Skew-symmetric augmented matrices and a characterization of virtual doodles

TL;DR

This paper introduces skew-symmetric augmented matrices to determine the non-classical nature of virtual doodles and characterizes the virtualization process of classical doodles, building on topological and algebraic frameworks.

Contribution

It presents a novel matrix-based method for classifying virtual doodles and characterizes their virtualization, advancing the understanding of doodle equivalence and virtual knot theory.

Findings

Skew-symmetric augmented matrices can detect non-classical virtual doodles.

The paper provides a characterization of virtualization in classical doodles.

Connections between homology intersection numbers and doodle classification are established.

Abstract

In this paper, we present a brief overview of the concept of doodles from the perspective of J.S. Carter's work on classifying immersed curves and the work of J.S. Carter, S. Kamada, and M. Saito on stable equivalence of knots on surfaces and virtual knot cobordisms. We use the homology intersection number and the work of G. Cairns and D. Elton on the Gauss word problem to introduce the concept of skew-symmetric augmented matrices for determining whether a virtual doodle is non-classical. We also provide a characterization of the virtualization of classical doodles.