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
.