Construction and Classification of Weaves

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The purpose of this research was to define, construct, and classify periodic weaves as new topological objects. To construct weaves, our approach has been improved using combinatorial arguments and we developed new weaving invariants to classify them using knot theory. In particular, we developed the 'polygonal link methods' which allow one to build and classify weaves using tiling theory. More specifically, given a doubly periodic tiling of the plane and a polygonal link method, by instructing how to cover edges and vertices of the tiling by strands, we introduced a systematic algorithm to predict, distinguish, construct and classify weaving motifs and other entangled structures such as polycatenane and mixed motifs. Then, we started a classification of weaves by their symmetry groups.