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| In [[mathematics]], a '''Legendrian knot''' often refers to a smooth embedding of the circle into {{nowrap|<math>\mathbb R^3</math>,}} which is tangent to the standard [[contact structure]] on {{nowrap|<math>\mathbb R^3</math>.}} It is the lowest dimensional case of a [[Legendrian submanifold]], which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional that is always tangent to the contact hyperplane.
| | Gunsmith Houston Dorothy from Sackville, really loves cats, [http://comoficarrico.comoganhardinheiro101.com/ como ganhar dinheiro na internet] and post cards. Maintains a tour blog and has lots to write about after visiting Port of the Moon. |
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| Two Legendrian knots are equivalent if they are isotopic through a family of Legendrian knots. There can be inequivalent Legendrian knots that are isotopic as topological knots. Many inequivalent Legendrian knots can be distinguished by considering their [[Thurston-Bennequin invariant]]s and rotation number, which are together known as the "classical invariants" of Legendrian knots. More sophisticated invariants have been constructed, including one constructed combinatorially by Chekanov and using holomorphic discs by Eliashberg. This [[Chekanov-Eliashberg invariant]] yields an invariant for loops of Legendrian knots by considering the monodromy of the loops. This has yielded noncontractible loops of Legendrian knots which are contractible in the space of all knots.
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| Any Legendrian knot may be C^0 perturbed to a [[transverse knot]] (a knot transverse to a contact structure) by pushing off in a direction transverse to the contact planes. The set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilizations is in bijection with the set of transverse knots.
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| ==References==
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| {{reflist}}
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| *{{cite book
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| |title= An introduction to contact topology; Volume 109 of Cambridge studies in advanced mathematics
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| |last=Geiges
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| |first=Hansjörg
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| |authorlink=Hansjörg Geiges
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| |coauthors=
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| |year=2008
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| |publisher=Cambridge University Press
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| |location=
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| |isbn=0-521-86585-9
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| |page=94
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| |url=http://books.google.com/books?id=RERR4zMDYRgC&pg=PA94&dq=%22Legendrian+knot%22&hl=en&ei=plIDTf7GIYX7lweXnZjKCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCMQ6AEwAA#v=onepage&q=%22Legendrian%20knot%22&f=false }}
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| *{{cite book
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| |title=European Congress of Mathematics: Barcelona, July 10–14, 2000,
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| |last= Casacuberta
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| |first= Carlos
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| |authorlink=Carlos Casacuberta
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| |coauthors=
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| |year=2001
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| |publisher=Birkhäuser
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| |location=
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| |isbn= 3764364181
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| |page=526
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| |url=http://books.google.com/books?id=uFqQ5B5nAm0C&pg=PA526&dq=%22Legendrian+knot%22&hl=en&ei=plIDTf7GIYX7lweXnZjKCQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CCgQ6AEwAQ#v=onepage&q=%22Legendrian%20knot%22&f=false }}
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| *{{cite journal|first=J. |last=Epstein |first2=D. |last2=Fuchs |first3=M. |last3=Meyer |title=Chekanov–Eliashberg invariants and transverse approximations of Legendrian knots|journal=[[Pacific Journal of Mathematics]] | volume=201 |year=2001 | issue= 1 |pages= 89–106}}
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| * {{cite journal|last=Kalman |first=Tamas |title=Contact homology and one parameter families of Legendrian knots |journal=[[Geometry & Topology]] |volume=9 |year=2005| pages= 2013–2078}}
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| * {{citation | first = Joshua M. | last = Sabloff | title = What Is . . . a Legendrian Knot? | journal = AMS Notices | volume = 56 | issue = 10 | pages = 1282–1284 | year = 2009 | url = http://www.ams.org/notices/200910/rtx091001282p.pdf }}.
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| {{DEFAULTSORT:Legendrian Knot}}
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| [[Category:Knots and links]]
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| {{knottheory-stub}}
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Gunsmith Houston Dorothy from Sackville, really loves cats, como ganhar dinheiro na internet and post cards. Maintains a tour blog and has lots to write about after visiting Port of the Moon.