en>Bender2k14: added category

2013-07-21T17:40:15Z

added category

New page

[[File:Quadrisecant.jpg|thumb|240px|Physical model of a quadrisecant of a [[trefoil knot]]]]
In [[geometry]], a '''quadrisecant''' or '''quadrisecant line''' of a [[curve]] is a [[Line (geometry)|line]] that passes through four points of the curve.

==In knot theory==
In three-dimensional [[Euclidean space]], every [[Unknot|non-trivial]] [[Tame knot|tame]] [[Knot (mathematics)|knot]] or [[Link (knot theory)|link]] has a quadrisecant. Originally established in the case of knotted [[polygon]]s and [[smooth curve|smooth]] knots by [[Erika Pannwitz]],<ref name="pannwitz"/>
this result was extended to knots in suitably [[general position]] and links with nonzero [[linking number]],<ref name="mm"/>
and later to all nontrivial tame knots and links.<ref name="kuperberg"/>

Pannwitz proved more strongly that the number of distinct quadrisecants is lower bounded by a function of the minimum number of boundary singularities in a locally-flat open disk bounded by the knot.<ref name="pannwitz"/><ref name="jin"/> {{harvtxt|Morton|Mond|1982}} conjectured that the number of distinct quadrisecants of a given knot is always at least ''n''(''n'' − 1)/2, where ''n'' is the [[Crossing number (knot theory)|crossing number]] of the knot.<ref name="mm"/><ref name="jin"/> However, counterexamples to this conjecture have since been discovered.<ref name="jin"/>

Two-component links have quadrisecants in which the points on the quadrisecant appear in alternating order between the two components,<ref name="mm"/> and nontrivial knots have quadrisecants in which the four points, [[cyclic order|ordered cyclically]] as ''a'',''b'',''c'',''d'' on the knot, appear in order ''a'',''c'',''b'',''d'' along the quadrisecant.<ref name="denne"/> The existence of these alternating quadrisecants can be used to derive the [[Fary–Milnor theorem]], a [[lower bound]] on the [[total curvature]] of a nontrivial knot.<ref name="denne"/> Quadrisecants have also been used to find lower bounds on the [[ropelength]] of knots.<ref name="dds"/>

==In algebraic geometry==
[[Arthur Cayley]] derived a formula for the number of quadrisecants of an [[algebraic curve]] in three-dimensional [[complex projective space]], as a function of its [[Degree of a polynomial|degree]] and [[Geometric genus|genus]].<ref name="cayley"/> For a curve of degree ''d'' and genus ''g'', the number of quadrisecants is<ref name="gh11"/>
:<math>\frac{(d-2)(d-3)^2(d-4)}{12}-\frac{g(d^2-7d+13-g)}{2}.</math>

==Of skew lines==
[[File:Double six.svg|thumb|The [[Schläfli double six]]]]
In three-dimensional [[Euclidean space]], every set of four [[skew lines]] in [[general position]] either has two quadrisecants (also called in this context [[Transversal (geometry)|transversals]]) or none. Any three of the four lines determine a [[Ruled surface|doubly ruled surface]], in which one of the two sets of ruled lines contains the three given lines, and the other ruling consists of trisecants to the given lines. If the fourth of the given lines pierces this surface, its two points of intersection lie on the two quadrisecants; if it is disjoint from the surface, then there are no quadrisecants.<ref name="hcv"/>

The quadrisecants of sets of lines play an important role in the construction of the [[Schläfli double six]], a [[Configuration (geometry)|configuration]] of twelve lines intersecting each other in 30 crossings. If five lines {{nowrap|1=''a''''i'' (for ''i'' = 1,2,3,4,5)}} are given in a three-dimensional space, such that all five are intersected by a common line ''b''6 but are otherwise in general position, then each of the five quadruples of the lines ''a''''i'' has a second quadrisecant ''b''''i'', and the five lines ''b''''i'' formed in this way are all intersected by a common line ''a''6. These twelve lines and the 30 intersection points ''aibj'' form the double six.<ref name="schlafli"/><ref name="coxeter"/>

==References==
{{reflist|30em|refs=
<ref name="cayley">{{citation
| last = Cayley | first = Arthur | author-link = Arthur Cayley
| jstor = 108806
| pages = 453–483
| title = [[Philosophical Transactions of the Royal Society of London]]
| volume = 153
| year = 1863}}.</ref>
<ref name="coxeter">{{citation
| last = Coxeter | first = H. S. M. | authorlink = Harold Scott MacDonald Coxeter
| contribution = An absolute property of four mutually tangent circles
| doi = 10.1007/0-387-29555-0_5
| location = New York
| mr = 2191243
| pages = 109–114
| publisher = Springer
| series = Math. Appl. (N. Y.)
| title = Non-Euclidean geometries
| volume = 581
| year = 2006}}. Coxeter repeats Shläfli's construction, and provides several references to simplified proofs of its correctness.</ref>
<ref name="denne">{{citation
| last = Denne | first = Elizabeth Jane
| arxiv = math/0510561
| publisher = [[University of Illinois at Urbana-Champaign]]
| series = Ph.D. thesis
| title = Alternating quadrisecants of knots
| year = 2004}}.</ref>
<ref name="dds">{{citation
| last1 = Denne | first1 = Elizabeth
| last2 = Diao | first2 = Yuanan
| last3 = Sullivan | first3 = John M.
| doi = 10.2140/gt.2006.10.1
| journal = [[Geometry & Topology]]
| mr = 2207788
| pages = 1–26
| title = Quadrisecants give new lower bounds for the ropelength of a knot
| url = http://msp.warwick.ac.uk/gt/2006/10/p001.xhtml
| volume = 10
| year = 2006}}.</ref>
<ref name="gh11">{{citation
| last1 = Griffiths | first1 = Phillip
| last2 = Harris | first2 = Joseph
| isbn = 9781118030776
| page = 296
| publisher = John Wiley & Sons
| series = Wiley Classics Library
| title = Principles of Algebraic Geometry
| url = http://books.google.com/books?id=Sny48qKdW40C&pg=PA296
| volume = 52
| year = 2011}}.</ref>
<ref name="hcv">{{citation
| last1 = Hilbert | first1 = David | author1-link = David Hilbert
| last2 = Cohn-Vossen | first2 = Stephan | author2-link = Stephan Cohn-Vossen
| edition = 2nd
| isbn = 978-0-8284-1087-8
| location = New York
| page = 164
| publisher = Chelsea
| title = Geometry and the Imagination
| year = 1952}}.</ref>
<ref name="jin">{{citation
| last = Jin | first = Gyo Taek
| contribution = Quadrisecants of knots with small crossing number
| doi = 10.1142/9789812703460_0025
| mr = 2197955
| pages = 507–523
| publisher = World Sci. Publ., Singapore
| series = Ser. Knots Everything
| title = Physical and numerical models in knot theory
| url = http://knot.kaist.ac.kr/~trefoil/papers/quad-small-xing/quad-small-xing(AMS-SMM).pdf
| volume = 36
| year = 2005}}.</ref>
<ref name="kuperberg">{{citation
| last = Kuperberg | first = Greg | author-link = Greg Kuperberg
| arxiv = math/9712205
| doi = 10.1142/S021821659400006X
| journal = [[Journal of Knot Theory and Its Ramifications]]
| mr = 1265452
| pages = 41–50
| title = Quadrisecants of knots and links
| volume = 3
| year = 1994}}.</ref>
<ref name="mm">{{citation
| last1 = Morton | first1 = Hugh R.
| last2 = Mond | first2 = David M. Q.
| doi = 10.1016/0040-9383(82)90007-6
| journal = [[Topology (journal)|Topology]]
| mr = 0649756
| pages = 235–243
| title = Closed curves with no quadrisecants
| volume = 21
| year = 1982}}.</ref>
<ref name="pannwitz">{{citation
| last = Pannwitz | first = Erika | author-link = Erika Pannwitz
| doi = 10.1007/BF01452857
| issue = 1
| journal = [[Mathematische Annalen]]
| pages = 629–672
| title = Eine elementargeometrische Eigenschaft von Verschlingungen und Knoten
| volume = 108
| year = 1933}}.</ref>
<ref name="schlafli">{{citation
| last = Schläfli | first = Ludwig | authorlink = Ludwig Schläfli
| editor-last = Cayley | editor-first = Arthur | editor-link = Arthur Cayley
| journal = Quarterly journal of pure and applied mathematics
| pages = 55–65, 110–120
| title = An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface
| url = http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0002
| volume = 2
| year = 1858}}.</ref>
}}

[[Category:Knot theory]]
[[Category:Algebraic geometry]]

BHT algorithm - Revision history

en>Bender2k14: added category