View source for Unknot

{{refimprove|date=November 2021}}
{{Short description|Loop seen as a trivial knot}}
{{Infobox knot theory
 | name=              Unknot
 | practical name=    Circle
 | image=             Blue Unknot.png
 | caption=           
 | arf invariant=     0
 | braid number=      1
 | bridge number=     0
 | crossing number=   0
 | genus=             0
 | linking number=    0
 | stick number=      3
 | unknotting number= 0
 | tunnel number=     0
 | conway_notation=   -
 | ab_notation=       0<sub>1</sub>
 | d-t_name =         0a<sub>1</sub>
 | dowker notation=   -
 | thistlethwaite=    
 | last crossing=     
 | last order=        
 | next crossing=     3
 | next order=        1
 | alternating=      
 | class=             torus
 | fibered=           fibered
 | prime=             prime
 | slice=             slice
 | symmetry=          fully amphichiral
 | tricolorable=      
 | twist=             
}}
[[Image:unknots.svg|right|150px|thumb|Two simple diagrams of the unknot]]
In the [[knot theory|mathematical theory of knots]], the '''unknot''', '''not knot''', or '''trivial knot''', is the least knotted of all knots.  Intuitively, the unknot is a closed loop of rope without a [[Knot (mathematics)|knot]] tied into it, unknotted.  To a knot theorist, an unknot is any [[embedding|embedded]] [[Topological sphere|topological circle]] in the [[3-sphere]] that is [[ambient isotopy|ambient isotopic]] (that is, deformable) to a geometrically round [[circle]], the '''standard unknot'''.

The unknot is the only knot that is the boundary of an embedded [[disk (mathematics)|disk]], which gives the characterization that only unknots have [[Seifert surface|Seifert genus]] 0.  Similarly, the unknot is the [[identity element]] with respect to the [[knot sum]] operation.

== Unknotting problem ==
{{main|Unknotting problem}}

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]]. Unknot recognition is known to be in both [[NP (complexity)|NP]] and [[co-NP]].

It is known that [[Heegaard Floer homology|knot Floer homology]] and [[Khovanov homology]] detect the unknot, but these are not known to be efficiently computable for this purpose.  It is not known whether the Jones polynomial or [[finite type invariant]]s can detect the unknot.

== Examples ==
It can be difficult to find a way to untangle string even though the fact it started out untangled proves the task is possible.  Thistlethwaite and Ochiai provided many examples of diagrams of unknots that have no obvious way to simplify them, requiring one to temporarily increase the diagram's [[crossing number (knot theory)|crossing number]]. Such cases are known as [[hard unknot]]s.

<gallery>
Image:thistlethwaite_unknot.svg | [[Morwen Thistlethwaite|Thistlethwaite]] unknot
Image:Ochiai unknot.svg | One of Ochiai's unknots
</gallery>

While rope is generally not in the form of a closed loop, sometimes there is a canonical way to imagine the ends being joined together.  From this point of view, many useful practical knots are actually the unknot, including those that can be tied in a [[bight (knot)|bight]].<ref name="knotty">{{cite web|url=http://www.volkerschatz.com/knots/knots.html|title=Knotty topics|author=Volker Schatz|access-date=2007-04-23|url-status=dead|archive-url=https://web.archive.org/web/20110717230520/http://www.volkerschatz.com/knots/knots.html|archive-date=2011-07-17}}</ref>

Every [[wild knot|tame knot]] can be represented as a [[linkage (mechanical)|linkage]], which is a collection of rigid line segments connected by universal joints at their endpoints.  The [[stick number]] is the minimal number of segments needed to represent a knot as a linkage, and a [[stuck unknot]] is a particular unknotted linkage that cannot be reconfigured into a flat convex polygon.<ref>{{cite journal |author=Godfried Toussaint |author-link=Godfried Toussaint |title=A new class of stuck unknots in Pol-6|journal=Contributions to Algebra and Geometry|date=2001|volume=42|issue=2|pages=301–306|url=http://www.emis.de/journals/BAG/vol.42/no.2/b42h2to1.pdf |archive-url=https://web.archive.org/web/20030512075528/http://www.emis.de/journals/BAG/vol.42/no.2/b42h2to1.pdf |archive-date= 2003-05-12}}</ref>  Like crossing number, a linkage might need to be made more complex by subdividing its segments before it can be simplified.

== Invariants ==
The [[Alexander-Conway polynomial|Alexander–Conway polynomial]] and [[Jones polynomial]] of the unknot are trivial:
: <math>\Delta(t) = 1,\quad \nabla(z) = 1,\quad V(q) = 1.</math>

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.

The unknot is the only knot whose [[knot group]] is an infinite [[cyclic group]], and its [[knot complement]] is [[homeomorphism|homeomorphic]] to a [[solid torus]].

== See also ==
* {{annotated link|Knot (mathematics)}}
* {{annotated link|Unlink}}
* {{Annotated link|Unknotting number}}

== References ==
{{reflist}}

== External links ==
* {{Knot Atlas|0 1|Unknot|date=May 7, 2013}}
* {{MathWorld|Unknot}}

{{Knot theory|state=collapsed}}

[[Category:Circles]]