Knot exterior ($E_K$)

The knot exterior is $E_K=S^3\setminus \mathrm{int}\nu K$, where $\nu K$ is an open tubular neighborhood of the knot. It is homotopy equivalent to the knot complement $S^3\setminus K$.

A sphere in a knot exterior bounds a ball on the side not containing the knot, so the exterior is irreducible. A compression disk for its boundary torus would cut the exterior to a manifold with sphere boundary; irreducibility makes that manifold a ball, and reversing the compression gives a solid torus exterior. Its meridian disk has boundary a preferred knot longitude, so adjoining the boundary annulus gives a spanning disk for the knot, which is then an unknot. The Loop theorem therefore makes the peripheral fundamental group injective for every nontrivial knot. In particular its knot group contains an injected copy of the torus group and cannot be cyclic.

The knot group is the fundamental group of the knot exterior.

The abelianization of a knot group is infinite cyclic. Nielsen changes of a two-generator free basis reduce its exponent vector to $(1,0)$. In the cyclic-cover cellular chain complex, the first boundary is then $(t-1,0)$, whose kernel is a free rank-one Laurent module. Quotienting by the lifted relator boundaries gives a cyclic module, regardless of the number of relators. A noncyclic Alexander module therefore obstructs every two-generator knot-group presentation.

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.

For a knot diagram with at least one crossing, take one oriented meridional group generator for every arc between consecutive undercrossings. At each crossing impose the conjugacy relation between the incoming and outgoing underpassing generators, conjugating by the overpassing generator with exponent fixed by the crossing sign. These generators and relations present the knot group.

One crossing relation is redundant, giving a group presentation with one more generator than relations. The redundancy follows by following the diagram's regions, or by applying the Seifert-van Kampen theorem to the crossing neighborhoods and projection complement. For a crossing-free unknot there are no crossing relations to delete: its presentation is simply $⟨\mu \mid ⟩\cong \mathbb{Z}$. This treats the zero-crossing case without claiming it has a redundant relation.

A Wirtinger generator is the meridian of a knot associated with an arc of a knot diagram between successive undercrossings. The overpassing meridian conjugates the incoming meridian to the outgoing meridian, with exponent determined by the crossing sign.

A meridian is an oriented simple closed curve on the boundary of a knot exterior that bounds a disk in the removed tubular neighborhood and has linking number one with the knot.

Viewed inside the removed tubular neighborhood, this is a meridian of a solid torus.

The infinite cyclic cover corresponds to the kernel of the abelianization ${\pi }_1(E_K)\to \mathbb{Z}$. Its deck group is generated by $t$, so its cellular chains and homology are modules over $\mathbb{Z}[t^{\pm 1}]$.