Seifert surface

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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.

Contents 1 Examples 2 Existence and Seifert matrix 3 Genus of a knot 4 See also 5 External links

[edit] 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

[edit] 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 H₁(S) is free abelian on 2g generators, where

g = (2 + d − f − m)/2

is the genus of S. The intersection form Q on H₁(S) is skew-symmetric, and there is a basis of 2g cycles

a₁,a₂,...,a_2g

with

Q=(Q(a_i,a_j))

the direct sum of g copies of

.

The 2g2g integer Seifert matrix

V=(v(i,j)) has

v(i,j) the linking number in Euclidean 3-space (or in the 3-sphere) of a_i and the pushoff of a_j out of the surface, with

V − V^* = Q

where V^*=(v(j,i)) the transpose matrix. Every integer 2g2g 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 . 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 . It is again an invariant of the knot or link.

[edit] 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 — 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:

[edit] See also

• Crosscap number
• Arf invariant (knot)

[edit] External links

• The SeifertView programme of Jack van Wijk visualizes the Seifert surfaces of knots constructed using Seifert's algorithm.