Editing Seifert surface

{{Short description|Orientable surface whose boundary is a knot or link}}
[[File:Borromean Seifert surface.png|thumb|A Seifert surface bounded by a set of [[Borromean rings]].]]

In [[mathematics]], a '''Seifert surface''' (named after [[Germany|German]] [[mathematician]] [[Herbert Seifert]]<ref>{{Cite journal |first=H. |last=Seifert |title=Über das Geschlecht von Knoten |journal=[[Mathematische Annalen|Math. Annalen]] |volume=110 |issue=1 |pages=571–592 |year=1934 |doi=10.1007/BF01448044 |s2cid=122221512 |language=de}}</ref><ref>{{Cite journal |first1=Jarke J. |last1=van Wijk | author1-link = Jack van Wijk |first2=Arjeh M. |last2=Cohen |title=Visualization of Seifert Surfaces |journal=IEEE Transactions on Visualization and Computer Graphics |volume=12 |issue=4 |pages=485–496 |year=2006 |doi=10.1109/TVCG.2006.83 |pmid=16805258 |s2cid=4131932 }}</ref>) is an orientable [[Surface (topology)|surface]] whose [[boundary of a manifold|boundary]] is a given [[knot (mathematics)|knot]] or [[link (knot theory)|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 knot|tame]] [[oriented]] knot or link in [[Euclidean space|Euclidean 3-space]] (or in the [[3-sphere]]). A Seifert surface is a [[compact space|compact]], [[connected space|connected]], [[oriented]] [[Surface (topology)|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==
[[File:Hopf band wikipedia.png|150px|right|thumb|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
:<math>V = \begin{pmatrix}1 & -1 \\ 0 & 1\end{pmatrix}.</math>

==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 [[Lev Pontryagin|Pontryagin]] in 1930.<ref>{{Cite journal |first1=F. |last1=Frankl |first2=L. |last2=Pontrjagin |year=1930 |title=Ein Knotensatz mit Anwendung auf die Dimensionstheorie |journal=Math. Annalen |volume=102 |issue=1 |pages=785–789 |doi=10.1007/BF01782377 |s2cid=123184354 |language=de}}</ref> 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 <math>S</math>, given a projection of the knot or link in question.

Suppose that link has ''m'' components ({{nowrap|''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 <math>S</math> is constructed from ''f'' disjoint disks by attaching ''d'' bands. The [[homology group]] <math>H_1(S)</math> is free abelian on 2''g'' generators, where

:<math>g = \frac{1}{2}(2 + d - f - m)</math>

is the [[Genus (mathematics)|genus]] of <math>S</math>. The [[Intersection form (4-manifold)|intersection form]] ''Q'' on <math>H_1(S)</math> is [[skew-symmetric matrix|skew-symmetric]], and there is a basis of 2''g'' cycles <math>a_1, a_2, \ldots, a_{2g}</math> with
<math>Q = (Q(a_i, a_j))</math> equal to a direct sum of the ''g'' copies of the matrix

:<math>\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}</math>

[[File:Seiferthomologypushoff.png|thumb|right| 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 2''g'' × 2''g'' integer '''Seifert matrix'''

:<math>V = (v(i, j))</math>

has <math>v(i, j)</math> the [[linking number]] in [[Euclidean 3-space]] (or in the [[3-sphere]]) of ''a''<sub>''i''</sub> and the "pushoff" of ''a''<sub>''j''</sub> in the positive direction of <math>S</math>. More precisely, recalling that Seifert surfaces are bicollared, meaning that we can extend the embedding of <math>S</math> to an embedding of <math>S \times [-1, 1]</math>, given some representative loop <math>x</math> which is homology generator in the interior of <math>S</math>, the positive pushout is <math>x \times \{1\}</math> and the negative pushout is <math>x \times \{-1\}</math>.<ref>Dale Rolfsen. Knots and Links. (1976), 146-147.</ref>

With this, we have

:<math>V - V^* = Q,</math>

where ''V''<sup>&lowast;</sup> = (''v''(''j'', ''i'')) the transpose matrix. Every integer 2''g'' × 2''g'' matrix <math>V</math> with <math>V - V^* = Q</math> arises as the Seifert matrix of a knot with genus ''g'' Seifert surface.

The [[Alexander polynomial]] is computed from the Seifert matrix by <math>A(t) = \det\left(V - tV^*\right),</math> which is a polynomial of degree at most 2''g'' in the indeterminate <math>t.</math>  The Alexander polynomial is independent of the choice of Seifert surface <math>S,</math> and is an invariant of the knot or link.

The [[signature of a knot]] is the [[symmetric bilinear form#Signature and Sylvester's law of inertia|signature]] of the symmetric Seifert matrix <math>V + V^\mathrm{T}.</math>  It is again an invariant of the knot or link.

==Genus of a knot==

{{anchor|knot genus}}
Seifert surfaces are not at all unique: a Seifert surface ''S'' of genus ''g'' and Seifert matrix ''V'' can be modified by a [[surgery theory|topological surgery]], resulting in a Seifert surface ''S''&prime; of genus ''g'' + 1 and Seifert matrix

:<math>V' = V \oplus \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

The '''genus''' of a knot ''K'' is the [[knot invariant]] defined by the minimal [[genus (mathematics)|genus]] ''g'' of a Seifert surface for ''K''.

For instance:
* An [[unknot]]—which is, by definition, the boundary of a [[disk (mathematics)|disc]]—has genus zero. Moreover, the unknot is the {{em|only}} knot with genus zero. 
* The [[trefoil knot]] has genus 1, as does the [[figure-eight knot (mathematics)|figure-eight knot]].
* The genus of a (''p'',{{nnbsp}}''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]]:
:<math>g(K_1 \mathbin{\#} K_2) = g(K_1) + g(K_2)</math>

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''' <math>g_c</math> of a knot is the least genus of all Seifert surfaces that can be constructed by the Seifert algorithm, and the '''free genus''' <math>g_f</math> is the least genus of all Seifert surfaces whose complement in <math>S^3</math> is a [[Handlebody#3-dimensional handlebodies|handlebody]]. (The complement of a Seifert surface generated by the Seifert algorithm is always a handlebody.) For any knot the inequality <math>g \leq g_f \leq g_c</math> obviously holds, so in particular these invariants place upper bounds on the genus.<ref>{{cite arXiv | eprint=math/9809142 | title=Bounding canonical genus bounds volume | date=24 September 1998 | author=Brittenham, Mark}}</ref>

The knot genus is [[NP-completeness|NP-complete]] by work of [[Ian Agol]], [[Joel Hass]] and [[William Thurston]].<ref>{{Cite book|last1=Agol|first1=Ian|author-link=Ian Agol|last2=Hass|first2=Joel|author-link2=Joel Hass|last3=Thurston|first3=William|title=Proceedings of the thiry-fourth annual ACM symposium on Theory of computing |chapter=3-manifold knot genus is NP-complete |author-link3=William Thurston|date=2002-05-19|chapter-url=https://doi.org/10.1145/509907.510016|series=STOC '02|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=761–766|arxiv=math/0205057|doi=10.1145/509907.510016|isbn=978-1-58113-495-7|s2cid=10401375|via=author-link}}</ref>

It has been shown that there are Seifert surfaces of the same genus that do not become [[Homotopy#Isotopy|isotopic]] either topologically or smoothly in the 4-ball.<ref>{{cite arXiv |last1=Hayden |first1=Kyle |last2=Kim |first2=Seungwon |last3=Miller |first3=Maggie |last4=Park |first4=JungHwan |last5=Sundberg |first5=Isaac |date=2022-05-30 |title=Seifert surfaces in the 4-ball |class=math.GT |language=en |eprint=2205.15283}}</ref><ref>{{Cite web |date=2022-06-16 |title=Special Surfaces Remain Distinct in Four Dimensions |url=https://www.quantamagazine.org/special-surfaces-remain-distinct-in-four-dimensions-20220616/ |access-date=2022-07-16 |website=Quanta Magazine |language=en}}</ref>

==See also==
*[[Crosscap number]]
*[[Arf invariant of a knot]]
*[[Murasugi sum]]
*[[Slice genus]]

==References ==
{{Reflist}}

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

{{Knot theory|state=collapsed}}

[[Category:Geometric topology]]
[[Category:Knot theory]]
[[Category:Surfaces]]