The Number of Reidemeister Moves Needed for Unknotting

A knot is an embedding of a circle S in a 3-manifold M , usually taken to be R or S. In the 1920’s Alexander and Briggs [2, §4] and Reidemeister [23] observed that questions about ambient isotopy of polygonal knots in R can be reduced to combinatorial questions about knot diagrams. These are labeled planar graphs with overcrossings and undercrossings marked, representing a projection of the knot onto a plane. They showed that any ambient isotopy of a polygonal knot can be achieved by a finite sequence of piecewise-linear moves which slide the knot across a single triangle, which are called elementary moves (or ∆-moves). They also showed that two knots were ambient isotopic if and only if their knot diagrams were equivalent under a finite sequence of local combinatorial changes, now called Reidemeister moves; see §7. A triangle in M defines a trivial knot, and a loop in the plane with no crossings is said to be a trivial knot diagram. A knot diagram D is unknotted if it is equivalent to a trivial knot diagram under Reidemeister moves. We measure the complexity of a knot diagram D by using its crossing number, the number of vertices in the planar graph D; see §7. A problem of long standing is to determine an upper bound for the number of Reidemeister moves needed to transform an unknotted knot diagram D to the trivial knot diagram, as an explicit function of the crossing number n; see Welsh [30, p. 95]. This paper obtains such a bound.