Seifert surface

A Seifert surface bounded by a set of Borromean rings.

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.

Examples

A Seifert surface for the Hopf link. This is an annulus, not a Möbius strip. It has two half-twists and is thus orientable.

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

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

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. ^[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 ${\displaystyle S}$, 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 ${\displaystyle S}$ is constructed from f disjoint disks by attaching d bands. The homology group ${\displaystyle H_{1}(S)}$ is free abelian on 2g generators, where

${\displaystyle g={\frac {1}{2}}(2+d-f-m)}$

is the genus of ${\displaystyle S}$. The intersection form Q on ${\displaystyle H_{1}(S)}$ is skew-symmetric, and there is a basis of 2g cycles ${\displaystyle a_{1},a_{2},\ldots ,a_{2g}}$ with ${\displaystyle Q=(Q(a_{i},a_{j}))}$ equal to a direct sum of the g copies of the matrix

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

An illustration of (curves isotopic to) the pushoffs of a homology generator a in the positive and negative directions for a Seifert surface of the figure eight knot.

The 2g × 2g integer Seifert matrix

${\displaystyle V=(v(i,j))}$

has ${\displaystyle v(i,j)}$ the linking number in Euclidean 3-space (or in the 3-sphere) of a_i and the "pushoff" of a_j in the positive direction of ${\displaystyle S}$. More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of ${\displaystyle S}$ to an embedding of ${\displaystyle S\times [-1,1]}$, given some representative loop ${\displaystyle x}$ which is homology generator in the interior of ${\displaystyle S}$, the positive pushout is ${\displaystyle x\times \{1\}}$ and the negative pushout is ${\displaystyle x\times \{-1\}}$. ^[4]

With this, we have

${\displaystyle V-V^{*}=Q,}$

where V^∗ = (v(j, i)) the transpose matrix. Every integer 2g × 2g matrix ${\displaystyle V}$ with ${\displaystyle V-V^{*}=Q}$ arises as the Seifert matrix of a knot with genus g Seifert surface.

The Alexander polynomial is computed from the Seifert matrix by ${\displaystyle A(t)=\det \left(V-tV^{*}\right),}$ which is a polynomial of degree at most 2g in the indeterminate ${\displaystyle t.}$ The Alexander polynomial is independent of the choice of Seifert surface ${\displaystyle S,}$ and is an invariant of the knot or link.

The signature of a knot is the signature of the symmetric Seifert matrix ${\displaystyle V+V^{\mathrm {T} }.}$ It is again an invariant of the knot or link.

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 topological surgery, resulting in a Seifert surface S′ of genus g + 1 and Seifert matrix

${\displaystyle V'=V\oplus {\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}$

The genus of a knot K is the knot invariant defined by the minimal genus g of a Seifert surface for K.

For instance:

• 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 1, as does the figure-eight knot.
• The genus of a (p, q)-torus knot is (p − 1)(q − 1)/2
• The degree of a knot's Alexander polynomial is a lower bound on twice its genus.

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

${\displaystyle g(K_{1}\mathbin {\#} K_{2})=g(K_{1})+g(K_{2})}$

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${\displaystyle g_{c}}$ of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus${\displaystyle g_{f}}$ is the least genus of all Seifert surfaces whose complement in ${\displaystyle S^{3}}$ is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality ${\displaystyle g\leq g_{f}\leq g_{c}}$ obviously holds, so in particular these invariants place upper bounds on the genus. ^[5]

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]}

See also

• Crosscap number
• Arf invariant of a knot
• Slice genus

References

1. Seifert, H. (1934). "Über das Geschlecht von Knoten". Math. Annalen (in German). 110 (1): 571–592. doi:10.1007/BF01448044. S2CID   122221512.
2. van Wijk, Jarke J.; Cohen, Arjeh M. (2006). "Visualization of Seifert Surfaces". IEEE Transactions on Visualization and Computer Graphics. 12 (4): 485–496. doi:10.1109/TVCG.2006.83. PMID   16805258. S2CID   4131932.
3. Frankl, F.; Pontrjagin, L. (1930). "Ein Knotensatz mit Anwendung auf die Dimensionstheorie". Math. Annalen (in German). 102 (1): 785–789. doi:10.1007/BF01782377. S2CID   123184354.
4. Dale Rolfsen. Knots and Links. (1976), 146-147.
5. Brittenham, Mark (24 September 1998). "Bounding canonical genus bounds volume". arXiv: math/9809142 .
6. Agol, Ian; Hass, Joel; Thurston, William (2002-05-19). "3-manifold knot genus is NP-complete". Proceedings of the thiry-fourth annual ACM symposium on Theory of computing. STOC '02. New York, NY, USA: Association for Computing Machinery. pp. 761–766. arXiv: math/0205057 . doi:10.1145/509907.510016. ISBN   978-1-58113-495-7. S2CID   10401375 – via author-link.
7. Hayden, Kyle; Kim, Seungwon; Miller, Maggie; Park, JungHwan; Sundberg, Isaac (2022-05-30). "Seifert surfaces in the 4-ball". arXiv: 2205.15283 [math.GT].
8. "Special Surfaces Remain Distinct in Four Dimensions". Quanta Magazine. 2022-06-16. Retrieved 2022-07-16.

External links

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