Seifert surface

In mathematics, a Seifert surface is a surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research.

Specifically, let L be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface S whose boundary is L such that the orientation on L is just the induced orientation from S.

It is a theorem that there always exists such a surface. This theorem was first published by F. Frankl and Lev Pontrjagin in 1930. A different proof, which is algorithmic (given a knot diagram for L), was published in 1934 by Herbert Seifert.

Note that any compact, connected, oriented surface with nonempty boundary is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. It is important to note that a Seifert surface must be oriented. It is possible to associated unoriented surfaces to knots as well. For example, the standard Möbius strip has the unknot for a boundary but is not considered to be a Seifert surface for the unknot.

Genus of a knot

One example of a knot invariant which is computed from a Seifert surface is the genus of a knot. The genus of a knot K is defined as minimal genus of all Seifert surfaces for K.

For instance:

• An unknot — which is, by definition, the boundary of a disc — has genus zero. Moreover, the unknot is the only knot with genus zero.
• The trefoil knot has genus one, as does the figure-eight knot.
• The genus of a (p,q)-torus knot is (p − 1)(q − 1)/2

A fundamental property of the genus is that it is additive with respect to the knot sum: