knot

A knot, in mathematics, is an embedding of the circle S^1 into three-dimensional Euclidean space (often R^3) considered up to ambient isotopy. Physical knots are tied with rope or string; they can be untied only if the string is allowed to pass through itself. In mathematics, a knot has no thickness or clasp and is studied as an idealized closed curve.

In knot theory, knots are studied via projections to a plane, forming knot diagrams with over- and under-crossings. Two diagrams represent the same knot if they can be related by Reidemeister moves of types I, II, and III.

Knot invariants assign quantities that stay constant under ambient isotopy. Examples include the knot group (the fundamental group), the Alexander polynomial, and the Jones polynomial, along with more modern quantum and finite-type invariants. No single invariant fully classifies all knots.

Common simple knots include the trefoil (3_1), the figure-eight (4_1), and the cinquefoil (5_1). Knots can be connected via connected sum to form composite knots; primes are knots that cannot be decomposed.

Related objects are links, consisting of two or more knots that do not intersect. In applications, knot theory informs DNA topology, molecular chemistry, and physics, where knotted fields and excitations can arise.

Historically, knot theory arose in the 19th century from practical knot tying and Kelvin’s vortex theory, with foundational work by Reidemeister. The discovery of the Alexander polynomial (1928) and the Jones polynomial (1984) spurred rapid development and ongoing research.