 Wirtinger presentation

For a knot diagram with at least one crossing, take one oriented meridional group generator for every arc between consecutive undercrossings. At each crossing impose the conjugacy relation between the incoming and outgoing underpassing generators, conjugating by the overpassing generator with exponent fixed by the crossing sign. These generators and relations present the knot group.

One crossing relation is redundant, giving a group presentation with one more generator than relations. The redundancy follows by following the diagram's regions, or by applying the Seifert-van Kampen theorem to the crossing neighborhoods and projection complement. For a crossing-free unknot there are no crossing relations to delete: its presentation is simply $⟨\mu \mid ⟩\cong \mathbb{Z}$. This treats the zero-crossing case without claiming it has a redundant relation.

The Wirtinger presentation is a method used in algebraic topology, particularly in the study of the fundamental group of a braid or a link in three-dimensional space. It was introduced by Wilhelm Wirtinger in the early 20th century. The Wirtinger presentation gives a way to describe the fundamental group of a knot or link by associating it with a specific set of generators and a set of relations derived from a diagram of the knot or link.