The Alexander invariant of a knot is the homology of the infinite cyclic cover of the complement of , considered as a module over , the ring of integral laurent polynomials. The Alexander invariant for a classical tame knot is finitely presentable, and only is significant.

For any knot in whose complement has the homotopy type of a finite CW-complex, the Alexander invariant is finitely generated and therefore finitely presentable. Because the Alexander invariant of a tame knot in has a square presentation matrix, its Alexander ideal is principal and it has an Alexander polynomial denoted .