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 $S^3\setminus K$.

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

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 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.

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