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]

An Alexander numbering assigns an integer to each region of an oriented link diagram so that crossing an oriented arc from right to left increases the number by one. Normalize the unbounded region to zero; this fixes the otherwise arbitrary additive constant. Equivalently, the number is the total winding number of the projected oriented curves about a point of that region.

In the corresponding Dehn presentation of a knot group, a region generator of number $m$ maps under abelianization to $t^m$, when each positively oriented meridian of a knot maps to $t$. The exterior normalization is necessary: the unbounded-region generator is the identity, so its exponent must be zero. Number differences across adjacent regions recover the meridional exponent one.