Editing Unknot (section)

== 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]].

<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.