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

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.