= Dehn presentation of a knot group
{c}
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 \b[overpassing strand is horizontal]. Name the northwest, southwest, southeast and northeast regions $a,b,c,d$, respectively. In a consistent Dehn-generator convention, impose
$$
ab^{-1}cd^{-1}=1.
$$
The overpassing meridian can be read on either side of the crossing as $ab^{-1}$ or $dc^{-1}$; 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 $\mathbb Z$.<Lecture 3. Fundamental groups and knot groups by S. Kim and V. O. Manturov>[Dehn-presentation figure and region generators, slide 21]{ref}
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