Knots!

When you think of a knot, you may think of tying a string, rope, or shoe. But those knots are a little different than the mathematical knots we will be discussing today because those all have ends to the knot.

A knot defined in mathematical terms is a closed curve in a 3D space that doesn’t intersect itself. The knot I will be focusing on in this blog post is knot 8_18 (according to Rolfsen Knot table from The Knot Atlas). The 3D print of the knot looks like this.

We represent knots in a 2D space by drawing knot projections. One knot can have multiple projections, but here is the projection of the knot we will be working with.

(http://katlas.math.toronto.edu/wiki/8_18)

We can determine how many crossings a knot has by looking at its projection. A crossing is the place where a knot crosses itself in a knot projection. Therefore, this is an eight-crossing knot, hence why it is labeled knot 8_18. 

Now that we know the basic information for this knot, we will focus on determining the unknot number, a colorability, the writhe, and the first two steps of polynomial Q.  

A knot k has an unknotting number n if there is a projection of the knot where changing n crossings gives the unknot and no smaller number of crossings on any projection. To determine this, I started with my original knot projection and found a sequence of crossing switches, then performing Reidemeister moves until I found the unknot. I started switching the two crossings circled in red (original knot of the left and knot after switching on the right). This resulted in the unknotting number being two. You can see the flow of my work below.

 Next, we will look at colorabilty. A knot is 3-colorable if 1) each strand in the knot is giving one of three colors, 2) at each crossing either all three colors are present or all the colors are the same, and 3) at least two colors are used in the knot. This knot is 3-colorable because this criterion fits for all 8 crossings as seen below.

The steps to determine this successful colorabilty is choosing a color for the first strand and crossing. Starting in the top left with crossing one, I first choose blue for the top strand, then red and green for the other two strands (since all three colors need to be present). I then moved onto crossing four where blue was already present. So, I labeled the other two strands red and green. I continued this for the rest of the crossings, assigning three colors at each crossing or two strands had the same color, then keeping the third strand the same color (as seen in crossing six and eight) until the entire projection was colored. This resulted in a successful tri-colorable knot.

Next to determine the writhe of the knot, we first orientate the knot. You can see how the knot projections is orientated based on the arrows along each strand. Next, we assign a +1 and -1 to all crossing based on their direction of crossing. The assignments resulted in the following.

Finally, the writhe will be the sum of all these +1 and -1. The writhe of this knot projection is 0.

Finally, we will look at the first steps of solving for the polynomial Q using this equation.

Q(L) = (-A³) ^-w(L) P(L)

First, we need to determine P(L). Which can be solved by the following.

Then plug in P(L) and the writhe to Q to get the following.

If we were to continue these steps, we would keep going until it was reduced to projections that produced the unknot. However, this could be very many steps, so we will only look at the first two steps which resulted in four knots remaining. 

We now know more about the characteristics of this knot!