Realizing exterior Cromwell moves on rectangular diagrams by Reidemeister moves (Q2878656)

Reidemeister moves are local changes in a link diagram. It has long been known that any two diagrams of the same link differ by a finite sequence of Reidemeister moves. If an explicit bound can be given on the length of this sequence then this gives an algorithm to test whether two diagrams represent the same link. A key sub-question is to find such a bound when one of the diagrams is the zero-crossing diagram of the unknot.NEWLINENEWLINEA rectangular diagram is a link diagram formed of arcs that are horizontal or vertical, and such that, at each crossing, the vertical arc is the overcrossing. \textit{I. A. Dynnikov} [Fundam. Math. 190, 29--76 (2006; Zbl 1132.57006)] showed (using rectangular diagrams and the related concept of arc presentations) that a rectangular diagram of the unknot can be changed to the zero-crossing diagram by monotonic simplification. That is, there is a finite sequence of moves, drawn from a fixed list (and referred to as `Cromwell moves' by the authors of the paper under review), between the two diagrams that never increases the complexity of the diagram. However, this sequence may contain many moves that do not change the complexity of the diagram.NEWLINENEWLINEBy noting that there are only finitely many rectangular diagrams of a fixed complexity, Dynnikov therefore gave a bound on the number of Cromwell moves needed to reach the zero-crossing diagram. \textit{A. Henrich} and \textit{L. Kauffman} [Unknotting Unknots, \url{arXiv: 1006.4176}] explicitly calculated the corresponding bound on Reidemeister moves (for link diagrams with a high number of crossings, this bound does not improve on that previously given by \textit{J. Hass} and \textit{J. C. Lagarias} [J. Am. Math. Soc. 14, No. 2, 399--428 (2001; Zbl 0964.57005)]). One step in this is counting the number of Reidemeister moves needed to achieve a Cromwell move. In the paper under review, such calculations are given for some types of Cromwell move not considered by Henrich and Kauffman. One result given also corrects an earlier result by \textit{C. Hayashi} [Math. Ann. 332, No. 2, 239--252 (2005; Zbl 1068.57005)].NEWLINENEWLINE\textit{M. Lackenby} [A polynomial upper bound on Reidemeister moves, \url{arXiv: 1302.0180}] has since extended Dynnikov's work to give a polynomial bound on the number of Reidemeister moves needed to carry an unknot diagram to the zero-crossing diagram. This still leaves open the question of whether there exists a polynomial-time algorithm for this.