Free Knots and Parity

TL;DR

This paper explores the concept of parity in knot theories, enabling the construction of new invariants and proving properties of knots, with a focus on virtual and free knots, and demonstrates the existence of non-trivial free knots.

Contribution

It introduces a parity-based framework satisfying axioms that lead to new invariants and results in virtual and free knot theories, including counterexamples to previous conjectures.

Findings

Existence of non-trivial free knots countering Turaev's conjecture

Construction of invariants valued in graphs and groups using parity

Application of parity to extend invariants in virtual knots

Abstract

We consider knot theories possessing a {\em parity}: each crossing is decreed {\em odd} or {\em even} according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, this leads to a possibility of constructing new invariants and proving minimality and non-triviality theorems for knots from these classes, and constructing maps from knots to knots. Our main example is virtual knot theory and its simplifaction, {\em free knot theory}. By using Gauss diagrams, we show the existence of non-trivial free knots (counterexample to Turaev's conjecture), and construct simple and deep invariants made out of parity. Some invariants are valued in graph-like objects and some other are valued in groups. We discuss applications of parity to virtual knots and ways of extending well-known invariants. The existence of a non-trivial parity for classical knots remains an open problem.