Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves

Abstract

Arnold introduced invariants $J^{+}$, $J^{-}$ and $St$ for generic planar curves. It is known that both $J^{+}/2+St$ and $J^{-}/2+St$ are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves for uknotting. $J^{-}/2+St$ works well for unmatched RII moves. However, it works only by halves for RI moves. Let $w$ denote the writhe for a knot diagram. We show that $J^{-}/2+St\pm w/2$ works well also for RI moves, and demonstrate that it gives a precise estimation for a certain knot diagram of the unknot with the underlying curve $r=2+\cos (n\theta /(n+1)),(0\le \theta \le 2(n+1)\pi$).

10 pages, 11 figures