Seifert area

The Seifert surface , named after the mathematician Herbert Seifert , referred to in the knot theory , a branch of topology , one from a node or entanglement bounded area . These surfaces can be used to investigate properties of the associated links (or nodes). For example, invariants of links or nodes can be determined using Seifert surfaces.

definition

Be the 3- sphere and an oriented link . A Seifert surface is a compact , oriented , contiguous , in given area that the entanglement has as oriented edge. ${\displaystyle \mathrm {S} ^{3}}$ ${\displaystyle L\subset \mathrm {S} ^{3}}$ ${\displaystyle \mathrm {S} ^{3}}$ ${\displaystyle L}$

The existence of a Seifert surface for each polygonal node was proven in 1930 by Frankl and Pontryagin .

construction

Seifert surface of the clover leaf loop (gender 1)

There is an algorithm that constructs a Seifert surface for every oriented link. This is called the Seifert algorithm . In the next section, a simpler method is shown first to construct a compact, coherent surface to form a loop, which, however, is generally not orientable.

With the algorithm, which is reminiscent of the construction of a chessboard, a compact, coherent, but not always orientable surface can be constructed. To do this, the areas of a node diagram are alternately colored black and white so that adjacent areas have different colors. This is always possible with entanglements. Thereafter, all areas of the same color connected by half-twisted stripes are regarded as one area. This is then an area with the loop as the edge, but which in general cannot be oriented.

The Seifert algorithm delivers a Seifert surface belonging to a given node, i.e. an orientable surface whose edge corresponds to the given node. This is generally not clearly determined by the node. For this purpose, the crossover points are selected on the node diagram and the diagram is cut at these points . As a result, the node diagram breaks down into 2d “sections”, if d is the number of crossover points. The sections are all provided with a fixed orientation that corresponds to a certain cycle of the node diagram. Furthermore, two assigned partial points, that is, those points that lie over the same colon of the projection, are connected by a straight “connecting line”. The route complex consisting of the sections and connecting routes is now divided into "circles" in the following way. One traverses a section, as indicated by the fixed orientation, then the connecting section adjoining the end point of the section, then the section starting from the new junction, then another connecting section and so on. Finally you come back to the starting point. If there is a section that has not yet been traversed, it gives rise to a new circle. Thus, on the whole, f circles can be formed. Each segment occurs in exactly one circle, each connecting route, on the other hand, in two, of which it is traversed in opposite directions. These circles are projected into the plane in polygons free of colon, which apparently do not intervene. An elementary surface piece is now clamped into each circle, whereby it can be assumed that each elementary surface piece, apart from the connecting lines, is projected one-to-one into the plane and that two different elementary surface pieces have no central point in common. The f elementary patches together form a surface clamped in the node. It can be orientated, as each connecting line is traversed by the two associated circles in opposite directions.

For alternating node diagrams , the Seifert algorithm returns an area of ​​minimal gender .

Seifert matrix

Be a Seifert area and its gender . The cut shape on the homology group of is skew symmetrical and there is a base of cycles ${\displaystyle S}$ ${\displaystyle g}$ ${\displaystyle Q}$ ${\displaystyle H_{1}(S)}$ ${\displaystyle S}$ ${\displaystyle 2g}$

${\displaystyle a_{1},a_{2},\ldots ,a_{2g}}$,

so the direct sum of copies of ${\displaystyle Q=(Q(a_{i},a_{j}))}$ ${\displaystyle g}$

${\displaystyle {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$

is. A matrix is then defined , the entries of which are calculated as the number of entanglements from and the pushoff of from the area. The matrix is called the Seifert matrix of the Seifert surface. ${\displaystyle 2g\times 2g}$ ${\displaystyle V}$ ${\displaystyle v(i,j)}$ ${\displaystyle a_{i}}$ ${\displaystyle a_{j}}$ ${\displaystyle V=(v(i,j))}$

The following applies , where denotes the transposed matrix . The Alexander polynomial of the node can be calculated with the help of the Seifert matrix , because it applies ${\displaystyle V-V^{*}=Q}$ ${\displaystyle V^{*}=(v(j,i))}$ ${\displaystyle \Delta _{K}}$ ${\displaystyle K}$

${\displaystyle \Delta _{K}(t)=\det(V-tV^{*})}$.

The signature of the symmetrical bilinear shape is called the signature of the node . ${\displaystyle V+V^{*}}$ ${\displaystyle K}$

Gender of a knot

The gender of a knot g (K) is the minimal gender of a Seifert surface of the node K. Examples:

• The trivial knot is the only knot with g (K) = 0 (it is the edge of a disk, so gender = 0).
• For the trefoil knot and the figure eight knot g (K) = 1.
• For the (p, q) - torus knot , g (K) = (p-1) (q-1) / 2.

It follows immediately from the formula that the degree of the Alexander polynomial is at most . Equality applies to alternating nodes , in general, equality does not always have to apply. Friedl and Vidussi have shown, however, that g (K) can be calculated using twisted Alexander polynomials: ${\displaystyle \Delta _{K}(t)=\det(V-tV^{*})}$ ${\displaystyle 2g(K)}$

${\displaystyle g(K)=\max \left\{{\frac {1}{k}}\operatorname {deg} \Delta _{K,\rho }\mid \rho \colon \pi _{1}(S^{3}-K)\rightarrow U(k)\right\}}$

where runs through all unitary representations of the node group and is the twisted Alexander polynomial. ${\displaystyle \rho }$ ${\displaystyle \pi _{1}(S^{3}-K)}$ ${\displaystyle \Delta _{K,\rho }}$ ${\displaystyle \rho }$

literature

• Charles Livingston: Knot Theory for Beginners . Vieweg, Braunschweig [a. a.] 1995, ISBN 3-528-06660-1 .
• Dale Rolfsen: Knots and Links. Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, 1990. xiv + 439 pp. ISBN 0-914098-16-0
• Alexander Stoimenow: Diagram genus, generators, and applications. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2016. ISBN 978-1-4987-3380-9

Web links

Commons : Seifert surfaces  - collection of pictures, videos and audio files

Individual evidence

1. ↑ ^{a b} W. B. Raymond Lickorish: An introduction to knot theory . Graduate Texts in Mathematics, 175. Springer-Verlag, New York, 1997. x + 201 pp. ISBN 0-387-98254-X , p. 15.
2. ↑ F. Frankl, L. Pontrjagin: A set of knots with applications to the dimension theory . Math. Ann. 102 (1930), no. 1, 785-789. online (PDF; 440 kB)
3. ↑ Herbert Seifert: About the sex of knots . In: Math. Annals . 110, No. 1, 1935, pp. 571-592. doi : 10.1007 / BF01448044 .
4. ↑ Kunio Murasugi : On the genus of the alternating knot. I, II. J. Math. Soc. Japan 10 1958 94-105, 235-248.
5. ^ Richard Crowell : Genus of alternating link types. Ann. of Math. (2) 69 1959 258-275.
6. ↑ David Gabai : Genera of the alternating links. Duke Math. J. 53 (1986) no. 3, 677-681.
7. ↑ Stefan Friedl, Stefano Vidussi: The Thurston norm and twisted Alexander polynomials