There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c 1 n 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 triangulated orientable PL 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 2 t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M . We obtain explicit values for c1 and c2 . Keywords: knot theory, knot diagram, Reidemeister move, normal surfaces, computational complexity This paper grew out of work begun while the authors were visiting the Mathematical Sciences Research Institute in Berkeley in 1996/7. Research at MSRI is supported in part by NSF grant DMS-9022140. The first au...

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.