Research ArticleNo Access

Classification of doubly periodic untwisted $(p, q)$ -weaves by their crossing number and matrices

Mizuki Fukuda

Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

Mathematics for Advanced Materials Open Innovation Laboratory, AIST, Tohoku University, Sendai 980-8577, Japan

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Motoko Kotani

Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

Mathematics for Advanced Materials Open Innovation Laboratory, AIST, Tohoku University, Sendai 980-8577, Japan

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, and

Sonia Mahmoudi

https://orcid.org/0000-0003-2095-4066

Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

Center for Functional Fabrics, Drexel University, Philadelphia, PA 19104, USA

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA

E-mail Address: sonia.mahmoudi.d8@tohoku.ac.jp

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https://doi.org/10.1142/S0218216523500323Cited by:4 (Source: Crossref)

Abstract

A weave is the lift to the thickened Euclidean plane of a particular type of quadrivalent planar connected graph with an over or under crossing information to each vertex and such that the lifted components are non-intersecting simple open curves. In this paper, we introduce a formal topological definition of weaves as three-dimensional entangled structures and characterize the equivalence classes of doubly periodic untwisted $(p, q)$ -weaves by introducing a new invariant, called crossing matrix. Finally, we suggest a combinatorial approach to classify this specific class of weaves by their crossing number.

Keywords:

AMSC: 57K10, 57K12, 57M15, 05A05

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