undercrossings

An undercrossing is a crossing in a knot diagram where the strand passes beneath another strand at that location. In planar projections of knots and links, each crossing is designated as either an overcrossing or an undercrossing according to which strand lies above the other.

In knot theory, the pair of overcrossing and undercrossing information encodes the three-dimensional structure of the knot as it is projected to two dimensions. In any diagram of a knot or link, every crossing involves one strand going over and one going under, and the particular arrangement of these crossings determines the knot or link type.

Crossings are manipulated through Reidemeister moves, which preserve the ambient isotopy class of the knot. The undercrossing information is part of the local data affected by these moves; changing an undercrossing to an overcrossing at a single location can change the topology of the diagram unless compensated by other moves.

Alternating knot diagrams feature a pattern where the strand alternates between over and under crossing as one travels along the knot. In such diagrams, undercrossings occur at regular intervals along the projection, contributing to the diagram’s alternating structure.

Outside pure knot theory, undercrossings appear in related contexts such as weaving and braid theory. In braids, for example, an undercrossing corresponds to the inverse of the standard generator, reflecting the directional nature of the crossing.

Notes of caution: the designation of undercrossing is diagram-specific; different projections of the same knot can assign different under/over patterns while representing the same knot type. The concept is essential for preserving the information needed to reconstruct the knot’s three-dimensional structure from its two-dimensional projection.