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| {{Infobox knot theory
| | Myrtle Benny is how I'm called and I feel comfortable when people use the complete name. To play baseball is the hobby he will by no means stop performing. South Dakota is exactly where me and my husband live and my family members enjoys it. Managing people has been his working day job for a while.<br><br>my web page :: over the counter std test ([http://wooribanking.com/xe/?document_srl=2163546 their explanation]) |
| | name= Unknot
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| | practical name= Torus
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| | image= Blue Unknot.png
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| | caption=
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| | arf invariant=
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| | bridge number= 0
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| | crossing number= 0
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| | linking number= 0
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| | stick number= 3
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| | unknotting number= 0
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| | conway_notation= -
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| | ab_notation= 0<sub>1</sub>
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| | dowker notation= -
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| | thistlethwaite=
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| | last crossing=
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| | last order=
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| | next crossing= 3
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| | next order= 1
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| | alternating=
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| | class= torus
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| | fibered= fibered
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| | prime= prime
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| | slice= slice
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| | symmetry= fully amphichiral
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| | tricolorable= tricolorable
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| | twist=
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| }}
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| [[Image:unknots.svg|right|165px|thumb|Two simple diagrams of the unknot]]
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| [[Image:thistlethwaite_unknot.svg|right|200px|thumb|A tricky unknot diagram by [[Morwen Thistlethwaite]]]]
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| The '''unknot''' arises in the [[knot theory|mathematical theory of knots]]. Intuitively, the unknot is a closed loop of rope without a [[knot]] in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. [[ambient isotopy|ambient-isotoped]], to the '''standard unknot''', i.e. the embedding of the [[circle]] as a geometrically round circle. The unknot is also called the '''trivial knot'''. An unknot is the [[identity element]] with respect to the [[knot sum]] operation.
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| == Unknotting problem ==
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| {{main|Unknotting problem}}
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| Deciding if a particular knot is the unknot was a major driving force behind [[knot invariant]]s, since it was thought this approach would possibly give an efficient algorithm to [[unknotting problem|recognize the unknot]] from some presentation such as a knot diagram. Currently there are several well-known unknot recognition algorithms (not using invariants), but they are either known to be inefficient or have no efficient implementation. It is not known whether many of the current invariants, such as [[finite type invariant]]s, are a complete invariant of the unknot, but [[Floer homology#Heegaard Floer homology|knot Floer homology]] is known to detect the unknot. Even if they were, the problem of computing them efficiently remains.
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| == Examples ==
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| Many useful practical knots are actually the unknot, including all knots which can be tied in the [[bight (knot)|bight]].<ref name="knotty">{{cite web|url=http://www.volkerschatz.com/knots/knots.html|title=Knotty topics|author=Volker Schatz|accessdate=2007-04-23}}</ref> Other noteworthy unknots are those that consist of rigid line segments connected by universal joints at their endpoints (linkages), that yet cannot be reconfigured into a convex polygon, thus acquiring the name [[stuck unknot|''stuck'' unknots]].<ref>[[Godfried Toussaint]], "A new class of stuck unknots in Pol-6," ''Contributions to Algebra and Geometry'', Vol. 42, No. 2, 2001, pp. 301-306.</ref>
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| ==Invariants==
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| The [[Alexander-Conway polynomial]] and [[Jones polynomial]] of the unknot are trivial:
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| :<math>\Delta(t) = 1,\quad \nabla(z) = 1,\quad V(q) = 1.</math>
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| No other knot with 10 or fewer [[crossing number (knot theory)|crossings]] has trivial Alexander polynomial, but the [[Kinoshita-Terasaka knot]] and [[Conway knot]] (both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot.
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| The [[knot group]] of the unknot is an infinite [[cyclic group]], and the [[knot complement]] is [[homeomorphism|homeomorphic]] to a [[solid torus]].
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| ==See also==
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| *[[Knot (mathematics)]]
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| *[[Unlink]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{Knot Atlas|0 1|Unknot|date=May 7, 2013}}
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| *{{MathWorld|Unknot}}
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| {{Knot theory|state=collapsed}}
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Myrtle Benny is how I'm called and I feel comfortable when people use the complete name. To play baseball is the hobby he will by no means stop performing. South Dakota is exactly where me and my husband live and my family members enjoys it. Managing people has been his working day job for a while.
my web page :: over the counter std test (their explanation)