Knot theory
Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R3. This is basically equivalent to a conventional knotted string with the ends joined together to prevent it from becoming undone. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram. But any given knot can be drawn in many different ways using a planar diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot.
The concept of a knot has been extended to higher dimensions by considering n-dimensional spheres in m-dimensional Euclidean space. This was investigated most actively in the period 1960-1980, when a number of breakthroughs were made. In recent years, low dimensional phenomena have garnered the most interest.
Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals.
In the last 30 years, knot theory has also become a tool in applied mathematics. Chemists and biologists use knot theory to understand, for example, chirality of molecules and the actions of enzymes on DNA.