The untangling number of 3-periodic tangles
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911698904940544 |
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| author | Andriamanalina, Toky Mahmoudi, Sonia Evans, Myfanwy E. |
| author_facet | Andriamanalina, Toky Mahmoudi, Sonia Evans, Myfanwy E. |
| contents | The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological, chemical and physical systems. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states are crystallographic rod packings, well known in structural chemistry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The untangling number of 3-periodic tangles Andriamanalina, Toky Mahmoudi, Sonia Evans, Myfanwy E. Geometric Topology 57K10, 57K12, 57K35 The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological, chemical and physical systems. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states are crystallographic rod packings, well known in structural chemistry. |
| title | The untangling number of 3-periodic tangles |
| topic | Geometric Topology 57K10, 57K12, 57K35 |
| url | https://arxiv.org/abs/2504.01747 |