2022 Fiscal Year Annual Research Report
Construction and Classification of Weaves
| Project/Area Number |
22J13397
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| Research Institution | Tohoku University |
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Principal Investigator |
MAHMOUDI Sonia 東北大学, 材料科学高等研究所, 客員研究員
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| Project Period (FY) |
2022-04-22 – 2024-03-31
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| Keywords | knot theory / low-dimensional topology / weave / link thickened torus / periodic tilings / topological invariants / entanglement |
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| Outline of Annual Research Achievements |
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.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
I was able to achieve my study and to obtain my PhD degree. I now have new directions and collaborations. I had the chance to interact with many experts in my field both in Japan and abroad. I attended very useful presentations during international and interdisciplinary conferences. I gave talks to present my results and created new connections. I also received useful comments, solved some of my research problems and learned more about interdisciplinary research. I started a collaboration with experts and progressed on a very interesting math problem that connects our two topics. Finally, with the support of Tohoku University, I organized an international and interdisciplinary workshop on mathematics and materials science entitled Visualization and Ideal Embeddings of Entangled Structures.
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| Strategy for Future Research Activity |
First, I will continue the classification of doubly periodic weaves according to their symmetry and highlight key features in terms of ideal embeddings that could be useful for applications in materials science and systems.
Besides, from a purely mathematical point of view, my objective is to introduce a new algebraic structure to describe and classify weaves using braid theory as well as by using the relation between polynomial invariants of knots and links with algebra. Moreover, the objective is to study other doubly periodic entangled structures and to compare them with our weaves in order to develop interesting applications in materials science. I also plan to develop more projects in collaboration within this topic in Japan and abroad to extend my research and knowledge.
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