Seifert matrix
A matrix associated with knots and links in order to investigate their topological properties by algebraic methods (cf. Knot theory). Named after H. Seifert [1], who applied the construction to obtain algebraic invariants of one-dimensional knots in . Let
be an
-dimensional
-component link, i.e. a pair consisting of an oriented sphere
and a differentiable or piecewise-linear oriented submanifold
of this sphere which is homeomorphic to the disconnected union of
copies of the sphere
. There exists a compact
-dimensional orientable submanifold
of
such that
; it is known as the Seifert manifold of the link
. The orientation of the Seifert manifold
is determined by the orientation of its boundary
; since the orientation of
is fixed, the normal bundle to
in
turns out to be oriented, so that one can speak of the field of positive normals to
. Let
be a small displacement along this field, where
is the complement to an open tubular neighbourhood of
in
. If
is odd, one defines a pairing
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associating with an element the linking coefficient of the classes
and
. This
is known as the Seifert pairing of the link
. If
and
are of finite order, then
. The following formula is valid:
![]() |
where the right-hand side is the intersection index (in homology) of the classes and
on
.
Let be a basis for the free part of the group
. The
-matrix
with integer entries is called the Seifert matrix of
. The Seifert matrix of any
-dimensional knot has the following property: The matrix
is unimodular (cf. Unimodular matrix), and for
the signature of the matrix
is divisible by
(
is the transpose of
). Any square matrix
with integer entries is the Seifert matrix of some
-dimensional knot if
, and the matrix
is unimodular.
The Seifert matrix itself is not an invariant of the link ; the reason is that the construction of the Seifert manifold
and the choice of the basis
are not unique. Matrices of the form
![]() |
where is a row-vector and
a column-vector, are known as elementary expansions of
, while
itself is called an elementary reduction of its elementary expansions. Two square matrices are said to be
-equivalent if one can be derived from the other via elementary reductions, elementary expansions and unimodular congruences (i.e. transformations
, where
is a unimodular matrix). For higher-dimensional knots
and one-dimensional links
the
-equivalence class of the Seifert matrix is an invariant of the type of the link
. In case
is a knot, the Seifert matrix
uniquely determines a
-module
, where
is an infinite cyclic covering of the complement of the knot. The polynomial matrix
is the Alexander matrix (see Alexander invariants) of the module
. The Seifert matrix also determines the
-dimensional homology and the linking coefficients in the cyclic coverings of the sphere
that ramify over the link.
References
[1] | H. Seifert, "Ueber das Geschlecht von Knoten" Math. Ann. , 110 (1934) pp. 571–592 |
[2] | R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963) |
[3] | J. Levine, "Polynomial invariants of knots of codimension two" Ann. of Math. , 84 (1966) pp. 537–554 |
[4] | J. Levine, "An algebraic classification of some knots of codimension two" Comment. Math. Helv. , 45 (1970) pp. 185–198 |
Comments
For a description of the Seifert manifold in the case , i.e. the Seifert surface of a link, cf. Knot and link diagrams.
Seifert matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Seifert_matrix&oldid=48646