The number of Reidemeister moves needed for unknotting

There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1-skeleton of the interior of a compact, orientable, triangulated P L 3-manifold M. There is a positive constant c2 such that for each t � 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c 2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.