Skip to main content

*Insert Knot-Based Pun Here*

 A knot is a closed loop which is a transformation of the circle into 3D space! Unlike a physical knot, a mathematical knot has no ends, and thus isn't tied and can't be untied. The fundamental knot is the circle, which is the unknot. 


For a knot more complex than the circle to be formed, the threads of the knot must be crossed a finite number of times. These can be undone through the Reidemeister moves, which move threads past each other. Certain operations do not change the not, and these create invariants of a knot. 


It cannot be trivially proven that a given knot is not actually a clever twist of the circle. Coloring the knot could reveal if a not is not the unknot. However, the failure of coloring a certain number of times does not always mean a knot is the unknot, as different starting points for the coloring may affect the outcome as well as using more colors. The simplest coloring is three-coloring. For this process, three colors are used and the threads of the knots are colored such that at least three colors are used, and that at every crossing either all three colors are present or only one color is present. 

Below I show an attempt at a three coloring of my knot, as well as my uncolored knot. I chose 10-47 which has a crossing number of 10 and is the 47th variation of the 10 crossing knot. 


Uncolored Knot


As circled, my 3-coloring failed! However. this does not mean it is the unknot. This failed because there was a crossing which had 2 colors present. The only options are at a crossing that all 3 colors must be present, or only one can be. 


My knot has a writhe of 4 given the orientation I define below. "Above" crossings are +1 while "negative" crossings have a value of -1, and the sum of the crossings gives the total writhe.







The writhe will change with the number of crossings, and how they are presented in the projection (2D representation, such as the images above). Using the python library, I decided to plot the writhes of the first 150 knots with 10 crossings, and the plot is shown below.


Interestingly, it appears that the writhe is somewhat cyclic! I assigned a value of -1 for knots pyknotid didn't have value for. The maximum writhe was 10 and the minimum was 0. It must attempt to orient the knots such that they have a maximal number of positive crossings. 


Additionally, every knot has an attribute known as its unknotting number. The unknotting number asserts the minimum number of times the knot must be passed through itself to become the unknot. This is not trivial to see, and it is often more intuitive to present the upper bound of this number, since only an especially insightful mathematician could see the moves such that they will produce the minimum number. According to the knot atlas entry for 10-47, it has an unknotting number of 2! I have no Earthly idea of what processes they went through to obtain that, but I applaud their dedication. Below I present what I believe is an upper bound of this process as well as show the steps I took in order to obtain the unknot. 

I suggest that the upper bound of my unknotting number is 11, since it took 11 moves in order to arrive at the unknot. Surely, a more clever mathematician could reduce this much further since the minimal number is 2. Below I show my process










Finally, every knot has an associated polynomial! Below I give an expansion of the polynomial for my knot. 













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

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
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