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.

Give every region of a knot diagram a generator and set the generator of the unbounded region to the identity. At a crossing, reading the four incident regions cyclically gives a relation $a{b}^{-1}c{d}^{-1}=1$, with the cyclic order reversed if the opposite convention is chosen. These relations form the Dehn presentation of the knot group.

An Alexander numbering assigns an integer to every region of an oriented link diagram so that crossing an oriented arc from right to left increases the integer by one. Under abelianization, a Dehn region generator with number $m$ maps to $t^m$.

A Wirtinger presentation has one meridional generator for every arc of a knot diagram and one conjugacy relator at every crossing. For a connected diagram one crossing relator is redundant, giving a deficiency-one presentation of the knot group.

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 $\mathbb{Z}[t^{\pm 1}]$.