A Wirtinger presentation has one meridional generator for every arc of a knot diagram and one conjugacy relator at every crossing. For a connected diagram one crossing relator is redundant, giving a deficiency-one presentation of the knot group.
A Wirtinger generator is the meridian of a knot associated with an arc of a knot diagram between successive undercrossings. The overpassing meridian conjugates the incoming meridian to the outgoing meridian, with exponent determined by the crossing sign.
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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.