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 maps under abelianization to , when each positively oriented meridian of a knot maps to . 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.
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