|A trefoil knot, the most simple non-trivial knot|
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|
|An example of traditional Inca quipu|
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|
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.