Seifert Matrix

Given a Seifert Form $f(x,y)$, choose a basis $e_1$, ..., $e_{2g}$ for $H_1(\hat M)$ as a Z-module so every element is uniquely expressible as

$$n_1e_1+\ldots+n_{2g}e_{2g}$$

with $n_i$ integer, define the Seifert matrix $V$ as the $2g\times 2g$ integral Matrix with entries

$$v_{ij}={\rm lk}(e_i,e_j^+).$$

The right-hand Trefoil Knot has Seifert matrix

$$V=\left[{\matrix{-1 & 1\cr 0 & -1\cr}}\right].$$

A Seifert matrix is not a knot invariant, but it can be used to distinguish between different Seifert Surfaces for a given knot.

See also Alexander Matrix

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200-203, 1976.