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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.15252 |
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| _version_ | 1866915765274279936 |
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| author | Andriamanalina, Toky Mahmoudi, Sonia Evans, Myfanwy E. |
| author_facet | Andriamanalina, Toky Mahmoudi, Sonia Evans, Myfanwy E. |
| contents | Periodic networks serve as models for the structural organisation of biological and chemical crystalline systems. Single or multiple networks can have different configurations in space, where entanglement may arise due to the way the (possibly curvilinear) edges weave around each other. This entanglement influences the functional, physical, and chemical properties of the materials modelled by the networks, which highlights the need to quantify its complexity. In this paper, we define the least tangled embeddings of 3-periodic networks that we call ground states, through the use of knot-theoretic crossing diagrams. The concept of a ground state permits the definition of a measure of entanglement complexity called the untangling number that quantifies the distance between a given 3-periodic structure and its least tangled version. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15252 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measuring the entanglement complexity of 3-periodic networks through the untangling number Andriamanalina, Toky Mahmoudi, Sonia Evans, Myfanwy E. Geometric Topology Periodic networks serve as models for the structural organisation of biological and chemical crystalline systems. Single or multiple networks can have different configurations in space, where entanglement may arise due to the way the (possibly curvilinear) edges weave around each other. This entanglement influences the functional, physical, and chemical properties of the materials modelled by the networks, which highlights the need to quantify its complexity. In this paper, we define the least tangled embeddings of 3-periodic networks that we call ground states, through the use of knot-theoretic crossing diagrams. The concept of a ground state permits the definition of a measure of entanglement complexity called the untangling number that quantifies the distance between a given 3-periodic structure and its least tangled version. |
| title | Measuring the entanglement complexity of 3-periodic networks through the untangling number |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2506.15252 |