Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves

Details Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves

2010-11-17

arxiv.org/abs/1011.3963v1

Mathematics

Geometric Topology

10 pages, 11 figures

Scientific paper

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 $r = 2 + \cos (n \theta/(n+1)),\ (0 \le \theta \le 2(n+1)\pi$).

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