Invariants—Coloring

We’ll develop a few tools for distinguishing , including the simple, yet useful, coloring invariants.

It’s generally too hard to tell when two knots are the same or not by looking at their diagrams. In fact, if you’re lucky you can prove they are the same, but you can never prove they are different because there are infinitely many ways you can alter the diagram through Reidemeister moves, and you can never know if you just missed the route to get from one to another. For example, to start to study knots, people started making table of “prime” knots by drawing all possible diagrams with a given number of crossings and using Reidemeister to try to figure out which were different. One of the main such projects was done by Tait and Little in the 1890’s. They included these two knots:

Unfortunately, these two are the same, though many people missed this fact—until 1973 when it was discovered by a lawyer named Kenneth Perko who did math for fun in his spare time. These two are now called the “Perko pair” even though they are really the same. So what we need to do is to find something that is easy to calculate from a diagram, but that does not change when using Reidemeister moves so that different versions of the same know possess the same property. Such a thing is called a . Ideally we could calculate something simple and it could tell any two different knots apart. We haven’t found anything yet, but who knows? Here is an example of a property that is easy to tell from a knot diagram, and doesn’t change when we do Reidemeister moves. It’s called 3-colorability and all it really asks is if we can paint the strands of a knot diagram according to certain rules.

Definition: A knot is 3 colorable if we can paint the strands of its diagram using three different colors, with the stipulation that at any crossing, either all three strands must be the same color, or else all three must be different. We also require the use of more tha one color, because painting all the strands the same color is a trivial way to solve the problem!

It is actually pretty obvious that the trefoil is 3-colorable, by painting each strand a different color. Note that at each of the three crossings, all three colors are used. What

1 happens if we were to try to paint the figure-of-eight with only three colors? Try! There’s no way to make it work! This means that the trefoil and the figure-of-eight must be different knots. Before we say this for sure, let’s verify that if one diagram for a knot is 3-colorable, then all of them are. This is easy for twisting. For poking, everything is OK if both strands are the same color. If they aren’t, then the part that gets “poked under” the other part will acquire the third color, so again the result is 3-colorable. The slide move is a little trickier and breaks down into cases depending on how many different colors there are in the part of the diagram being examined. But going through the cases shows that, indeed, 3-colorability in invariant under Reidemeister moves. So it is a knot invariant. We could extend the definition to other numbers:

Definition: A knot or is n-colorable if the strands of a diagram can be painted with no more than n different colors (numbered 1–n, or more usually 0–n − 1), using at least two to avoid trivial monochromatic diagrams, and so that at each crossing the colors of the NW and SE strands add to the same amount, modulo n, as the colors of the SW and NE strands.

The figure-of-eight knot happens to be 5-colorable, as is easily seen just by trying. We only need to check colorability for n a prime. Showing this will be a homework problem. Also, no knot is 2-colorable (although every link of two or more pieces is—just color one piece with one color, and the rest with a different color). Try the problems on the handout!

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