Selfcrossing

Selfcrossing refers to the phenomenon where a path or curve intersects itself. This concept is encountered in various mathematical disciplines, including geometry, topology, and graph theory. In geometry, a self-intersecting curve is one where at least two non-consecutive points on the curve occupy the same location. For example, a figure-eight curve is a simple illustration of a self-crossing shape.

In the context of knot theory, which is a branch of topology, self-crossing is fundamental. A knot is defined as an embedding of a circle in three-dimensional Euclidean space. If this embedding is not trivial (i.e., it cannot be continuously deformed into an unknotted circle), it may exhibit self-crossings when projected onto a plane. These crossings are crucial for distinguishing between different knots, with the number and type of crossings contributing to the knot's invariant properties.

In graph theory, a self-crossing can occur in a planar embedding of a graph. A planar graph is one that can be drawn on a plane such that no two edges cross each other. However, if a graph is drawn in a way that violates this condition, it is said to have self-crossing edges. The study of planar embeddings and their crossings is important in algorithms related to network design and visualization. The term can also be used more generally to describe any situation where a defined path or trajectory revisits a previously occupied point.