Thirty-two equivalence relations on knot projections

TL;DR

This paper classifies 32 homotopy equivalence relations on knot projections based on Reidemeister move restrictions, identifying 20 distinct non-trivial cases and introducing new invariants as tools.

Contribution

It introduces a comprehensive classification of homotopy relations on knot projections and develops new invariants to distinguish these cases.

Findings

20 non-trivial equivalence classes identified

32 classifications analyzed with move restrictions

New invariants developed for classification

Abstract

We consider 32 homotopy classifications of knot projections (images of generic immersions from a circle into a 2-sphere). These 32 equivalence relations are obtained based on which moves are forbidden among the five type of Reidemeister moves. We show that 32 cases contain 20 non-trivial cases that are mutually different. To complete the proof, we obtain new tools, i.e., new invariants.