Minimal unknotting sequences of Reidemeister moves containing unmatched RII moves

TL;DR

This paper introduces a refined invariant combining Arnold's invariants and writhe to accurately estimate the minimal number of Reidemeister moves, including unmatched RII and RI moves, needed to unknot certain diagrams.

Contribution

It extends the application of Arnold's invariants by incorporating writhe to precisely estimate unknotting sequences involving unmatched RII and RI moves.

Findings

The invariant $J^- /2 + St \, ext{plus or minus} \, w/2$ effectively estimates the minimal Reidemeister moves.

The method provides exact bounds for specific knot diagrams of the unknot with complex underlying curves.

The approach improves understanding of unknotting complexity in knot theory.

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