Knotentyps

Knotentyps, or knot types, are the classification of knots used in knot theory. They are defined as equivalence classes of embeddings of a circle into three-dimensional space under ambient isotopy. In practical terms, two closed curves that can be continuously deformed into one another without cutting or passing through themselves belong to the same knot type.

A knot diagram provides a 2D projection with crossing information. The Reidemeister moves show how diagrams of equivalent knots are related. If one diagram can be transformed into another via Reidemeister moves, the underlying knot types are the same.

Distinguishing knot types relies on invariants. The minimal crossing number is a simple measure. More powerful invariants include knot polynomials such as the Alexander polynomial, the Jones polynomial, and the HOMFLY-PT polynomial; the genus; hyperbolic volume for hyperbolic knots; and properties like chirality and primeness.

Typical examples include the unknot, the trefoil knot, and the figure-eight knot; torus knots T(p,q) and composite knots formed by connected sums of prime knot types. Knot tables organize prime knot types by increasing minimal crossing number.

Computation and research: complete classification of all knot types is not achieved. Researchers and software tools (for example KnotInfo and SnapPy) compute invariants to identify knot types and study their geometric properties. The study of knot types intersects topology, geometry, and mathematical physics.

Applications and broader use: knot types appear in mathematical models of DNA and other polymers, in the study of fluid vortices, and in education as a framework for understanding topology and spatial reasoning.