The unknot, and a knot equivalent to it
The unknot is a loop of rope without a knot in it (in knot theory, ropes have no ends; they are loops).
A knot theory might describe the unknot as the standard embedding of the circle (i.e. what the layman would call a circle), though any knot that can be 'untied' to give this would also be considered as the unknot. Deciding if a particular knot is the unknot was a major driving force behind knot invariants and it is not known whether the current invariants are sufficient to answer this question.
An unknot is the identity element with respect to the connected sum of knots operation.
This object is also called the not knot or the trivial knot. The latter is the more natural name for it, since it fits in with the usual mathematical concept of 'trivial', but the punning term unknot has stuck. Nevertheless, it is common to say a knot 'is trivial' to mean it is the unknot.
Some useful rope shapes are actually unknots, such as the slip knot and the sheepshank.
See also: