Signature of a knot explained
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.
The Alexander module formulation
Knot signatures can also be defined in terms of the Alexander module 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
-module whose underlying
-module is
but where
acts by the inverse covering transformation. Blanchfield's formulation of
Poincaré duality for
gives a canonical isomorphism
H1(X;Q)\simeq\overline{H2(X;Q)}
where
denotes the 2nd cohomology group of
with compact supports and coefficients in
. The universal coefficient theorem for
gives a canonical isomorphism with
\operatorname{Ext}Q[Z](H1(X;Q),Q[Z])
(because the Alexander module is
-torsion). Moreover, just like in the
quadratic form formulation of Poincaré duality, there is a canonical isomorphism of
-modules
\operatorname{Ext}Q[Z](H1(X;Q),Q[Z])\simeq\operatorname{Hom}Q[Z](H1(X;Q),[Q[Z]]/Q[Z])
, where
denotes the field of fractions of
. This isomorphism can be thought of as a sesquilinear duality pairing
H1(X;Q) x H1(X;Q)\to[Q[Z]]/Q[Z]
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 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
—i.e.: 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.
See also
References
- 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)