Seifert surface

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A Seifert surface bounded by a set of Borromean rings.

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[edit]

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 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

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

Existence and Seifert matrix[edit]

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 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 + dfm)/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

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.

Genus of a knot[edit]

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 & 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:

g(K_1 \# K_2) = g(K_1) + g(K_2)

See also[edit]

References[edit]

  1. ^ Seifert, H. (1934). "Über das Geschlecht von Knoten". Math. Annalen 110 (1): 571–592. doi:10.1007/BF01448044.  (German)
  2. ^ van Wijk, Jarke J.; Cohen, Arjeh M. (2006). "Visualization of Seifert Surfaces". IEEE Trans. on Visualization and Computer Graphics 12 (4): 485–496. doi:10.1109/TVCG.2006.83. 
  3. ^ Frankl, F.; Pontrjagin, L. (1930). "Ein Knotensatz mit Anwendung auf die Dimensionstheorie". Math. Annalen 102 (1): 785–789. doi:10.1007/BF01782377.  (German)

External links[edit]