Skip to main content

Knot 9-31

Knot 9-31
A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into.
The Crossing Number
For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings!
The Unknotting Number
The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still want to perform the correct moves to untie the knot. From the Knot Atlas, knot 9-31 has an unknotting number of 2. Below are the steps/moves that I took to "untie" my knot.
From the image, both the moves are marked with a circle. By switching these crossings, the knot can easily be untwisted, resulting in the unknot. This is better understood with the 3D model. The model almost already wants to unknot itself, but these two crossings restrict that. This confirms that the minimum number of times the knot must be passed through itself to untie it is 2!
Colorability
Next, we will find the Colorability of my knot. For a simple definition, tricolorability, is the ability of a knot to be colored with three colors subject to certain rules. These rules include: Each strand in the knot is given 1 of 3 colors, at least two colors are used on the knot, and at each crossing there is either all 3 colors or all the colors are the same. Now, this is just for a knot that is three colorable. A knot can be N colorable where N is any number that is less than or equal to the crossing number. Below is my attempt at seeing if my knot is three colorable!
From my image, it is clear that my attempt to see if my knot was three-colorable failed. This does not mean that my knot is not entirely three-colorable, but it means that the route I chose or the decisions that I made were not the correct ones. I started in the top right corner of the knot. I ended up having two crossings that resulted in repetition of colors.
Writhe
In simple terms, the writhe is the total number of positive (+1) crossings summed with the total number of negative (-1). The way that we determine if a crossing is positive or negative is using a variation of the right-hand rule. This is shown here:
Now, I am going to calculate the writhe for my knot! The image below shows my process.
From my computation, I found the writhe of my knot to be: -1
Knot Polynomial
The knot polynomial is one of the hardest things to compute for a knot. The knot polynomial is a knot invariant in the form of a polynomial, in our case Q, in which the coefficients encode some of the properties of a given knot. Below is my computation for the first 2 steps of my knot.
This computation was very challenging, but I am confident with how both my image came out from a computation perspective and an artistic perspective. I had to hand draw all of my knots from this blog, so I am pretty satisfied with how they all came out. In conclusion, knots are very unique topics of mathematics and are far one of the most interesting things I have learned. I never knew the number of properties knots had and how intriguing they are to study.

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more