Knot theory is the study of how circles
can be embedded in 3 dimensional space.
Two knots are equivalent if they can
be pushed and stretched without tearing
them or pulling them through themselves
so that they are identical. The simplest
knot that is not trivial is the trefoil knot.

A mathematical knot is a simple closed curve in three dimensions. The zero knot (also called the unknot), which is simply a ring, is one example of a knot, as is the trefoil knot. To visualize a trefoil knot, think of a clover shape made of a single connected string. Two knots are considered equivalent or ambiently isotopic if one can be bent, twisted, stretched, or pulled into the shape of the other. Cutting the knot or allowing it to intersect itself is not allowed. Any ambient isotopy can be expressed using only three types of moves, known as Reidemeister Moves.

Though knots are three-dimensional objects, they are often represented in two-dimensional diagrams. The crossing number of a knot is one property of a knot that describes the minimum number of crossings that a given knot can have. Another knot property is bridge number, or the number of bridges possible for the knot, a bridge being an arc in a knot diagram between two undercrossings (where a piece of the knot crosses under another piece) with at least one overcrossing (where a piece of the knot crosses over another piece) between them. The unknotting number of a knot is the minimum number of crossings that must be changed in order for the knot to become an unknot.

So now the question, why do we care? Knot theory has several important scientific applications, a notable one being in molecular biology in the study of DNA. DNA often becomes knotted, and there are enzymes called topoisomerases that perform topological manipulations on strands of DNA. In statistical mechanics, knots can be used to model systems, and it is theorized that the Jones polynomial has applications to quantum field theory. All very groovy stuff, I'm sure.

If is difficult to describe knots and knot theory without the help of embedded diagrams, so if further clarification is needed, these are the websites I used for this writeup and they all contain some nifty graphics that are very useful:

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