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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]

S. Kim (knot theory) coauthored lectures on fundamental groups and knot groups with V. O. Manturov. The primary title page identifies the author only as S. Kim; a fuller personal identity is not inferred from those initials.^[ref]

The lectures explain the van Kampen theorem and derive diagrammatic group presentations. A Dehn presentation of a knot group assigns generators to regions and a relation at each crossing, converting topological gluing into algebra. This supplies a reusable route from knot diagrams to invariants of their complements.^[ref]

V. O. Manturov works on knot theory. With S. Kim (knot theory), he authored lectures connecting fundamental groups with diagrams of knots. The displayed title page verifies both sets of initials.^[ref]

Their presentation of knot groups explains how loops in a knot complement can be encoded by generators and relations. Wirtinger presentations use arcs, whereas a Dehn presentation of a knot group uses regions and crossing relations. The two constructions show how a picture supplies algebraic information about an otherwise difficult three-dimensional complement.^[ref]