Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves (Q2909502)

Let \(D\) be an oriented knot diagram. A Reidemeister move of type \(II\) is called matched if the arcs of the knot spanning the corresponding bigon are oriented in the same direction, otherwise it is called unmatched. For a trigon face of a diagram a sign can be defined, and a Reidemeister move of type \(III\) always changes a negative trigon to a positive one or vice versa; in the first case the \(RIII\) move is called positive.NEWLINENEWLINEThe main result of the paper shows for each \(n\geq 3\) an explicit diagram \(D_n\) of the unknot with \(n^2\) crossings, which can be deformed to a trivial diagram by a sequence of \(n(n^2+5)/6\) Reidemeister moves, consisting of positive \(RIII\) moves, unmatched \(RII\) moves deleting bigons and \(RI\) moves deleting positive crossings. Moreover, it is shown that any sequence of Reidemeister moves bringing \(D_n\) to the trivial diagram must contain at least \(n(n^2+5)/6\) unmatched \(RII\) moves deleting bigons, \(RI\) moves deleting positive crossings or positive \(RIII\) moves. Hence the given sequence is minimal.NEWLINENEWLINE\textit{V. I. Arnold} defined certain invariants \(J^{+}\), \(J^{-}\) and \(St\) for generic plane curves [Adv. Sov. Math. 21, 33--91 (1994; Zbl 0864.57027)]. A combination of these invariants is then applied to the curve determined by a knot diagram to get bounds on the number of Reidemeister moves needed to trivialize a diagram. Namely, let \(w\) be the writhe of a diagram, then it is shown that \(J^{-}+St+w/2\) does not change under an \(RI\) move creating a positive crossing or a matched \(RII\) move, but decreases by 1 under an \(RI\) move creating a negative crossing, an unmatched \(RII\) move creating a bigon face or a negative \(RIII\) move. A similar result is proved for \(J^{-}+St-w/2\).