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In the 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 tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot.
| Unknot | |
|---|---|
 | |
| Common name | Circle |
| Arf invariant | 0 |
| Braid no. | 1 |
| Bridge no. | 0 |
| Crossing no. | 0 |
| Genus | 0 |
| Linking no. | 0 |
| Stick no. | 3 |
| Tunnel no. | 0 |
| Unknotting no. | 0 |
| Conway notation | - |
| A–B notation | 01 |
| Dowker notation | - |
| Next | 31 |
| Other | |
| torus, fibered, prime, slice, fully amphichiral | |
The unknot is the only knot that is the boundary of an embedded disk, which gives the characterization that only unknots have Seifert genus 0. Similarly, the unknot is the identity element with respect to the knot sum operation.
Unknotting problem
Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP.
It is known that 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 invariants 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.
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![image]()
Thistlethwaite unknot -
![image]()
One of Ochiai's unknots
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.
Every tame knot can be represented as a 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. 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 and Jones polynomial of the unknot are trivial:
No other knot with 10 or fewer 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 homeomorphic to a solid torus.
See also
- Knot (mathematics) – Embedding of the circle in three dimensional Euclidean space
- Unlink – Link that consists of finitely many unlinked unknots
- Unknotting number – Minimum number of times a specific knot must be passed through itself to become untied
References
- Volker Schatz. "Knotty topics". Archived from the original on 2011-07-17. Retrieved 2007-04-23.
- Godfried Toussaint (2001). "A new class of stuck unknots in Pol-6" (PDF). Contributions to Algebra and Geometry. 42 (2): 301–306. Archived from the original (PDF) on 2003-05-12.
External links
- "Unknot", The Knot Atlas. Accessed: May 7, 2013.
- Weisstein, Eric W. "Unknot". MathWorld.
This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Unknot news newspapers books scholar JSTOR November 2021 Learn how and when to remove this message In the 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 tied into it unknotted To a knot theorist an unknot is any embedded topological circle in the 3 sphere that is ambient isotopic that is deformable to a geometrically round circle the standard unknot UnknotCommon nameCircleArf invariant0Braid no 1Bridge no 0Crossing no 0Genus0Linking no 0Stick no 3Tunnel no 0Unknotting no 0Conway notation A B notation01Dowker notation Next31Othertorus fibered prime slice fully amphichiralTwo simple diagrams of the unknot The unknot is the only knot that is the boundary of an embedded disk which gives the characterization that only unknots have Seifert genus 0 Similarly the unknot is the identity element with respect to the knot sum operation Unknotting problemDeciding if a particular knot is the unknot was a major driving force behind knot invariants since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram Unknot recognition is known to be in both NP and co NP It is known that 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 invariants can detect the unknot ExamplesIt 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 Thistlethwaite unknot One of Ochiai s unknots 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 Every tame knot can be represented as a 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 Like crossing number a linkage might need to be made more complex by subdividing its segments before it can be simplified InvariantsThe Alexander Conway polynomial and Jones polynomial of the unknot are trivial D t 1 z 1 V q 1 displaystyle Delta t 1 quad nabla z 1 quad V q 1 No other knot with 10 or fewer 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 homeomorphic to a solid torus See alsoKnot mathematics Embedding of the circle in three dimensional Euclidean space Unlink Link that consists of finitely many unlinked unknots Unknotting number Minimum number of times a specific knot must be passed through itself to become untiedReferencesVolker Schatz Knotty topics Archived from the original on 2011 07 17 Retrieved 2007 04 23 Godfried Toussaint 2001 A new class of stuck unknots in Pol 6 PDF Contributions to Algebra and Geometry 42 2 301 306 Archived from the original PDF on 2003 05 12 External links Unknot The Knot Atlas Accessed May 7 2013 Weisstein Eric W Unknot MathWorld
