Wilhelm Wirtinger observed that the complements of knots in 3-space have fundamental groups with presentations of this form.
A knot K is an embedding of the one-sphere S1 in three-dimensional space R3.
(Alternatively, the ambient space can also be taken to be the three-sphere S3, which does not make a difference for the purposes of the Wirtinger presentation.)
A Wirtinger presentation is derived from a regular projection of an oriented knot.
The fundamental group is generated by loops winding around each arc.
More generally, co-dimension two knots in spheres are known to have Wirtinger presentations.
Michel Kervaire proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied: Conditions (3) and (4) are essentially the Wirtinger presentation condition, restated.
Kervaire proved in dimensions 5 and larger that the above conditions are necessary and sufficient.
Characterizing knot groups in dimension four is an open problem.
For the trefoil knot, a Wirtinger presentation can be shown to be