Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert[1][2]) is an orientable 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 embedded in 3-space whose boundary is L such that the orientation on L is just the induced orientation from S. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link.

A single knot or link can have many different inequivalent Seifert surfaces.

The standard Möbius strip has the unknot for a boundary but is not a Seifert surface for the unknot because it is not orientable.

The "checkerboard" coloring of the usual minimal crossing projection of the trefoil knot gives a Mobius strip with three half twists.

As with the previous example, this is not a Seifert surface as it is not orientable.

This theorem was first published by Frankl and Pontryagin in 1930.

The algorithm produces a Seifert surface

, given a projection of the knot or link in question.

Suppose that link has m components (m = 1 for a knot), the diagram has d crossing points, and resolving the crossings (preserving the orientation of the knot) yields f circles.

is constructed from f disjoint disks by attaching d bands.

is free abelian on 2g generators, where is the genus of

the linking number in Euclidean 3-space (or in the 3-sphere) of ai and the "pushoff" of aj in the positive direction of

More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of

The Alexander polynomial is computed from the Seifert matrix by

The Alexander polynomial is independent of the choice of Seifert surface

Seifert surfaces are not at all unique: a Seifert surface S of genus g and Seifert matrix V can be modified by a topological surgery, resulting in a Seifert surface S′ of genus g + 1 and Seifert matrix The genus of a knot K is the knot invariant defined by the minimal genus g of a Seifert surface for K. For instance: A fundamental property of the genus is that it is additive with respect to the knot sum: In general, the genus of a knot is difficult to compute, and the Seifert algorithm usually does not produce a Seifert surface of least genus.

of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus

is the least genus of all Seifert surfaces whose complement in

obviously holds, so in particular these invariants place upper bounds on the genus.

[5] The knot genus is NP-complete by work of Ian Agol, Joel Hass and William Thurston.

[6] It has been shown that there are Seifert surfaces of the same genus that do not become isotopic either topologically or smoothly in the 4-ball.