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
given by taking the linking number
indicate the translates of a and b respectively in the positive and negative directions of the normal bundle to S. Given a basis
(where g is the genus of the surface) the Seifert form can be represented as a 2g-by-2g Seifert matrix V,
The signature of the matrix
, thought of as a symmetric bilinear form, is the signature of the knot K. Slice knots are known to have zero signature.
Knot signatures can also be defined in terms of the Alexander module of the knot complement.
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:
acts by the inverse covering transformation.
Blanchfield's formulation of Poincaré duality for
denotes the 2nd cohomology group of
with compact supports and coefficients in
The universal coefficient theorem for
(because the Alexander module is
Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of
{\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} ])}
denotes the field of fractions of
This isomorphism can be thought of as a sesquilinear duality pairing
denotes the field of fractions of
This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a
be any linear function which is invariant under the involution
, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on
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
: the prime power decomposition gives an orthogonal decomposition of
Cherry Kearton has shown how to compute the Milnor signature invariants from this pairing, which are equivalent to the Tristram-Levine invariant.