Functorial maps and weak parities

TL;DR

This paper explores the concept of weak parities in knot diagrams, introduces the maximal weak parity, and demonstrates how it induces a projection from virtual to classical knots, enhancing understanding of knot invariants.

Contribution

It introduces the notion of maximal weak parity and describes its application to knots in closed oriented surfaces, linking virtual and classical knot theories.

Findings

Maximal weak parity is characterized for knots in surfaces.

Weak parity provides a projection from virtual to classical knots.

Functorial maps and weak parities are shown to be equivalent descriptions.

Abstract

Functorial maps and weak parities are equivalent descriptions of rules of substitution virtual crossings for classical in diagrams of a knot in a way compatible with Reidemeister moves. We introduce the notion of maximal weak parity and describe it for knots in a given closed oriented surface. This weak parity defines a projection from virtual knots to classical knots.