Parity and Relative Parity in Knot Theory

Abstract

In the present paper we give a simple proof of the fact that the set of virtual links with orientable atoms is closed. More precisely, the theorem states that if two virtual diagrams $K$ and $K'$ have orientable atoms and they are equivalent by Reidemeister moves, then there is a sequence of diagrams $K = K_1 \to...\to K_n=K'$ all having orientable atoms where $K_i$ is obtained from $K_{i-1}$ by a Reidemeister move. The initial proof heavily relies on the topology of virtual links and was published in \cite{IM}. Our proof is based on the notion of parity which was introduced by the second named author in 2009. We split the set of crossings of a virtual link diagram into sets of {\it odd} and {\it even} in accordance with a fixed rule. The rule must only satisfy several conditions of Reidemeister's type. Then one can construct functorial mappings of link diagrams by using parity. The concept of parity allows one to introduce new invariants and strengthen well-known ones \cite{Ma1}.