undercrossing

Undercrossing is a term used principally in knot theory to describe the way two strands cross in a knot or link diagram. At a crossing, one strand passes over the other; the strand that goes beneath is called the undercrossing, while the strand that lies on top is the overcrossing. In standard graphical conventions, the overcrossing strand appears uninterrupted as it crosses the other, whereas the undercrossing strand continues beneath the top strand, often drawn with the top strand appearing to pass over it.

The designation of overcrossing and undercrossing is local to the crossing and is preserved under ambient isotopy; along with the other crossings, it defines the diagram of a knot or link. The collection of all undercrossings and overcrossings in a diagram determines its projection, and changes to these statuses—crossing changes—can transform one knot into a different knot or link. Such changes are central to many skein relations used to define knot invariants, such as the Jones polynomial and the Kauffman bracket.

In practical terms, undercrossings are part of the descriptive language used when drawing knots, weaving patterns, or describing braided structures. They do not, by themselves, determine the three-dimensional geometry of a knot but encode essential topological information about how strands interlace.

See also: overcrossing, knot theory, Reidemeister moves, skein relation.