Signature of a knot

The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.
Given a knot "K" in the 3-sphere, it has a Seifert surface "S" whose boundary is "K". The Seifert form of "S" is the pairing formula_1 given by taking the linking number formula_2 where formula_3 and formula_4 indicate the translates of "a" and "b" respectively in the positive and negative directions of the normal bundle to "S".
Given a basis formula_5 for formula_6 (where "g" is the genus of the surface) the Seifert form can be represented as a "2g"-by-"2g" Seifert matrix "V", formula_7. The signature of the matrix formula_8, thought of as a symmetric bilinear form, is the signature of the knot "K".
Slice knots are known to have zero signature.
The Alexander module formulation.
Knot signatures can also be defined in terms of the Alexander module of the knot complement. Let formula_9 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: formula_10. Given a formula_11-module formula_12, let formula_13 denote the formula_11-module whose underlying formula_15-module is formula_12 but where formula_17 acts by the inverse covering transformation. Blanchfield's formulation of Poincaré duality for formula_9 gives a canonical isomorphism formula_19 where formula_20 denotes the 2nd cohomology group of formula_9 with compact supports and coefficients in formula_15. The universal coefficient theorem for formula_20 gives a canonical isomorphism with formula_24 (because the Alexander module is formula_11-torsion). Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of formula_11-modules formula_27, where formula_28 denotes the field of fractions of formula_11. This isomorphism can be thought of as a sesquilinear duality pairing formula_30 where formula_28 denotes the field of fractions of formula_11. This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a formula_11-module is isomorphic to formula_34. Let formula_35 be any linear function which is invariant under the involution formula_36, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on formula_37 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 formula_37—i.e.: the prime power decomposition gives an orthogonal decomposition of formula_39. Cherry Kearton has shown how to compute the "Milnor signature invariants" from this pairing, which are equivalent to the "Tristram-Levine invariant".