A polynomial upper bound on Reidemeister moves (Q499087)

The main result of this paper is that any \(c\)-crossing diagram of the unknot may be reduced to the trivial diagram by a sequence of at most \((236c)^{11}\) Reidemeister moves, with all intermediate diagrams having at most \(49c^2\) crossings. There is a parallel result for diagrams of split links. These upper bounds are huge (even for small values of \(c\)). Nevertheless the argument provides a new proof that the problem of recognizing the unknot is in the class \(NP\) of problems for which correct solutions can be verified ``quickly''. Whether this problem is in the class \(P\) of problems with quick decision algorithms remains open. (Note also the lower bound due to \textit{J. Hass} and \textit{T. Nowik} [Discrete Comput. Geom. 44, No. 1, 91--95 (2010; Zbl 1191.57006)] that there are \(c\)-crossing diagrams of the unknot which require at least \(c^2/25\) Reidemeister moves to trivialize.) The argument is based on the notion of arc presentation, due to \textit{I. A. Dynnikov} [Acta Appl. Math. 69, No. 3, 243--283 (2001; Zbl 1006.57001); Fundam. Math. 190, 29--76 (2006; Zbl 1132.57006)], and the theory of normal surfaces.