knotheorie

Knot theory, or knotheorie in German, is a branch of topology that studies mathematical knots: embeddings of a circle into three-dimensional space considered up to ambient isotopy. A knot can be represented by a knot diagram, a planar projection with information at each crossing about which strand passes over the other.

Origins date to the 19th century in the study of molecular models and the problem of distinguishing knots. The subject was later formalized by mathematicians such as Kurt Reidemeister, J. W. Alexander, and Vaughan Jones, who introduced algebraic and combinatorial tools for distinguishing knots. Reidemeister moves describe three local changes on knot diagrams that do not alter knot type.

Core ideas include knot invariants: quantities or objects assigned to a knot that are preserved under ambient isotopy. Early invariants include the Alexander polynomial; later, the Jones polynomial and HOMFLY-PT polynomial provided powerful discriminants. Other invariants arise from the knot group, knot genus, and hyperbolic geometry of the knot complement, with Thurston’s work linking most knots to hyperbolic structures.

Methods combine diagrammatic techniques with algebra and geometry. Diagrammatic moves, Braids, and Dehn surgery connect knot theory to three-manifolds. Computational approaches address knot recognition, knot equivalence, and invariant calculations.

Applications span science and mathematics: in biology, DNA knotting and enzymatic processes; in chemistry, construction of molecular knots; in physics, connections to quantum field theory and topological quantum invariants. Knot theory remains active, with many problems open, including the complete classification of knots via invariants and the behavior of knots under various operations.