On Type II Reidemeister moves of links

Abstract

Östlund (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 Östlund 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.