguish between two knots. In addition, tricolorability is an isotopy invariant of a knot.

1 Rules of Tricolorability
2 How to Color a Knot
- 2.1 Example of a Tricolorable Knot
- 2.2 Example of a Non-Tricolorable Knot
3 Isotopy Invariant
4 References
5 See Also

[edit] Rules of Tricolorability

A knot is tricolorable if each strand of the knot diagram can be colored one of three colors according to two rules:

1. At least two colors must be used

2. At each crossing either every strand is the same color, or no color is repeated

[edit] How to Color a Knot

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.

[edit] Example of a Tricolorable Knot

$\longrightarrow$ $\longrightarrow$ $\longrightarrow$ $\longrightarrow$

This is our beginning knot
(known as the False lover’s
knot).

Choose and color one entire
strand until reach crossings
in both directions.

At one of the crossings
that is connected to the colored
strand, choose another strand,
connected to that crossing,
and color it with another
color.

Once again choose another
strand that isn’t colored and
is connected to one of these
crossings, and color it a
different color than the other
strands that meet at that
crossing.

Repeat in this manner
until the whole knot is
colored, or you run into
an error (see next example).

This False lover's knot is tricolorable.

[edit] Example of a Non-Tricolorable Knot

$\longrightarrow$ $\longrightarrow$ $\longrightarrow$ $\longrightarrow$

This is our beginning knot
(known as the Figure eight
knot).

Begin by choosing and coloring
one entire strand until reach
crossings in both directions.

At one of the crossings
that is connected to the colored
strand, choose another strand,
cennected to that crossing,
and color it with another
color.

Once again choose another
strand and color it a different
color than the other strands at
that crossing.

NOT Tricolorable
We have reached an error
because at two of the crossings
it does not meet the 2^nd
requirement (See below).

This figure eight knot is NOT tricolorable. Looking at the 2^nd to last figure, one can see that one has run into a dilemna: each of the three crossings needs a different color to complete its tricolorability. The top crossing needs a green strand, the middle one needs a blue strand, and the bottom crossing needs a red strand. Since one cannot color this knot in accordence with the rules of tricolorability, it is considered NOT tricolorable.

[edit] 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.