Knot signatures can also be defined in terms of the
Alexander module of the knot complement. Let X be the universal abelian cover of the knot complement. Consider the Alexander module to be the first
homology group of the universal abelian cover of the knot complement: H_1(X;\mathbb Q). Given a \mathbb Q[\mathbb Z]-module V, let \overline{V} denote the \mathbb Q[\mathbb Z]-module whose underlying \mathbb Q-module is V but where \mathbb Z acts by the inverse covering transformation. Blanchfield's formulation of
Poincaré duality for X gives a canonical isomorphism H_1(X;\mathbb Q) \simeq \overline{H^2(X;\mathbb Q)} where H^2(X;\mathbb Q) denotes the 2nd cohomology group of X with compact supports and coefficients in \mathbb Q. The universal coefficient theorem for H^2(X;\mathbb Q) gives a canonical isomorphism with \operatorname{Ext}_{\mathbb Q[\mathbb Z]}(H_1(X;\mathbb Q),\mathbb Q[\mathbb Z]) (because the Alexander module is \mathbb Q[\mathbb Z]-torsion). Moreover, just like in the
quadratic form formulation of Poincaré duality, there is a canonical isomorphism of \mathbb Q[\mathbb Z]-modules \operatorname{Ext}_{\mathbb Q[\mathbb Z]}(H_1(X;\mathbb Q),\mathbb Q[\mathbb Z]) \simeq \operatorname{Hom}_{\mathbb Q[\mathbb Z]}(H_1(X;\mathbb Q),[\mathbb Q[\mathbb Z/\mathbb Q[\mathbb Z] ), where [\mathbb Q[\mathbb Z denotes the field of fractions of \mathbb Q[\mathbb Z]. This isomorphism can be thought of as a
sesquilinear duality pairing H_1(X;\mathbb Q) \times H_1(X;\mathbb Q) \to [\mathbb Q[\mathbb Z/\mathbb Q[\mathbb Z] where [\mathbb Q[\mathbb Z denotes the field of fractions of \mathbb Q[\mathbb Z]. This form takes value in the rational polynomials whose denominators are the
Alexander polynomial of the knot, which as a \mathbb Q[\mathbb Z]-module is isomorphic to \mathbb Q[\mathbb Z]/\Delta K. Let tr : \mathbb Q[\mathbb Z]/\Delta K \to \mathbb Q be any linear function which is invariant under the involution t \longmapsto t^{-1}, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on H_1 (X;\mathbb Q) whose signature is an invariant of the knot. All such signatures are concordance invariants, so all signatures of
slice knots are zero. The sesquilinear duality pairing respects the prime-power decomposition of H_1 (X;\mathbb Q)—i.e.: the prime power decomposition gives an orthogonal decomposition of H_1 (X;\mathbb R). Cherry Kearton has shown how to compute the
Milnor signature invariants from this pairing, which are equivalent to the
Tristram-Levine invariant. ==See also==