Knoopvlak

Knoopvlak is a term used in knot theory to denote the plane onto which a knot is projected to produce a knot diagram. It refers to the two-dimensional projection surface, typically a plane in three-dimensional space, used for visualizing and studying the knot by drawing its crossings and strands in the plane.

To obtain a knoopvlak diagram, a knot K embedded in three-dimensional Euclidean space is projected orthogonally (or by another chosen projection) onto a chosen plane P. The resulting image is a planar curve with a finite number of transverse double points, called crossings. At each crossing, information is added to indicate which strand passes over and which passes under. In general position, the projection yields a regular diagram, and different planes may produce different but equivalent diagrams of the same knot.

Knoopvlak diagrams are foundational in knot theory because many knot invariants are defined from diagrams, such as crossing number, writhe, and polynomial invariants. While the diagram depends on the chosen knoopvlak, ambient isotopy of the knot in space corresponds to Reidemeister moves on the diagrams. Thus, invariants computed from a diagram are properties of the knot itself, not of the particular knoopvlak used.

Examples include projecting the unknot onto a plane to obtain a simple circle with no crossings, or projecting a trefoil knot to yield a diagram with three crossings. The concept emphasizes the role of the projection plane in translating three-dimensional knot geometry into two-dimensional combinatorial data for study. See also knot theory, knot diagram, Reidemeister moves, and projection plane.