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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.

Thicken the planar graph underlying a knot diagram. Its boundary supports a Heegaard diagram with face disks on one side and crossing-tunnel disks on the other. The face generators and crossing curves give a Dehn presentation of a knot group.

Use the region labels and curve orientations from part (a). A Dehn presentation of a knot group generator goes from the basepoint above the projection plane through its region and returns through the unbounded region. Equivalently it is dual to the correspondingly labeled $\alpha$ disk of the Heegaard diagram. A crossing relator records the signed intersections of its $\beta$ curve with these disks. Setting $o=1$ givesFor example, the overpassing strand at $A$ equates the meridian of a knot words $d$ and $a{e}^{-1}$; at $B$ it equates $a{d}^{-1}$ and $b{c}^{-1}$. The other two crossings equate $b{a}^{-1}$ with $c{e}^{-1}$ and $c{d}^{-1}$ with $e$.

Choose the region-generator direction so that $e$ is the shown meridian of a knot $\mu$. The Alexander numbering of the regions, starting from zero outside and increasing by one on crossing an oriented strand from right to left, isOne can also obtain this directly by abelianization of the relators: $a=d+e$, $c=d+e$, $b=a-d+c$, and $b-a+e-c=0$. The last two imply $d=e$, and the resulting first homology group is freely generated by $e$. Thus $H_1(X_8;\mathbb{Z})\cong \mathbb{Z}$, with no extra torsion.

Label the unbounded region $r_0$ and the three lobe regions $a,b,c$ in cyclic order, and label the central region $d$. The Dehn presentation of a knot group Heegaard diagram is obtained by thickening the projection graph: take one alpha compressing curve for each of $a,b,c,d$, one beta curve for each crossing, and place the basepoint in $r_0$. Orient and order the crossing curves so that their cyclic region words areThis intersection data, together with $r_0=1$, specifies the requested pointed Heegaard diagram up to the usual isotopies and handle slides.

Give each bounded region $R_i$ a generator $a_i$ and give the unbounded region $R_0$ the identity generator. At every crossing, read the four incident regions cyclically as $a,b,c,d$ and imposeUsing the opposite cyclic convention inverts all such relators and gives the same group. One crossing relation is redundant, leaving $n$ generators and $n-1$ relators; this is the Dehn presentation of a knot group.

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]