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

• Link concordance

• C.Gordon, Some aspects of classical knot theory. Springer Lecture Notes in Mathematics 685. Proceedings Plans-sur-Bex Switzerland 1977.
• J.Hillman, Algebraic invariants of links. Series on Knots and everything. Vol 32. World Scientific.
• C.Kearton, Signatures of knots and the free differential calculus, Quart. J. Math. Oxford (2), 30 (1979).
• J.Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44, 229-244 (1969)
• J.Milnor, Infinite cyclic coverings, J.G. Hocking, ed. Conf. on the Topology of Manifolds, Prindle, Weber and Schmidt, Boston, Mass, 1968 pp. 115–133.
• K.Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117, 387-482 (1965)
• A.Ranicki On signatures of knots Slides of lecture given in Durham on 20 June 2010.
• H.Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76, 464-498 (1962)