Parity on based matrices

TL;DR

This paper introduces a new parity concept for based matrices, an algebraic structure related to virtual knots, expanding the tools for knot theory analysis.

Contribution

It defines the reduced stable parity on based matrices, providing a novel example of parity for virtual knots.

Findings

Introduces the notion of parity for based matrices.

Defines the reduced stable parity on based matrices.

Provides a new example of parity for virtual knots.

Abstract

A parity is a labeling of the crossings of knot diagrams which is compatible with Reidemeister moves. We define the notion of parity for based matrices -- algebraic objects introduced by V. Turaev in his research of virtual strings. We present the reduced stable parity on based matrix which gives a new example of a parity of virtual knots.