The unknot, also called the trivial knot (Rolfsen 1976, p. 51), is a closed loop that is not knotted. In the 1930s Reidemeister first proved that knots exist which are distinct from the unknot by inventing and making use of the so-called Reidemeister moves and coloring each part of a knot diagram with one of three colors.
The unknot is implemented in the Wolfram Language as KnotData["Unknot"].
The knot sum of two unknots is another unknot.
The Jones polynomial of the unknot is defined to give the normalization
 |
(1)
|
The unknot has Alexander polynomial 
and Conway
polynomial 
 |  |  |
(2)
|
 |  |  |
(3)
|
Surprisingly, there are known examples of nontrivial knots with Alexander polynomial 1, although no
such examples occur among the knots of 10 or fewer crossings. An example is the 
-pretzel
knot (Adams 1994, p. 167). Rolfsen (1976, p. 167) gives four other
such examples.
Haken (1961) devised an algorithm to tell if a knot projection is the unknot. The algorithm is so complicated, however, that it has never been implemented.
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