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.

Examples

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The standard Mobius 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 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

A =

1 1 0 1

The SeifertView programme http://www.win.tue.nl/~vanwijk/seifertview/ of Jack van Wijk visualizes the Seifert surfaces of knots constructed using Seifert's algorithm.

Existence and Seifert matrix

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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 $H\_1(S)$ is free abelian on 2g generators, where g=(2+d-f-m)/2 is the genus of $S$. The 2g$\backslash times$2g integer Seifert matrix A of the linking numbers of 2g linearly independent cycles generating $H\_1(S)$ has $(i,j)$ entry the linking number in Euclidean 3-space (or in the 3-sphere) of the i^th cycle and the pushoff of the j^th cycle out of the surface. The Alexander polynomial may now be computed from the Seifert matrix.

Genus of a knot

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One example of a knot invariant which is computed from a Seifert surface is the genus of a knot. The genus of a knot K is defined as minimal genus g of all Seifert surfaces 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

A fundamental property of the genus is that it is additive with respect to the knot sum:

$g(K\_1\; \backslash \#\; K\_2)\; =\; g(K\_1)\; +\; g(K\_2)$

Crosscap Number of a knot

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Every knot K in 3-space bounds a compact, connected non-orientable surface of genus $g$. The minimum of such $g$ is called the Crosscap Number or non-orientable genus of K.

For instance:

• The Crosscap number of the unknot is defined to be Zero.
• The Crosscap number of the trefoil is One, as it bounds a Mobius strip and is not trivial.
• The Crosscap number of a torus knot was determined by M.Teragaito.

The formula for the knot sum is

$C(k\_1)+C(k\_2)-1\; \backslash leq\; C(k\_1\; \backslash \#\; k\_2)\; \backslash leq\; C(k\_1)+C(k\_2).$

References

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• Clark, B.E. "Crosscaps and Knots" , Int. J. Math and Math. Sci, Vol 1, 1978, pp 113-124
• Murakami, Hitoshi and Yasuhara, Akira. "Crosscap number of a knot," Pacific J. Math. 171 (1995), no. 1, 261--273.
• Teragaito, Masakazu. "Crosscap numbers of torus knots," Topology Appl. 138 (2004), no. 1-3, 219--238.
• Teragaito, Masakazu and Hirasawa, Mikami. "Crosscap numbers of 2-bridge knots," Arxiv:math.GT/0504446.
• J.Uhing. "Zur Kreuzhaubenzahl von Knoten", diploma thesis, 1997, University of Dortmund, (German language)

Geometric topology | Knot theory | Surfaces

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