Signature of a knot
Encyclopedia
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
where 
and 
indicate the translates of a and b respectively in the positive and negative directions of the normal bundle
to S.
Given a basis
for 
(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.
of the knot complement. Let
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: 
. Given a 
-module 
, let 
denote the 
whose underlying 
-module is 
but where 
acts by the inverse covering transformation. Blanchfield's formulation of Poincaré duality
for
gives a canonical isomorphism 
where 
denotes the 2nd cohomology group of 
with compact supports and coefficients in 
. The universal coefficient theorem for 
gives a canonical isomorphism with 
(because the Alexander module is 
-torsion). Moreover, just like in the quadratic form formulation of Poincaré duality
, there is a canonical isomorphism of
-modules 
, where 
denotes the field of fractions of 
. This isomorphism can be thought of as a sesquilinear duality pairing 
where 
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
-module is isomorphic to 
. Let 
be any 
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
-- ie: 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.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. It may be computed from the Seifert surface
Seifert surface
In mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
.
Given a knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
K in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
, it has a Seifert surface
Seifert surface
In mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
S whose boundary is K. The Seifert form of S is the pairing
given by taking the linking numberLinking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other...
where and indicate the translates of a and b respectively in the positive and negative directions of the normal bundleNormal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding .-Riemannian manifold:...
to S.
Given a basis
for (where g is the genus of the surface) the Seifert form can be represented as a 2g-by-2g Seifert matrix V, . The signatureSymmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
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.
The Alexander module formulation
Knot signatures can also be defined in terms of the Alexander moduleAlexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
of the knot complement. Let
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: . Given a -module , let denote the whose underlying -module is but where acts by the inverse covering transformation. Blanchfield's formulation of Poincaré dualityPoincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
for
gives a canonical isomorphism where denotes the 2nd cohomology group of with compact supports and coefficients in . The universal coefficient theorem for gives a canonical isomorphism with (because the Alexander module is -torsion). Moreover, just like in the quadratic form formulation of Poincaré dualityPoincaré duality
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
, there is a canonical isomorphism of
-modules , where denotes the field of fractions of . This isomorphism can be thought of as a sesquilinear duality pairing where denotes the field of fractions of . This form takes value in the rational polynomials whose denominators are the Alexander polynomialAlexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
of the knot, which as a
-module is isomorphic to . Let be any 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
-- ie: 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.