On Type II Reidemeister moves of links

TL;DR

This paper investigates the role of Type II Reidemeister moves in link homotopy, introduces a Gauss diagram formula for two-component links, and discusses implications for the Ostlund conjecture and link invariants.

Contribution

It provides a new Gauss diagram formula for two-component links, advancing the understanding of Reidemeister moves and link invariants.

Findings

Counterexamples to Ostlund conjecture are acknowledged.

A new computable Gauss diagram formula for two-component links is introduced.

Implications for link homotopy and invariants are discussed.

Abstract

\"Ostlund (2001) showed that all planar isotopy invariants of generic plane curves that are unchanged under cusp moves and triple point moves, and of finite degree (in self-tangency moves) are trivial. Here the term "of finite degree" means Arnold-Vassiliev type. It implies the conjecture, which was often called \"Ostlund conjecture: "Types I and III Reidemeister moves are sufficient to describe a homotopy from any generic immersion from the circle into the plain to the standard embedding of the circle". Although counterexamples are known nowadays, there had been no (easy computable) function that detects the difference between the counterexample and the standard embedding on the plain. However, we introduce a desired function (Gauss diagram formula) is found for the two-component case.