The number of Reidemeister moves needed for unknotting

Abstract:

There is a positive constant $c_1$ such that for any diagram $\mathcal {D}$ 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 $\mathcal {D}$. 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 $PL$ 3-manifold $M$. There is a positive constant $c_2$ such that for each $t \geq 1$, 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 $c_1$ and $c_2$.