Seifert surface

A Seifert surface bounded by a set of Borromean rings.

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) 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

A Seifert surface for the Hopf link. This is an annulus, not a Möbius strip. It has two half-twists and is thus orientable.

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

$V={\begin{pmatrix}1&-1\\0&1\end{pmatrix}}.$

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. 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 $S$, 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 $S$ is constructed from f disjoint disks by attaching d bands. The homology group $H_{1}(S)$ is free abelian on 2g generators, where

$g=(2+d-f-m)/2$

is the genus of $S$. The intersection form Q on $H_{1}(S)$ is skew-symmetric, and there is a basis of 2g cycles

$a_{1},a_{2},\ldots ,a_{2g}$

with

$Q=(Q(a_{i},a_{j}))$

the direct sum of g copies of

${\begin{pmatrix}0&-1\\1&0\end{pmatrix}}$.

An illustration of the "pushoff" of a homology generator a in the positive and negative directions for the Seifert surface of the figure eight knot.

The 2g × 2g integer Seifert matrix

$V=(v(i,j))$

has $v(i,j)$ 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 $S$. More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of $S$ to an embedding of $S\times [-1,1]$, given some representative loop $x$ which is homology generator in the interior of $S$, the positive pushout is $x\times \{1\}$ and the negative pushout is $x\times \{-1\}$.

With this, we have

$V-V^{*}=Q,$

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

The Alexander polynomial is computed from the Seifert matrix by $A(t)=\det(V-tV^{*}),$ which is a polynomial of degree at most 2g in the indeterminate $t.$ The Alexander polynomial is independent of the choice of Seifert surface $S,$ and is an invariant of the knot or link.

The signature of a knot is the signature of the symmetric Seifert matrix $V+V^{\mathrm {T} }.$ 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

$V'=V\oplus {\begin{pmatrix}0&1\\1&0\end{pmatrix}}.$

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:

$g(K_{1}\#K_{2})=g(K_{1})+g(K_{2})$

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 $g_{c}$ of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus $g_{f}$ is the least genus of all Seifert surfaces whose complement in $S^{3}$ is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality $g\leq g_{f}\leq g_{c}$ obviously holds, so in particular these invariants place upper bounds on the genus.

See also

• Crosscap number
• Arf invariant of a knot

External links

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