In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) 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 neither 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
V = ( 1 − 1 0 1 ) . {\displaystyle V={\begin{pmatrix}1&-1\\0&1\end{pmatrix}}.}
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 S {\displaystyle 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 {\displaystyle S} is constructed from f disjoint disks by attaching d bands. The homology group H 1 ( S ) {\displaystyle H_{1}(S)} is free abelian on 2g generators, where
g = 1 2 ( 2 + d − f − m ) {\displaystyle g={\frac {1}{2}}(2+d-f-m)}
is the genus of S {\displaystyle S} . The intersection form Q on H 1 ( S ) {\displaystyle H_{1}(S)} is skew-symmetric, and there is a basis of 2g cycles a 1 , a 2 , … , a 2 g {\displaystyle a_{1},a_{2},\ldots ,a_{2g}} with
Q = ( Q ( a i , a j ) ) {\displaystyle Q=(Q(a_{i},a_{j}))} equal to a direct sum of the g copies of the matrix
( 0 − 1 1 0 ) {\displaystyle {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}
The 2g × 2g integer Seifert matrix
V = ( v ( i , j ) ) {\displaystyle V=(v(i,j))}
has v ( i , j ) {\displaystyle v(i,j)} the linking number in Euclidean 3-space (or in the 3-sphere) of ai and the "pushoff" of aj in the positive direction of S {\displaystyle S} . More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of S {\displaystyle S} to an embedding of S × [ − 1 , 1 ] {\displaystyle S\times [-1,1]} , given some representative loop x {\displaystyle x} which is homology generator in the interior of S {\displaystyle S} , the positive pushout is x × { 1 } {\displaystyle x\times \{1\}} and the negative pushout is x × { − 1 } {\displaystyle x\times \{-1\}} . With this, we have
V − V ∗ = Q , {\displaystyle V-V^{*}=Q,}
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