# 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

$$\theta : H _ {q} V \otimes H _ {q} V \rightarrow \mathbf Z ,$$

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 (in homology) 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 $A = ( - 1 ) ^ {q} A ^ \prime$ is unimodular (cf. Unimodular matrix), and for $q = 2$ the signature of the matrix $A + A ^ \prime$ is divisible by $16$( $A ^ \prime$ 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 $A + ( - 1 ) ^ {q} A ^ \prime$ 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 $A \rightarrow P ^ \prime A P$, 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 $\mathbf Z [ t , t ^ {-} 1 ]$- module $H _ {q} \widetilde{X}$, where $\widetilde{X}$ is an infinite cyclic covering of the complement of the knot. The polynomial matrix $t A + ( - 1 ) ^ {q} A ^ \prime$ 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.

#### 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

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