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Thursday, April 9, 2015

Throwback Fact: Knot Theory

A trefoil knot, the most simple non-trivial knot
According to The Math Book by Clifford A. Pickover, "The use of knots may predate modern humans (Homo sapiens). This statement is absolutely fascinating, and prompted a research session on a portion of mathematics that the Center's math intern hadn't encountered yet.

A mathematical knot is different from what people normally think of as knots. Instead of a piece of string with two free ends and a tangle in the middle, mathematical knots come from embedding the unit circle into three dimensions, and twisting and disturbing the continuous line from there. Almost always, knot theorists are studying closed loops. From this point, it's important to note that a knot can be trivial (also known as an unknot), and in this case the loop can be unfurled to be a single unit circle loop. For a knot to be non-trivial, it will always have crossings no matter how the "string" is pulled.

An example of a traditional Celtic knot, made of two loops
Knots have come up in human history, as tools, as clothing techniques, as decoration, since it could be recorded. In fact, in some cultures, knots were used to make recordings. The Inca used quipus to make business records in a base 10 positional system as early as the third millenium BCE. The Chinese of the Tang and Song dynasties (around 1000 CE) used knots as decoration, making elaborate shapes out of a single strand of cloth. And of course, the Celtics used knots as artwork starting about 450 CE, and used different knots to represent spiritual or esoteric things. The Celtic knots are most closely related to knot theory because they are made from unending loops instead of physical rope, and many follow the definition of a non-trivial knot as known by knot theorists.

An example of traditional Inca quipu
Knot theory itself wasn't created until the late 1800s. Before the discovery of atoms, many scientists theorized that the universe was filled with a substance called "ether", and all matter was tangled in that mysterious ether. Naturally, the scientists' next thought was that they could understand elements and life itself by studying the knots that tangled around ether, so mathematicians began tabulating knots and their images. After the rise of atomic science, mathematicians continuing studying knots for the sake of learning, and knot theory was created as a field.

Today, knot theory is relevant in other branches of mathematics again. It is tied to DNA and molecular proteins, and the study of whether different knots express gene expression. Knot theory has also gained more depth as mathematics has progressed, and knots in four dimentions are somewhat concievable to even inexperienced mathematicians.

A chart with examples of knots, the larger number representing the number of crossovers
And we'll wrap up this Throwback history lesson with a joke from this Reddit comment:

Two college students are walking through campus. One asks, "What's your favorite branch of mathematics?"

The other replies, "Knot theory."

The first shakes his head and says, "Yeah, me neither."

To learn more about knot theory, visit this site. And on this website, you can see knots drawn out by selecting the features yourself.

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