In mathematics, a Seifert surface (named after German mathematician Herbert Seifert[1] [2]) is an orientable 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.
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.
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
V=\begin{pmatrix}1&-1\ 0&1\end{pmatrix}.
It is a theorem that any link always has an associated Seifert surface. This theorem was first published by Frankl and Pontryagin in 1930.[3] 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
Suppose that link has m components (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
H1(S)
g=
| 1 | |
| 2 |
(2+d-f-m)
is the genus of
S
H1(S)
a1,a2,\ldots,a2g
Q=(Q(ai,aj))
\begin{pmatrix}0&-1\ 1&0\end{pmatrix}
The 2g × 2g integer Seifert matrix
V=(v(i,j))
has
v(i,j)
S
S
S x [-1,1]
x
S
x x \{1\}
x x \{-1\}
With this, we have
V-V*=Q,
where V∗ = (v(j, i)) the transpose matrix. Every integer 2g × 2g matrix
V
V-V*=Q
The Alexander polynomial is computed from the Seifert matrix by
A(t)=\det\left(V-tV*\right),
t.
S,
The signature of a knot is the signature of the symmetric Seifert matrix
V+VT.
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
V'=V ⊕ \begin{pmatrix}0&1\ 1&0\end{pmatrix}.
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:
g(K1n{\#}K2)=g(K1)+g(K2)
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
gc
gf
S3
g\leqgf\leqgc
The knot genus is NP-complete by work of Ian Agol, Joel Hass and William Thurston.[6]
It has been shown that there are Seifert surfaces of the same genus that do not become isotopic either topologically or smoothly in the 4-ball.[7] [8]