A closed loop which is not Knotted. In the 1930s, by making use of Reidemeister Moves, Reidemeister first proved that Knots exist which are distinct from the unknot. He proved this by Coloring each part of a knot diagram with one of three colors.
The Knot Sum of two unknots is another unknot.
The Jones Polynomial of the unknot is defined to give the normalization
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. Although it is not immediately obvious, the unknot is a Prime Knot.
See also Colorable, Knot, Knot Theory, Link, Reidemeister Moves, Unknotting Number
Haken, W. ``Theorie der Normalflachen.'' Acta Math. 105, 245-375, 1961.