When are two knots considered the same?

When one can be continuously deformed into the other within space without cutting — formally, when they are related by an ambient isotopy, equivalently when their diagrams differ by a finite sequence of Reidemeister moves.

Is there a single invariant that classifies all knots?

No complete, easily computable invariant is known. Different invariants detect different features, and even strong ones like the Jones polynomial fail to distinguish all distinct knots, which keeps the classification problem open.

Knot Theory

Knot theory studies how circles can be embedded in three-dimensional space, seeking invariants that decide when two knots are the same and capture the subtle topology of low dimensions.

Definition

Knot theory is the branch of low-dimensional topology that studies embeddings of one or more circles in three-dimensional space up to ambient isotopy, classifying them by means of computable invariants.

Scope

This area covers knots and links as embeddings of circles in space, their diagrams and the Reidemeister moves that generate equivalence, and the hierarchy of invariants used to distinguish them — from classical invariants like the knot group, Seifert genus, and Alexander polynomial to the quantum invariants such as the Jones and HOMFLY polynomials and their categorifications. The braid groups, which present links through closures, and connections to three- and four-dimensional topology are included, while general algebraic-topology machinery is treated in its own area.

Sub-topics

Core questions

When are two knot diagrams equivalent, and how do the Reidemeister moves answer this?
Which invariants can distinguish knots, and how complete or incomplete are they?
How do algebraic structures like the braid group and the Temperley-Lieb algebra generate knot invariants?
How does knot theory in three dimensions connect to the topology of four-manifolds?

Key concepts

Knots, links, and ambient isotopy
Knot diagrams and Reidemeister moves
Classical invariants: knot group, genus, Alexander polynomial
Quantum invariants: Jones and HOMFLY polynomials
Braid groups and braid closures

Clinical relevance

Knot theory illuminates the topology of DNA and the action of topoisomerase enzymes, the statistical mechanics behind the Jones polynomial, and questions in quantum computing and topological field theory where knot invariants arise as physical quantities.

History

Originating in Tait's 19th-century tabulation of knots, the subject gained rigor with Reidemeister's moves and the Alexander polynomial in the 1920s and 1930s, and was transformed in 1984 by Jones's discovery of a new polynomial invariant from operator algebras, opening the era of quantum invariants.

Key figures

Kurt Reidemeister
John Conway
Vaughan Jones

Seminal works

lickorish1997
rolfsen1976

Frequently asked questions

When are two knots considered the same?: When one can be continuously deformed into the other within space without cutting — formally, when they are related by an ambient isotopy, equivalently when their diagrams differ by a finite sequence of Reidemeister moves.
Is there a single invariant that classifies all knots?: No complete, easily computable invariant is known. Different invariants detect different features, and even strong ones like the Jones polynomial fail to distinguish all distinct knots, which keeps the classification problem open.