Two-generator knot groups have cyclic Alexander modules

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.