- Seifert surface
mathematics, a Seifert surface is a surfacewhose 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 invariantsare 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
orientedknot 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-spaceis 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.
Möbius striphas the unknotfor 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 knotgives 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
theoremthat 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 1934by Herbert Seifertand relies on what is now called the Seifert algorithm. The algorithmproduces a Seifert surface , 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 is constructed from f disjoint disks by attaching d bands. The homology group is free abelian on 2g generators, where
:"g" = (2 + "d" − "f" − "m")/2
genusof . The intersection formQ on is skew-symmetric, and there is a basis of 2g cycles
the direct sum of g copies of
The 2g2g integer Seifert matrix
linking numberin Euclidean 3-space(or in the 3-sphere) of ai and the pushoff of aj out of the surface, with
where V*=(v(j,i)) the transpose matrix. Every integer 2g2g matrix with * arises as the Seifert matrix of a knot with genus g Seifert surface.
Alexander polynomialis computed from the Seifert matrix by *), which is a polynomial in the indeterminate of degree . The Alexander polynomial is independent of the choice of Seifert surface , and is an invariant of the knot or link.
signature of a knotis the signature of the symmetric Seifert matrix . 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
The genus of a knot "K" is the
knot invariantdefined by the minimal genus g of a Seifert surface for "K".
unknot— which is, by definition, the boundary of a disc — has genus zero. Moreover, the unknot is the "only" knot with genus zero.
trefoil knothas genus one, as does the figure-eight knot.
* The genus of a ("p","q")-
torus knotis ("p" − 1)("q" − 1)/2
* The degree of the
Alexander polynomialis 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
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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