Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) 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 surfaces to knots which are not oriented nor orientable, as well.
Examples.
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
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. 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 formula_2, 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 formula_2 is constructed from "f" disjoint disks by attaching "d" bands. The homology group formula_4 is free abelian on 2"g" generators, where
is the genus of formula_2. The intersection form "Q" on formula_4 is skew-symmetric, and there is a basis of 2"g" cycles
with
the direct sum of "g" copies of
The 2"g" × 2"g" integer Seifert matrix
has formula_12 the linking number in Euclidean 3-space (or in the 3-sphere) of "a""i" and the "pushoff" of "a""j" in the positive direction of formula_2. More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of formula_2 to an embedding of formula_15, given some representative loop formula_16 which is homology generator in the interior of formula_2, the positive pushout is formula_18 and the negative pushout is formula_19.
With this, we have
where "V"* = ("v"("j","i")) the transpose matrix. Every integer 2"g" × 2"g" matrix formula_21 with formula_22 arises as the Seifert matrix of a knot with genus "g" Seifert surface.
The Alexander polynomial is computed from the Seifert matrix by formula_23 which is a polynomial of degree at most 2"g" in the indeterminate formula_24 The Alexander polynomial is independent of the choice of Seifert surface formula_25 and is an invariant of the knot or link.
The signature of a knot is the signature of the symmetric Seifert matrix formula_26 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
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:
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 formula_29 of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus formula_30 is the least genus of all Seifert surfaces whose complement in formula_31 is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality formula_32 obviously holds, so in particular these invariants place upper bounds on the genus.