Seifert surface

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

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}}.

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

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

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

{\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

V=(v(i,j))

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

With this, we have

V-V^{*}=Q,

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

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

The signature of a knot is the signature of the symmetric Seifert matrix $V+V^{\mathrm {T} }.$ 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

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:

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 $g_{c}$ of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the free genus $g_{f}$ is the least genus of all Seifert surfaces whose complement in $S^{3}$ is a handlebody. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality $g\leq g_{f}\leq g_{c}$ 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]

References

^ Seifert, H. (1934). "Über das Geschlecht von Knoten". Math. Annalen (in German). 110 (1): 571–592. doi:10.1007/BF01448044. S2CID 122221512.
^ 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.
^ 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.
^ Dale Rolfsen. Knots and Links. (1976), 146-147.
^ Brittenham, Mark (24 September 1998). "Bounding canonical genus bounds volume". arXiv:math/9809142.
^ 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.
^ Hayden, Kyle; Kim, Seungwon; Miller, Maggie; Park, JungHwan; Sundberg, Isaac (2022-05-30). "Seifert surfaces in the 4-ball". arXiv:2205.15283 [math.GT].
^ "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.

[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.

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Knot theory (knots and links)
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