# 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.

A Seifert surface bounded by a set of Borromean rings.

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

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

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

${\displaystyle g=(2+d-f-m)/2}$

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

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

with

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

the direct sum of g copies of

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

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

${\displaystyle V=(v(i,j))}$

has ${\displaystyle v(i,j)}$  the linking number in Euclidean 3-space (or in the 3-sphere) of ai and the "pushoff" of aj in the positive direction of ${\displaystyle S}$ , i.e. the loop that sits in the complement of S above the positive side and projects down onto aj.[4]

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

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

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

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

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

The genus of a knot K is the knot invariant defined by the minimal genus g of a Seifert surface for K.

For instance:

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

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

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 ${\displaystyle g_{c}}$  of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus ${\displaystyle g_{f}}$  is the least genus of all Seifert surfaces whose complement in ${\displaystyle S^{3}}$  is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality ${\displaystyle g\leq g_{f}\leq g_{c}}$  obviously holds, so in particular these invariants place upper bounds on the genus.[5]