# Definition:Seifert Matrix

## Definition

For a knot $K$ with Seifert surface $S$, the Seifert matrix $V$ of $K$ is defined by its entries as:

$v_{ij} = \operatorname{lk} \left({ x_i, x_k^* }\right)$

where:

the $x_a$ are the generators of the fundamental group $\pi_1(S)$
$x_a^*$ is the positive push-off of $x_a$
$\operatorname{lk}$ is the linking number of the two loops.

## Source of Name

This entry was named for Karl Johannes Herbert Seifert.