- Seifert surface
In

mathematics , a**Seifert surface**is asurface 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, manyknot 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 the3-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", and every connected component of "S" has non-empty boundary.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. It is important to note that a Seifert surface must beoriented . It is possible to associate unoriented (and not necessarily orientable) surfaces to knots as well.**Examples**The standard

Möbius strip has theunknot for a boundary but is not considered to be a Seifert surface for the unknot because it is not orientable.The "checkerboard" coloring of the 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:$V\; =\; egin\{pmatrix\}1\; -1\; \backslash \; 0\; 1end\{pmatrix\}.$**Existence and Seifert matrix**It is a

theorem that there always exists a Seifert surface. This theorem was first published by F. Frankl and Lev Pontrjagin in1930 . A different proof was published in1934 byHerbert Seifert and relies on what is now called theSeifert algorithm . Thealgorithm 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 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" = (2 + "d" − "f" − "m")/2

is the

genus of $S$. Theintersection form *Q*on $H\_1(S)$ isskew-symmetric , and there is a basis of*2g*cycles:

*a*_{1},a_{2},...,a_{2g}with

:

*Q=(Q(a*_{i},a_{j}))the direct sum of

*g*copies of:$egin\{pmatrix\}\; 0\; -1\; \backslash \; 1\; 0end\{pmatrix\}$.

The

*2g*$imes$*2g*integer**Seifert matrix**:

*V=(v(i,j))*has$v(i,j)$ the

linking number inEuclidean 3-space (or in the3-sphere ) of*a*and the pushoff of_{i}*a*out of the surface, with_{j}:$V-V$

$=Q$^{*}where

*V*the transpose matrix. Every integer^{*}=(v(j,i))*2g*$imes$*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 $A(t)=det(V-tV$^{*}), which is a polynomial in the indeterminate $t$ of degree $leq\; 2g$. 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 $V+V^\; op$. 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 surgery, to be replaced by a Seifert surface*S'*of genus*g+1*and Seifert matrix:

*V'=V*$oplus\; egin\{pmatrix\}\; 0\; 1\; \backslash \; 1\; 0\; end\{pmatrix\}$.The

**genus**of a knot "K" is theknot invariant defined by the minimal genus*g*of a Seifert surface for "K".For instance:

* Anunknot — which is, by definition, the boundary of a disc — has genus zero. Moreover, the unknot is the "only" knot with genus zero.

* Thetrefoil knot has genus one, as does the figure-eight knot.

* The genus of a ("p","q")-torus knot is ("p" − 1)("q" − 1)/2

* The degree of theAlexander polynomial is a lower bound on twice the genus of the knot.A fundamental property of the genus is that it is additive with respect to the

knot sum ::$g(K\_1\; \#\; K\_2)\; =\; g(K\_1)\; +\; g(K\_2)$**ee also***

Crosscap number

*Arf invariant (knot) **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.

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