Knot theory

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File:Frame left_|thumb|Frame_left]]_]] Knot theory is a branch of topology 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 terms, a knot is an embedding of a circle in 3-dimensional Euclidean space, \(\mathbb{R}^3\). Two knots are considered equivalent if one can be transformed into the other via a continuous deformation, known as an isotopy.

History

The origins of knot theory can be traced back to the 19th century when Lord Kelvin proposed that atoms were knots in the aether. This idea led to the first systematic tabulation of knots by Peter Guthrie Tait, Charles Niven, and Thomas Kirkman. Although Kelvin's theory was eventually discarded, the mathematical study of knots continued to develop.

Basic Concepts

Knot Invariants

A key aspect of knot theory is the study of knot invariants, which are properties of knots that remain unchanged under isotopy. Examples of knot invariants include the knot group, the Alexander polynomial, and the Jones polynomial.

Types of Knots

Knots can be classified into various types based on their properties. Some common types include:

• The unknot, which is a simple loop.
• The trefoil knot, which is the simplest nontrivial knot.
• The figure-eight knot, which is the simplest knot with a crossing number of four.

Knot Diagrams

A knot diagram is a projection of a knot onto a plane, with information about the over and under crossings. Knot diagrams are useful for visualizing and manipulating knots.

Applications

Knot theory has applications in various fields including biology, where it is used to study the structure of DNA and other molecules, and in chemistry, where it helps in understanding the properties of certain chemical compounds. It also has applications in physics, particularly in the study of quantum field theory and statistical mechanics.

Related Pages

• Topology
• Braid theory
• Knot polynomial
• Link (knot theory)
• Knot group
• Reidemeister move