Knot group (${\pi }_1(E_K)$)

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.