Irregular smoothing and the number of Reidemeister moves

TL;DR

This paper introduces a new link diagram invariant based on irregular smoothing, which improves the estimation of Reidemeister moves needed for unlinking compared to previous methods using regular smoothing.

Contribution

The paper presents a novel link diagram invariant utilizing irregular smoothing, enhancing the estimation accuracy for Reidemeister moves required to unlink knots.

Findings

New invariant provides better estimates for certain unknot diagrams

Irregular smoothing can outperform regular smoothing in Reidemeister move estimation

Demonstrated with specific knot diagram examples

Abstract

In the previous paper, we considered a link diagram invariant of Hass and Nowik type using regular smoothing and unknotting number, to estimate the number of Reidemeister moves needed for unlinking. In this paper, we introduce a new link diagram invariant using irregular smoothing, and give an example of a knot diagram of the unknot for which the new invariant gives a better estimation than the old one.