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Seifert surface

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Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is a surface whose boundary is a given knot or link.

Contents

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 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 = ( 1 − 1 0 1 ) .

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 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 (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 + d − f − m)/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

( 0 − 1 1 0 ) .

The 2g × 2g 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 × 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 ( t ) = d e t ( V − t V *), which is a polynomial in the indeterminate t of degree ≤ 2 g . 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 ⊤ . 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; in order to be replaced by a Seifert surface S' of genus g+1 and Seifert matrix

V'=V ⊕ ( 0 1 1 0 ) .

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
  • 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 ( K 1 # K 2 ) = g ( K 1 ) + g ( K 2 )

    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 g c of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus g f is the least genus of all Seifert surfaces whose complement in S 3 is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality g ≤ g f ≤ g c obviously holds, so in particular these invariants place upper bounds on the genus.

    References

    Seifert surface Wikipedia
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