Seifert surface
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>) 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.
ExamplesEdit
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 matrixEdit
It is a theorem that any link always has an associated Seifert surface. This theorem was first published by Frankl and Pontryagin in 1930.<ref>Template:Cite journal</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 (Template:Nowrap 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 2g generators, where
- <math>g = \frac{1}{2}(2 + d - f - m)</math>
is the genus of <math>S</math>. The intersection form Q on <math>H_1(S)</math> is skew-symmetric, and there is a basis of 2g 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>
The 2g × 2g 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 ai and the "pushoff" of aj 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∗ = (v(j, i)) the transpose matrix. Every integer 2g × 2g 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 2g 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 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 knotEdit
Template:Anchor 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
- <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 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 Template:Em knot with genus zero.
- The trefoil knot has genus 1, as does the figure-eight knot.
- The genus of a (p,Template:Nnbspq)-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. (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>Template:Cite arXiv</ref>
The knot genus is NP-complete by work of Ian Agol, Joel Hass and William Thurston.<ref>Template:Cite book</ref>
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.<ref>Template:Cite arXiv</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
See alsoEdit
ReferencesEdit
External linksEdit
- The SeifertView programme of Jack van Wijk visualizes the Seifert surfaces of knots constructed using Seifert's algorithm.