 Dehn presentation of a knot group

Assign a group generator to every complementary region of a knot diagram, and set the unbounded-region generator to the identity. At a crossing, rotate the local picture so that the overpassing strand is horizontal. Name the northwest, southwest, southeast and northeast regions $a,b,c,d$, respectively. In a consistent Dehn-generator convention, imposeThe overpassing meridian can be read on either side of the crossing as $a{b}^{-1}$ or $d{c}^{-1}$; equating them gives the relation. Thus the over/under information determines which pair of adjacent regions appears first. Merely listing four regions cyclically without that information does not specify the relation.

Together these crossing relations present the knot group. Replacing region pairs by the associated meridians of a knot gives the equivalent Wirtinger presentation. The crossing-free unknot has two regions, and setting the outer generator to the identity leaves one free generator, as required for its group $\mathbb{Z}$.^[ref]