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In themathematical theory of knots, theunknot,not knot, ortrivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without aknot tied into it, unknotted. To a knot theorist, an unknot is anyembeddedtopological circle in the3-sphere that isambient isotopic (that is, deformable) to a geometrically roundcircle, thestandard unknot.
The unknot is the only knot that is the boundary of an embeddeddisk, which gives the characterization that only unknots haveSeifert genus 0. Similarly, the unknot is theidentity element with respect to theknot sum operation.
Deciding if a particular knot is the unknot was a major driving force behindknot invariants, since it was thought this approach would possibly give an efficient algorithm torecognize the unknot from some presentation such as aknot diagram. Unknot recognition is known to be in bothNP andco-NP.
It is known thatknot Floer homology andKhovanov homology detect the unknot, but these are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial orfinite type invariants can detect the unknot.
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'scrossing number.
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 abight.[1]
Everytame knot can be represented as alinkage, which is a collection of rigid line segments connected by universal joints at their endpoints. Thestick number is the minimal number of segments needed to represent a knot as a linkage, and astuck unknot is a particular unknotted linkage that cannot be reconfigured into a flat convex polygon.[2] Like crossing number, a linkage might need to be made more complex by subdividing its segments before it can be simplified.
TheAlexander–Conway polynomial andJones polynomial of the unknot are trivial:
No other knot with 10 or fewercrossings has trivial Alexander polynomial, but theKinoshita–Terasaka knot andConway 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 whoseknot group is an infinitecyclic group, and itsknot complement ishomeomorphic to asolid torus.
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