Knot diagram

A knot diagram is a generic projection of a knot to a plane together with overcrossing and undercrossing information at every double point.

A knot diagram is reduced when it has no nugatory crossing, equivalently no crossing can be isolated from the rest of the diagram by a circle meeting the diagram only at that crossing.

An alternating knot diagram alternates between overcrossings and undercrossings while traversing its component.

The crossing number $c(K)$ is the least number of crossings in any knot diagram of $K$.

Every reduced alternating diagram of a link has the minimum possible number of crossings. This result is one of the former Tait conjectures.

The three Reidemeister moves are local changes of a link diagram that create or remove a kink, create or remove two opposite crossings, or slide one strand past a crossing. Two diagrams represent isotopic links exactly when they are related by planar isotopy and finitely many Reidemeister moves.