Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves
2010
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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