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

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.