Knot Coloring Puzzles

Mathematicians sure must love crayons!

Coloring puzzles appear over and over again in mathematics.

Mathematicians have spent huge amounts of time studying how to color maps and the dots and lines of graphs.

So it's logical to see that mathematicians have also asked, "What interesting things will we discover if we try coloring knots?"

Here are some colored knots

One of these knots is not (knot?) colored according to the rules below. Which one is it?

Rules for Coloring Knots

Each place there is a crossing in the knot there are three things to color:

The strand that crosses over
One side of the strand that goes under
The other side of the strand that goes under

In this picture these three strands are colored red, yellow, and blue respectively:

Here are the coloring rules::

At each crossing you have two choices for the colors of the strands that meet:

You may color the strands so that they are all different colors at the crossing. (If you color a strand blue in for one crossing, you can use the color blue again somewhere else, but only as long as it will not meet another blue strand at a crossing.)
You may color the strands so that they are all the same color at the crossing.

What does coloring tell you?

Sometimes it can be tricky to figure out how to color a knot with as few colors as possibile and still follow the rules. But once you do, you can usually show that if you try to use one less color, it will be impossible to follow the rules.

Can you use fewer colors with any of the three knots that are colored correctly in the pictures above?

How many colors will you need to use when the knot that is colored wrong is colored right?

Knot theorists are particularly interested in knots that are 3-colorable, which is to say knots that can be colored according to these rules using exactly 3 colors. It turns out that if a knot is 3-colorable, no matter how you deform or redraw the diagram, it will always be 3 colorable.

Do you believe that?

Why do you think it works?

It is also true that if you have two knots and one is 3-colorable and the other one isn't, you can be absolutely certain that it is impossible to twist one of the knots around so that it looks like the other one.

Do you believe that?

Why do you think it works?