The Alexander invariant H_*(X^~) of a knot K is the homology of the infinite cyclic cover of the complement of K, considered as a module over Λ, the ring of integral laurent polynomials. The Alexander invariant for a classical tame knot is finitely presentable, and only H_1 is significant. For any knot K^n in S^(n + 2) 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 S^3 has a square presentation matrix, its Alexander ideal is principal and it has an Alexander polynomial denoted Δ(t).

Alexander ideal | Alexander matrix | Alexander polynomial