Introduction to Invariants

Hello!

First of all, thanks for keeping up with this blog, and the uneven distribution of posts.

This post will mostly be explaining some of the basics of knot theory.

So starting off, we talked about how there are many, many different kinds of knots, in last weeks post, and how, they're all basically "a closed, non-self-intersecting curve that is embedded in three dimensions" (read: circles). 

From this, we can consider the projections of a knot. Projections are ways which we can display the knot, and each knot can have many, many different projections. Think of it as a knot in a bubble that we can turn to look at. For example, you can find different projections of the KotW (Knot of the Week) here: 8_17 Projections. (If you click around, you'll see that the knot looks different, but stays the same,)

Another interesting knot property to consider is the crossing number of the knot. A crossing is a place where a knot crosses itself, and the crossing number, naturally, would be the (minimum) number of times a knot would have to cross with itself.

Figure 1 - The crossings of this 7_5 knot has been circled. 

The knot has 7 crossings which cannot be simplified. So even if you take a part of the knot, and you keep on twisting and twisting and twisting it, the crossing number would not change.

The crossing number, here, is what we would call an invariant of a knot.

A knot invariant is a property of a knots that will stay the same even when it is pulled around. Because of this, knot invariants are good ways, and the typically used ways, to classify knots. There are many different invariants used, including knot polynomials, bridge number, curvature, etc.

Hope you enjoyed this tidbit, and expect another post very soon!



Works Cited:

Knotilus.math.uwo.ca,. "Knotilus". N.p., 2016. Web. 13 Feb. 2016.

Mathworld.wolfram.com,. "Knot -- From Wolfram Mathworld". N.p., 2016. Web. 13 Feb. 2016.

Knot of the Week

The Knot of the Week: The 8_17 knot.

This knot is the simplest of the non-invertible knots, which means that you cannot deform the knot (move parts of it around) so that you can make it the mirror version of itself.

If you want to learn more about it, feel free to go here: 8_17.

References:

Commons.wikimedia.org,. "File:8 17 Knot.Svg - Wikimedia Commons". N.p., 2016. Web. 13 Feb. 2016.

Introducing the Knot

There's knot a better way to start off a knot theory blog than with a knot joke or a knot pun.

Marty Scharlemann tells the story of a calculus student who came in for help, and after Marty had worked some problems, the student said "So what kind of math do you like?"
Marty said "Knot theory."
The student said "Yeah, me neither."

(Joke credited to, and found in The Knot Book)

        

Starting off more properly, though--Hello! I'm Alice Yang, a high school senior at BASIS Scottsdale. I enjoy, as one can probably tell by now, math. In my free time, when I’m not doing math, I enjoy drinking tea, and playing violin.

My blog, this post the first of many, will be detailing my research the next few months in the area of knot theory. My fascination with knots probably started the previous summer, at SUMaC (Stanford University Math Camp). It started off after I unsuccessfully tried to figure out a puzzle, pictured below.

Figure 1 -- The puzzle that started it all. 

So, in hopes of trying to figure out the puzzle I decided to learn about knot theory. A summer later, I’ve learned more about the subject, through my research project, and it is a truly fascinating area. So, when I had to choose a Senior Research Project to complete in my third trimester of school, I decided to choose knot theory.

Figure 2 - A trefoil knot -- this is one of the simplest types of knots, with three crossings. 

In my senior research project, I'm going to be considering knots, and, specifically, ways to classify these knots, using knot invariants (explained later in the post).

So, by now, you (hopefully) will be wondering, what exactly a knot is. The best way to explain it is: if you take a string, jumble it up in the middle, and glue the ends together, you have a knot. This is extremely vague, and, therefore, there are many, many different kinds of knots.

Figure 3 – There are many different kinds of knots.

There are also, because of this, many different ways of classifying knots, including knot polynomials, crossing number. The invariant of a knot is a property that, when defined for two knots, holds true and stays the same for both. My project aims to create one using the conformal length of the knot (more about that in later posts).

I have the great opportunity of working under the guidance of Dr. Julien Paupert on this project, among others possibly, and hope, through his guidance and in the process of working, to learn more about knots, invariants, and math in general. Dr. Paupert’s personal page can be found here: Link to Dr. Paupert's Page, and the general ASU Department of Mathematics page can be found here: Link to ASU Math Department .

If you’re really curious, there’s a very conveniently located math dictionary to your right. My project, specifically is titled It’s a Knot (or Not?): Determining Knot Invariants Using  the Knot Conformal Length, and the project proposal can be found right here: Link to SRP Proposal.

I’m excited to start working, and I hope to keep you informed weekly, or sooner, on any updates.

Works Cited:

Adams, Colin Conrad. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W.H. Freeman, 1994. Print.

Haneyama Vortex. Digital image. Fun and Games Montville. Fun & Games, Montville, n.d. Web. <http://www.funandgamesmontville.com/wp-content/uploads/2014/11/Hanayama-Vortex.jpg>.

Knot-Crossings. Digital image. Math and the Art of MC Escher. Math and the Art of MC Escher, n.d. Web. <http://euler.slu.edu/escher/upload/3/37/Knots-9cross.gif>.

Trefoil. Digital image. The University of Scranton. The University of Scranton, n.d. Web. <http://www.scranton.edu/academics/ignite/issues/2013/spring/Images/interior/Trefoil_knot_arb.jpg>.