Knots

When describing a knot in math, one must first disregard what they previously knew. The average person will say a knot is what you make when you tie your shoes or what happens when your earbuds tangle, but this isn’t the mathematical definition of a knot. This knot, where a string is tied and two ends are left open, is referred to as a “braid.“ In math, however, a knot is a closed, non-intersecting curve in 3-dimensional space. This definition allows different types of knots that mathematicians study in Knot Theory.

There are some very basic knots where are listed below.

The unknot is the simplest of knots with zero “crossings.“ The next knot is called the Trifold knot more commonly. This has three crossings making it the simplest knot that still has crossings. The way that knots are labeled is first by how many crossings it has. A “crossing“ is where on pice of the knot overlaps another creating a weaving behavior. There can also be multiple knots with the same number of crossings.

The idea that multiple knots can have the same number of crossings is what drives the knot field. For example, no matter how hard you try to deform 7₄ into 7₅, you will never get the same knot.

Deforming a knot has the same restrictions as topology. The knot can knot be cut or glued. With this, knots can be manipulated so that their crossings change locations. Also, because the knot is embedded in a 3D space, this means that different angles of the knot may result in a different number of crossings. Once again, the simplest view of a knot is where those number of crossings is the smallest. A knot that is NOT in its basic form may even have more crossings than the simple. This is illustrated below.

All of these knots are the same, the trifold knot. They are all either rotated differently or are not in their simplest form. It becomes really challenging to differ on knot from another. Knot theorists will sometimes use computer modeling and sometimes a physical knot to see how far down they can simplify it.

Now, you might be wondering why this is important. The first time that knots became important was in the 1860s. Scientists proposed that the atom was made up of a knotted tube of something. This drew mathematicians in and kickstarted the study of knots. Even after knot theory separated from the study of atoms, mathematicians continued to study them. Knot theory has applications in string theory. Because of mathematicians' curiosity about knots, this field continued to expand.