Knot Theory: The Tangled Beauty of Mathematics

 

Tetsuo Argenton, R. (2017, September 29). Tabela de nós matemáticos. https://en.wikipedia.org/wiki/Knot_theory#/media/File:Tabela_de_n%C3%B3s_matem%C3%A1ticos_01,_crop.jpg

A Mathematical Introduction to Knots

If you were to take a piece of string, tie a knot in it, and then proceed to attach the ends with a piece of tape or glue, you have now created what in mathematics we define as, a knot. The difference between your everyday knot and a mathematical one, is that the latter will always be in a closed loop. No loose strings, no open ends, nothing. These knots can also be described as closed curves in three-dimensional space (ℝ³).

Knot theory is a relatively new field, with its origins dating back to the 19th century. It is an expanding field with several applications in physics, chemistry, and biology. One of the main questions that knot theory proposes is within the concept of equivalence. Can two knots ever be the same? What qualifies knots to be equivalent? How many types of knots are there?

Ambient Isotopy & Knot Invariants

Picture this... it's a Friday night, and you and I are spending some quality time tying some knots together:

my knot

your knot

We look at our distinctive knots and wonder... are these a variation of the same knot? can you take your knot, and transform it into mine? or are they completely different ones? Well, in order to find out we remember that we must use ambient isotopy. Ambient isotopy is the process of deforming a knot into the other via a sequence of bending, stretching, twisting, etc. without cutting or having the knot pass through itself. 

After , undergoing this process, we look at our knots and think that we're pretty successful. But there is only one way to truly confirm if we have succeeded, and that is using knot invariants. Knot invariants are characteristics of a knot that remain the same under certain transformations (i.e. mathematical quantities). Some examples of invariants include tricolorability (the ability of a knot to be colored with three colors subject to certain rules), the crossing number (the minimum number of crossings in a diagram of any isotopic knot), or even the unknotting number (the minimum number of times that a knot must be passed through itself to become the unknot).

To determine if our knots are the same, we draw out some diagrams and calculate a variation of knot polynomials, which are a class of these invariants. If both of our knots produce different polynomials, then we know they are not equal. In other words, we can determine if two knots are the same by checking if they have the same invariants. If the invariants are different, then the knots are different.

Why Do We Study Knots?

As mentioned earlier, knot theory has hellaaaa applications outside of math. For example, biologists are using knot theory to help us understand how enzymes can disentangle strands of linked DNA. Knot theory is also used in the field of quantum computing when we talk about the complexity of computing knot polynomials. And of course, in math, we don't just tie things for fun. We can use knot theory to help us solve some of the most difficult puzzles of higher-dimensional spaces.

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If you want a more math-heavy explanation of knot theory, here are some great papers I found that I think do a great job at explaining!

An Overview of Knot Invariants- University of Chicago

An Analysis and Comparison of Knot Polynomials- James Madison University