In mathematics, a **Seifert surface** (named after German mathematician Herbert Seifert^{[1]}^{[2]}) 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* embedded in 3-space whose boundary is *L* such that the orientation on *L* is just the induced orientation from *S*, and every connected component of *S* has non-empty boundary.

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. A Seifert surface must be oriented. It is possible to associate surfaces to knots which are not oriented nor orientable, as well.

## Examples

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. Applying Seifert's algorithm to this diagram, as expected, does produce a Seifert surface; in this case, it is a punctured torus of genus *g* = 1, and the Seifert matrix is

## Existence and Seifert matrix

It is a theorem that any link always has an associated Seifert surface. This theorem was first published by Frankl and Pontryagin in 1930.^{[3]} A different proof was published in 1934 by Herbert Seifert and relies on what is now called the Seifert algorithm. 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. Then the surface is constructed from *f* disjoint disks by attaching *d* bands. The homology group is free abelian on 2*g* generators, where

is the genus of . The intersection form *Q* on is skew-symmetric, and there is a basis of 2*g* cycles

with

the direct sum of *g* copies of

- .

The 2*g* × 2*g* integer **Seifert matrix**

has the linking number in Euclidean 3-space (or in the 3-sphere) of *a*_{i} and the "pushoff" of *a*_{j} in the positive direction of . More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of to an embedding of , given some representative loop which is homology generator in the interior of , the positive pushout is and the negative pushout is .^{[4]}

With this, we have

where *V*^{*} = (*v*(*j*,*i*)) the transpose matrix. Every integer 2*g* × 2*g* matrix with arises as the Seifert matrix of a knot with genus *g* Seifert surface.

The Alexander polynomial is computed from the Seifert matrix by which is a polynomial of degree at most 2*g* in the indeterminate The Alexander polynomial is independent of the choice of Seifert surface and is an invariant of the knot or link.

The signature of a knot is the signature of the symmetric Seifert matrix It is again an invariant of the knot or link.

## Genus of a knot

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:

- 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 1, as does the figure-eight knot.
- The genus of a (
*p*,*q*)-torus knot is (*p*− 1)(*q*− 1)/2 - The degree of a knot's Alexander polynomial is a lower bound on twice its genus.

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. For this reason other related invariants are sometimes useful. The **canonical 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 is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality obviously holds, so in particular these invariants place upper bounds on the genus.^{[5]}

## See also

## References

**^**Seifert, H. (1934). "Über das Geschlecht von Knoten".*Math. Annalen*(in German).**110**(1): 571–592. doi:10.1007/BF01448044.**^**van Wijk, Jarke J.; Cohen, Arjeh M. (2006). "Visualization of Seifert Surfaces".*IEEE Transactions on Visualization and Computer Graphics*.**12**(4): 485–496. doi:10.1109/TVCG.2006.83. PMID 16805258.**^**Frankl, F.; Pontrjagin, L. (1930). "Ein Knotensatz mit Anwendung auf die Dimensionstheorie".*Math. Annalen*(in German).**102**(1): 785–789. doi:10.1007/BF01782377.**^**Dale Rolfsen. Knots and Links. (1976), 146-147.**^**Brittenham, Mark (24 September 1998). "Bounding canonical genus bounds volume". arXiv:math/9809142.

## External links

- The SeifertView programme of Jack van Wijk visualizes the Seifert surfaces of knots constructed using Seifert's algorithm.