Skew-symmetric augmented matrices and a characterization of virtual doodles

Oscar Ocampo; José Gregorio Rodríguez-Nieto; Olga Patricia Salazar-Díaz

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arXiv · 2412.18959

Skew-symmetric augmented matrices and a characterization of virtual doodles

Oscar Ocampo·José Gregorio Rodríguez-Nieto·Olga Patricia Salazar-Díaz

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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.

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Oscar Ocampo, José Gregorio Rodríguez-Nieto, Olga Patricia Salazar-Díaz. 2024-12-25. Skew-symmetric augmented matrices and a characterization of virtual doodles. https://arxiv.org/abs/2412.18959

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