= 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.
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