Seifert Form

For $K$ a given Knot in $\Bbb{S}^3$, choose a Seifert Surface $M^2$ in $\Bbb{S}^3$ for $K$ and a bicollar $\hat M\times [-1,1]$ in $\Bbb{S}^3-K$. If $x\in H_1(\hat M)$ is represented by a 1-cycle in $\hat M$, let $x^+$ denote the homology cycle carried by $x\times 1$ in the bicollar. Similarly, let $x^-$ denote $x\times -1$. The function $f:H_1(\hat M)\times H_1(\hat M)\to Z$ defined by

$$f(x,y)={\rm lk}(x,y^+),$$

where lk denotes the Linking Number, is called a Seifert form for $K$.

See also Seifert Matrix

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 200-201, 1976.