The Mathematics of Knot Theory
Introduction
Knot theory is a branch of topology, a field of mathematics, that studies mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R³. 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).
History of Knot Theory
The study of knots dates back to the 19th century with the pioneering work of Carl Friedrich Gauss, who introduced the concept of linking number, a fundamental concept in knot theory. The first systematic study of knots was initiated by Peter Guthrie Tait in the late 19th century. Tait's work was motivated by his interest in the vortex atom theory of Lord Kelvin, who proposed that atoms were knots in the ether.
Mathematical Definition of Knots
In mathematical terms, a knot is defined as a smooth embedding of a circle in 3-dimensional Euclidean space, R³. This means that a knot is a loop that is embedded in space in such a way that it does not intersect itself. The simplest knot, called the unknot, is a perfect circle. More complex knots can be formed by twisting and looping the circle before joining the ends.
Knot Invariants
One of the central problems in knot theory is determining when two knots are equivalent, known as the knot equivalence problem. To tackle this problem, mathematicians have developed a variety of knot invariants, quantities that remain unchanged under certain transformations of the knot. Some of the most important knot invariants include the Jones polynomial, the Alexander polynomial, and the Conway polynomial.
Knot Diagrams
A common way to represent knots is through knot diagrams. A knot diagram is a 2-dimensional representation of a knot, where overcrossings and undercrossings are used to indicate the 3-dimensional structure of the knot. Knot diagrams are useful for visualizing and studying the properties of knots.
Applications of Knot Theory
While knot theory began as a purely theoretical field, it has found applications in a variety of areas. For example, in molecular biology, knot theory is used to study the structure of DNA and proteins. In physics, knot theory has been applied to study quantum field theory and string theory. In computer science, knot theory is used in the study of algorithms and data structures.