Knots

Introduction

Somehow it's the final blog of the semester! This time we're talking about knots. Knots are a topic in mathematics, specifically topology. Knots are exactly what they sound like; you make a knot when you tie your shoes, for example. The only difference is that mathematical knots have the ends of the strings tied together and they cannot be undone. We have simple knot examples, like the trefoil knot. This is a knot of three crossings. We can get even crazier by looking at knots of 8+ crossings, which will be discussed in the rest of this post.

(Image from centerofmathematics.blogspot.com)

Crossing number

Consider this knot. It has 10 crossings and is knot 10-139 in the knot atlas found at http://katlas.math.toronto.edu/wiki/10_139.

Unknotting number

The unknotting number is the sequence of crossing switches that give the unknot. From the knot atlas, I know the unknotting number of 10-139 is 3. This means that switching three crossings will give me the unknot. I have attempted to verify this using SnapPy. I was able to switch three crossings to produce the unknot, as seen below.

First, start with the knot projection.

Next switch the bottom left crossing seen in the above projection. This gives the knot below:

Now switch the bottom left crossing of the above projection. This gives:

Now switch the top right crossing of the above projection. After many, many Reidemeister moves, this is what we have. Note that this is the unknot!

Simply untwist that last crossing.

I'm pretty stoked that I was able to verify the unknotting number with the crossings I chose to switch. That was a lucky first try!

A coloring attempt

Next, I attempted to find a coloring for this knot projection. A knot projection is n-colorable if each strand has a number, at least two numbers are used, and at each crossing \(i+j-2k\equiv 0\), where \(i\) and \(j\) are the numbers on the undercrossings and \(k\) is the number on the overcrossings. Similarly, a knot projection is tricolorable if each strand has a color, at least two colors are used, and at each crossing either 1) all three colors come together or 2) all colors are the same. It was easier for me to attempt to color this knot with blues, pinks, and greens than to use modular arithmetic, but, as you can see, my tricolorability attempt failed given the choices I made for how to color this knot. I started with the right pink strand, then colored the top right crossing such that all three colors were present. From here, I did the same thing for the top left crossing. As you can see, I eventually ran into problems in two places. However, this does not mean that this knot isn't tricolorable. It means that there are a large number of ways to attempt to show tricolorability, but that this is not one of them. Because there are so many ways to color this projection, it can be hard to find one that works (or to prove a knot is not tricolorable by showing none work).

Writhe

Writhe is arguably the easiest computation for knots. We are assigning +1 and -1 to the crossings. Then, the writhe is simply the sum of these \(\pm\)1's divided by two.

Using the given orientation, I computed the writhe of this knot to be 5.

I think it's interesting that all of these crossings are +1's. I've stared at it and twisted my knot around so many times because it seems too coincidental, but hopefully I was able to visualize the crossings correctly!

Knot polynomials

The last thing we are going to calculate is the first two steps of the knot polynomial Q.

This part gave me a headache from staring at the projections so long trying to visualize what to do, but I am happy with how my drawings turned out. This project has been a fun dive into a brand new field of mathematics for me, and I think this was a good note to end the semester on. Math is really just learning how to learn new complicated things, so this was definitely very math-y in that sense.

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