2.242.158.186: small bit added

2014-07-19T14:59:47Z

small bit added

en>Panosth: /* Software */

2013-12-26T14:42:41Z

Software

← Older revision		Revision as of 16:42, 26 December 2013
Line 1:		Line 1:
	~~Hi there~~, ~~I am Sophia~~. ~~Playing badminton~~ is a ~~thing~~ that he is ~~totally addicted~~ to. I've always ~~loved residing~~ in ~~Mississippi~~. ~~Office supervising~~ is ~~my occupation~~.<br><br>~~my weblog~~; ~~psychic solutions~~ by ~~lynne~~ ([http://www.~~rusload~~.de/~~uprofile~~.~~php?UID~~=~~425591 clicking here~~])		[[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={{lang\|de\|Über das Geschlecht von Knoten}} \|journal=[[Mathematische Annalen\|Math. Annalen]] \|volume=110 \|issue=1 \|pages=571–592 \|year=1934 \|doi=10.1007/BF01448044 }} {{de icon}}</ref><ref>{{Cite journal \|first=Jarke J. \|last=van Wijk \| author1-link = Jack van Wijk \|first2=Arjeh M. \|last2=Cohen \|title=Visualization of Seifert Surfaces \|journal=IEEE Trans. on Visualization and Computer Graphics \|volume=12 \|issue=4 \|pages=485–496 \|year=2006 \|doi=10.1109/TVCG.2006.83 }}</ref>) is a [[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]] ''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. A Seifert surface must be [[oriented]]. It is possible to associate unoriented (and not necessarily orientable) surfaces to knots 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 considered to be 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 [[Felix Frankl\|Frankl]] and [[Lev Pontryagin\|Pontrjagin]] in 1930.<ref>{{Cite journal \|first=F. \|last=Frankl \|first2=L. \|last2=Pontrjagin \|year=1930 \|title={{lang\|de\|Ein Knotensatz mit Anwendung auf die Dimensionstheorie}} \|journal=Math. Annalen \|volume=102 \|issue=1 \|pages=785–789 \|doi=10.1007/BF01782377 }} {{de icon}}</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 (''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 ''2g'' generators, where

			:''g'' = (2 + ''d'' − ''f'' − ''m'')/2

			is the [[Genus (mathematics)\|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

			:''a<sub>1</sub>,a<sub>2</sub>,...,a<sub>2g</sub>''

			with

			:''Q=(Q(a<sub>i</sub>,a<sub>j</sub>))''

			the direct sum of ''g'' copies of

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



			The ''2g''<math>\times</math>''2g'' integer '''Seifert matrix'''

			:''V=(v(i,j))'' 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>'' out of the surface, with

			:<math>V-V</math>''<sup>*</sup>''<math>=Q</math>

			where ''V<sup></sup>=(v(j,i))'' the transpose matrix. Every integer ''2g''<math>\times</math>''2g'' matrix <math>V</math> with <math>V-V</math><sup></sup><math>=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(V-tV</math><sup>*</sup>), which is a polynomial in the indeterminate <math>t</math> of degree <math> \leq 2g</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^\top</math>. 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 theory\|surgery]], to be replaced by a Seifert surface ''S''' of genus ''g+1'' and Seifert matrix

			:''V'=V''<math>
			\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 ''only'' knot with genus zero.
			* The [[trefoil knot]] has genus one, as does the [[Figure-eight knot (mathematics)\|figure-eight knot]].
			* The genus of a (''p'',''q'')-[[torus knot]] is (''p'' − 1)(''q'' − 1)/2
			* The degree of the [[Alexander 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]]:
			:<math>g(K_1 \# K_2) = g(K_1) + g(K_2)</math>

			==See also==
			*[[Crosscap number]]
			*[[Arf invariant of a knot]]

			==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]]

en>Xqbot: r2.7.3) (Robot: Modifying it:Funzione trascendente di Lerch

2012-07-01T04:05:17Z

r2.7.3) (Robot: Modifying it:Funzione trascendente di Lerch

New page

Hi there, I am Sophia. Playing badminton is a thing that he is totally addicted to. I've always loved residing in Mississippi. Office supervising is my occupation.<br><br>my weblog; psychic solutions by lynne ([http://www.rusload.de/uprofile.php?UID=425591 clicking here])

Lerch zeta function - Revision history

2.242.158.186: small bit added

en>Panosth: /* Software */

en>Xqbot: r2.7.3) (Robot: Modifying it:Funzione trascendente di Lerch