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.
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
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.