Seifert surface

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In mathematics, a Seifert surface 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. It is important to note that a Seifert surface must be oriented. It is possible to associate unoriented (and not necessarily orientable) surfaces to knots as well.

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[edit] 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 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}.

[edit] Existence and Seifert matrix

It is a theorem that there always exists a Seifert surface. This theorem was first published by F. Frankl and Lev Pontrjagin 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 yields f circles. Then the surface S is constructed from f disjoint disks by attaching d bands. The homology group H1(S) is free abelian on 2g generators, where

g = (2 + d − f − m)/2

is the genus of S. The intersection form Q on H1(S) is skew-symmetric, and there is a basis of 2g cycles

a1,a2,...,a2g

with

Q=(Q(ai,aj))

the direct sum of g copies of

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

The 2g\times2g integer Seifert matrix

V=(v(i,j)) has

v(i,j) the linking number in Euclidean 3-space (or in the 3-sphere) of ai and the pushoff of aj out of the surface, with

V − V* = Q

where V*=(v(j,i)) the transpose matrix. Every integer 2g\times2g 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 in the indeterminate t of degree \leq 2g. 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^\top. It is again an invariant of the knot or link.

[edit] 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

V'=V \oplus \begin{pmatrix} 0 & 0 \\ 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 one, as does the figure-eight knot.
  • The genus of a (p,q)-torus knot is (p − 1)(q − 1)/2

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)

[edit] See also

[edit] External links

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