Knot Theory

Introduction

Knot theory is a branch of topological mathematics that studies mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone. In 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).

A close-up image of a complex knot made of a flexible material, demonstrating the intricacy and complexity of knot theory.

History

The origins of knot theory date back to the 19th century with the pioneering work of Carl Friedrich Gauss in the field of electromagnetism. However, it was not until the 20th century that knot theory became a significant part of mainstream mathematics, largely due to the work of J.W. Alexander and James Waddell Alexander II.

Mathematical Definition

In mathematical terms, a knot is defined as a smooth embedding of a circle into 3-dimensional Euclidean space, R³. This means that a knot can be thought of as a path in space that is closed - it loops back on itself so that it has no ends. The simplest example of a knot is the unknot or trivial knot - a simple loop.

Knot Invariants

Knot invariants are mathematical objects associated with a knot which remain unchanged under various transformations. These invariants, such as the Jones polynomial, the Alexander polynomial, and the Conway polynomial, are used to distinguish knots from one another.

Knot Diagrams

Knot diagrams are planar representations of knots which are particularly useful for computations in knot theory. They are obtained by projecting the knot onto a plane and introducing breaks to show which strand is over and which is under at each crossing.

Knot Groups

In knot theory, the knot group is a fundamental concept. It is the group of loops around the knot, under the operation of concatenation. The study of knot groups and their representations plays a crucial role in the classification of knots.

Applications

While knot theory originated in pure mathematics, it has found applications in various fields such as physics, chemistry, and biology. For instance, in molecular biology, knot theory is used to study the structure of DNA and proteins.