Seifert surface

Seifert surface

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.

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:

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 $H_1\left(S\right)$ is free abelian on 2g generators, where

:"g" = (2 + "d" − "f" − "m")/2

is the genus of $S$. The intersection form Q on $H_1\left(S\right)$ 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

:.

The 2g$imes$2g integer Seifert matrix

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

$v\left(i,j\right)$ 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$imes$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 $A\left(t\right)=det\left(V-tV$*), 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 $V+V^ op$. 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

:V'=V.

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 &mdash; which is, by definition, the boundary of a disc &mdash; 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::$g\left(K_1 # K_2\right) = g\left(K_1\right) + g\left(K_2\right)$

ee also

*Crosscap number
*Arf invariant (knot)

*The [http://www.win.tue.nl/~vanwijk/seifertview/ SeifertView programme] of Jack van Wijk visualizes the Seifert surfaces of knots constructed using Seifert's algorithm.

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