Knot theory

Knot Theory

Knot theory is a branch of topology, a field of mathematics that studies the properties of space that are preserved under continuous transformations. Knot theory specifically deals with the study of mathematical knots, which are embeddings of a circle in 3-dimensional Euclidean space, \(\mathbb{R}^3\). Unlike the knots in everyday life, mathematical knots have no ends and cannot be untied.

Definition

A knot is defined as a closed, non-self-intersecting curve that is embedded in three-dimensional space. Formally, a knot is a homeomorphism from the circle \(S^1\) to \(\mathbb{R}^3\). Two knots are considered equivalent if one can be transformed into the other via a continuous deformation, known as an ambient isotopy.

Knot Invariants

Knot invariants are quantities or algebraic objects that remain unchanged under ambient isotopies of the knot. They are crucial for distinguishing between different knots. Some important knot invariants include:

• Knot Polynomials: These include the Alexander polynomial, Jones polynomial, and HOMFLY polynomial. Each of these polynomials assigns a polynomial to a knot in such a way that equivalent knots have the same polynomial.

• Knot Group: The fundamental group of the knot complement, which is the set of all loops in the space surrounding the knot, modulo homotopy.

• Knot Genus: The minimum genus of any Seifert surface for the knot.

Types of Knots

Knot theory classifies knots into various types based on their properties:

• Trivial Knot: Also known as the unknot, it is a simple loop with no crossings or twists.

• Prime Knots: Knots that cannot be decomposed into simpler knots via connected sum operations.

• Composite Knots: Knots that can be expressed as the connected sum of two or more nontrivial knots.

Applications

Knot theory has applications in various fields, including:

• Biology: Understanding the structure of DNA and how it knots and unknots during cellular processes.

• Chemistry: Studying the topology of molecular structures and the synthesis of molecular knots.

• Physics: Analyzing the properties of quantum field theory and string theory.

History

The study of knots dates back to the 19th century, with significant contributions from mathematicians such as Carl Friedrich Gauss, who developed the Gauss linking integral, and Peter Guthrie Tait, who created the first systematic tables of knots. The modern development of knot theory was greatly influenced by the work of Vaughan Jones, who discovered the Jones polynomial in the 1980s.

Also see

• Braid theory
• Link (knot theory)
• Topology
• Knot polynomial