New Parities and Coverings over Free Knots

TL;DR

This paper extends parity theory to develop new, geometry-dependent parities for two-component virtual and free links, which are mutation-sensitive and enhance analysis capabilities in knot theory.

Contribution

It introduces novel parities for two-component links that depend on geometric properties and are mutation-sensitive, broadening the applicability of parity-based methods.

Findings

New parities depend on geometric properties of diagrams.

These parities are mutation-sensitive.

They can be used in all problems previously addressed by parities.

Abstract

In the present paper, we develop the parity theory invented in \cite{ManSb}; we construct new parities for two-component (virtual and free) links. New parities significantly depend on geometrical properties of diagrams; in particular, they are mutation-sensitive. New parities can be used practically in all problems, where parities were previously applied.