nulcrossings

Nulcrossings is a term used in knot theory and related graph-theoretic representations to describe certain crossing points in a planar projection that are neutral with respect to a chosen invariant under a restricted set of diagrammatic moves. In this context, a crossing is considered a nulcrossing if changing its over-under prescription does not alter the value of the invariant when the diagram remains equivalent under the specified moves (for example, a subset of Reidemeister moves). The exact identification of nulcrossings depends on the invariant and the move-set, so different theoretical frameworks may yield different classifications for the same diagram.

The concept serves as a tool for analyzing redundancy in knot diagrams and for developing simplification procedures in computational knot theory. By isolating nulcrossings, algorithms can simplify diagrams by toggling or removing crossings that do not affect the invariant, while preserving the diagram’s essential topological information. Nulcrossings can thus help distinguish between essential structure and incidental complexity within a projection.

Origin and status: The term has appeared in expository and educational materials as a way to illustrate how local changes can impact global invariants. It is not part of a standardized formal framework and is used primarily as a conceptual aid in teaching and exploratory research, rather than as a universally adopted definition.

See also: knot theory, Reidemeister moves, knot invariant, diagrammatic simplification.