Unknot  

Common name  Torus 
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   
AB notation  0_{1} 
Dowker notation   
Next  3_{1} 
Other  
torus, fibered, prime, slice, fully amphichiral 
The unknot arises in the 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. ambientisotoped, 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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How to Untangle Knotted Chains

Untangling a knot
Transcription
Hello, I'm Angelo at Arden Jewelers today we'll be learning how to untangle chains. I'm going to teach you three different techniques you can use at home. Let's get started. All right the first thing I like to do when I have a knot is spread out the chain so I try and get the knot in the middle. And normally it's easiest if you unhook the ends. Okay first technique with this knot is you're going to pick up the knot with your thumb finger and roll it to try and loosen it up so then you can get tweezers in it later. Okay, then I like to use tweezers that just have a little bit of spring and you put them in the knot and then let the spring from the tension kind of loosen it up a little bit more. Then you start get the knot loosened up. Paperclips A very professional tool. I actually use them quite a bit it helps a lot. Make sure you're on a hard surface like a table or something to be able to push down on and you're trying really hard not to grab the chain itself but just going through the knot. Got it. Now that you have your chains untangled let's talk about some ways to keep them from getting tangled in the future. The first method to keep your chains from getting tangled is a ziploc bag. Put the change in a ziplock baggie but leave the clasp out the top and then zip it. This works really well at keeping the chain from tying itself into a knot inside the ziploc bag. The next method you can use at home is a Kleenex. Please do not throw this Kleenex away after you put your nice jewelry in it. But if you lay it out fold it over and then roll it, that will also work. Then place this in your jewelry box to help keep the chain from being tangled. Remember if you get a knot that you can't get untangled, or you notice your chain is damaged, feel free to bring it by Arden Jewelers anytime and we'd be happy to help you fix it. Thank you for watching!
Contents
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. Currently there are several wellknown 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 invariants, are a complete invariant of the unknot, but knot Floer homology is known to detect the unknot. Even if they were, the problem of computing them efficiently remains.
Examples
Many useful practical knots are actually the unknot, including all knots which can be tied in the bight.^{[1]} 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 unknots.^{[2]}
Invariants
The AlexanderConway polynomial and Jones polynomial of the unknot are trivial:
No other knot with 10 or fewer crossings has trivial Alexander polynomial, but the KinoshitaTerasaka 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 nontrivial knot has the same Jones polynomial as the unknot.
The knot group of the unknot is an infinite cyclic group, and the knot complement is homeomorphic to a solid torus.
See also
References
 ^ Volker Schatz. "Knotty topics". Retrieved 20070423.
 ^ Godfried Toussaint (2001). "A new class of stuck unknots in Pol6" (PDF). Contributions to Algebra and Geometry. 42 (2): 301–306. Archived from the original (PDF) on 20030512.
External links
 "Unknot", The Knot Atlas. Accessed: May 7, 2013.
 Weisstein, Eric W. "Unknot". MathWorld.