Seifert surface

File:Hopf band wikipedia.png. 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

:

It is a theorem that any link always has an associated Seifert surface. This theorem was first published by Frankl and Pontryagin in 1930.{{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}} 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 ({{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 $S$ is constructed from f disjoint disks by attaching d bands. The homology group $H\_1(S)$ is free abelian on 2g generators, where

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

is the genus of $S$. The intersection form Q on $H\_1(S)$ is skew-symmetric, and there is a basis of 2g cycles $a\_1,\; a\_2,\; \backslash ldots,\; a\_\{2g\}$ with

$Q\; =\; (Q(a\_i,\; a\_j))$ equal to a direct sum of the g copies of the matrix

:

File:Seiferthomologypushoff.png

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 $S\; \backslash times\; [-1,\; 1]$, given some representative loop $x$ which is homology generator in the interior of $S$, the positive pushout is $x\; \backslash times\; \backslash \{1\backslash \}$ and the negative pushout is $x\; \backslash times\; \backslash \{-1\backslash \}$.Dale Rolfsen. Knots and Links. (1976), 146-147.

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 $A(t)\; =\; \backslash det\backslash left(V\; -\; tV^*\backslash right),$ 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 $V\; +\; V^\backslash mathrm\{T\}.$ It is again an invariant of the knot or link.

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

:

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 {{em|only}} knot with genus zero.
• The trefoil knot has genus 1, as does the 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:

:$g(K\_1\; \backslash mathbin\{\backslash \#\}\; K\_2)\; =\; g(K\_1)\; +\; g(K\_2)$

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 $g\; \backslash leq\; g\_f\; \backslash leq\; g\_c$ obviously holds, so in particular these invariants place upper bounds on the genus.{{cite arXiv | eprint=math/9809142 | title=Bounding canonical genus bounds volume | date=24 September 1998 | author=Brittenham, Mark}}

The knot genus is NP-complete by work of Ian Agol, Joel Hass and William Thurston.{{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}}

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.{{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}}{{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}}

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

{{Reflist}}

• 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