The number of Reidemeister moves needed for unknotting

Abstract

There is a positive constant c 1 c_1 such that for any diagram D \mathcal {D} representing the unknot, there is a sequence of at most 2 c 1 n 2^{c_1 n} Reidemeister moves that will convert it to a trivial knot diagram, where n n is the number of crossings in D \mathcal {D} . A similar result holds for elementary moves on a polygonal knot K K embedded in the 1-skeleton of the interior of a compact, orientable, triangulated P L PL 3-manifold M M . There is a positive constant c 2 c_2 such that for each t ≥ 1 t \geq 1 , if M M consists of t t tetrahedra and K K is unknotted, then there is a sequence of at most 2 c 2 t 2^{c_2 t} elementary moves in M M which transforms K K to a triangle contained inside one tetrahedron of M M . We obtain explicit values for c 1 c_1 and c 2 c_2 .