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 $S^{3}$. Let $ L = ( S ^ {n+2} , l ^ {n} ) $ be an $n$-dimensional $m$- component link, i.e. a pair consisting of an oriented sphere $ S ^ {n+2} $ and a differentiable or piecewise-linear oriented submanifold $ l ^ {n} $ of this sphere which is homeomorphic to the disconnected union of $ m $ copies of the sphere $ S^{n}$. There exists a compact $ ( n+ 1) $- dimensional orientable submanifold $ V $ of $ S ^ {n+2} $ such that $ \partial V = l $; it is known as the Seifert manifold of the link $ L $. The orientation of the Seifert manifold $ V $ is determined by the orientation of its boundary $ \partial V = l $; since the orientation of $ S ^ {n+2} $ is fixed, the normal bundle to $ V $ in $ S ^ {n+2} $ turns out to be oriented, so that one can speak of the field of positive normals to $ V $. Let $ i _ {+} : V \rightarrow Y $ be a small displacement along this field, where $ Y $ is the complement to an open tubular neighbourhood of $ V $ in $ S ^ {n+2} $. If $ n = 2 q - 1 $ is odd, one defines a pairing

$$ \theta : H_q V \otimes H_q V \rightarrow \ZZ, $$

associating with an element $ z _ {1} \otimes z _ {2} $ the linking coefficient of the classes $ z _ {1} \in H _ {q} V $ and $ i _ {+} * z _ {2} \in H _ {q} Y $. This $ \theta $ is known as the Seifert pairing of the link $ L $. If $ z _ {1} $ and $ z _ {2} $ are of finite order, then $ \theta ( z _ {1} \otimes z _ {2} ) = 0 $. The following formula is valid:

$$ \theta ( z _ {1} \otimes z _ {2} ) + ( - 1 ) ^ {q} \theta ( z _ {2} \otimes z _ {1} ) = z _ {1} \cdot z _ {2} , $$

where the right-hand side is the intersection index of the classes $ z _ {1} $ and $ z _ {2} $ on $ V $.

Let $ e _ {1} \dots e _ {k} $ be a basis for the free part of the group $ H _ {q} V $. The $ ( k \times k ) $- matrix $ A = \| \theta ( e _ {i} \otimes e _ {j} ) \| $ with integer entries is called the Seifert matrix of $ L $. The Seifert matrix of any $ ( 2 q - 1 ) $- dimensional knot has the following property: The matrix $ A = ( - 1 ) ^ {q} A^t $ is unimodular (cf. Unimodular matrix), and for $ q = 2 $ the signature of the matrix $A + A^t$ is divisible by $16$ ($A^t$ is the transpose of $A$). Any square matrix $ A $ with integer entries is the Seifert matrix of some $ ( 2 q - 1 ) $- dimensional knot if $ q \neq 2 $, and the matrix $ A + ( - 1 ) ^ {q} A^t $ is unimodular.

The Seifert matrix itself is not an invariant of the link $ L $; the reason is that the construction of the Seifert manifold $ V $ and the choice of the basis $ e _ {1} \dots e _ {k} $ are not unique. Matrices of the form

$$ \left \| \begin{array}{lcc} A &{} & 0 \\ \alpha & 0 & 1 \\ 0 & 1 & 0 \\ \end{array} \right \| ,\ \ \left \| \begin{array}{lll} A &\beta & 0 \\ 0 & 0 & 1 \\ {} & 0 & 0 \\ \end{array} \right \| , $$

where $ \alpha $ is a row-vector and $ \beta $ a column-vector, are known as elementary expansions of $ A $, while $ A $ itself is called an elementary reduction of its elementary expansions. Two square matrices are said to be $ S $- equivalent if one can be derived from the other via elementary reductions, elementary expansions and unimodular congruences (i.e. transformations $ A \rightarrow P^t A P $, where $ P $ is a unimodular matrix). For higher-dimensional knots $ ( m = 1 ) $ and one-dimensional links $ ( n = 1 ) $ the $ S $- equivalence class of the Seifert matrix is an invariant of the type of the link $ L $. In case $ L $ is a knot, the Seifert matrix $ A $ uniquely determines a $\ZZ[t, t^{-1} ] $- module $ H _ {q} \widetilde{X} $, where $ \widetilde{X} $ is an infinite cyclic covering of the complement of the knot. The polynomial matrix $ t A + ( - 1 ) ^ {q} A ^t $ is the Alexander matrix (see Alexander invariants) of the module $ H _ {q} \widetilde{X} $. The Seifert matrix also determines the $ q $- dimensional homology and the linking coefficients in the cyclic coverings of the sphere $ S ^ {2q+1}$ that ramify over the link.

Comments

For a description of the Seifert manifold in the case $n = 1$, i.e. the Seifert surface of a link, see Knot and link diagrams.

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