# Seifert surface

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 unoriented (and not necessarily orientable) surfaces to knots as well.

## Examples

The standard Möbius strip has the unknot for a boundary but is not considered to be 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 Pontrjagin 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 *2g* generators, where

*g*= (2 +*d*−*f*−*m*)/2

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

*a*_{1},a_{2},...,a_{2g}

with

*Q=(Q(a*_{i},a_{j}))

the direct sum of *g* copies of

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

*V=(v(i,j))*has

the linking number in Euclidean 3-space (or in the 3-sphere) of *a _{i}* and the pushoff of

*a*out of the surface, with

_{j}where *V ^{*}=(v(j,i))* the transpose matrix. Every integer

*2g*

*2g*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 in the indeterminate of degree . 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 surgery, to be replaced by 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 one, as does the figure-eight knot.
- The genus of a (
*p*,*q*)-torus knot is (*p*− 1)(*q*− 1)/2 - The degree of the Alexander polynomial is a lower bound on twice the genus of the knot.

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

## See also

## References

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## External links

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