Diagrammatic representations of 3-periodic entanglements
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917975260397568 |
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| author | Andriamanalina, Toky Evans, Myfanwy E. Mahmoudi, Sonia |
| author_facet | Andriamanalina, Toky Evans, Myfanwy E. Mahmoudi, Sonia |
| contents | Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of simple curves in $\mathbb{R}^3$ that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as preimages of links embedded in the 3-torus $\mathbb{T}^3 = \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ in its universal cover $\mathbb{R}^3$, where two non-isotopic links in $\mathbb{T}^3$ may possess the same TP tangle preimage. We consider the equivalence of TP tangles in $\mathbb{R}^3$ through the use of diagrams representing links in $\mathbb{T}^3$. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in $\mathbb{T}^3$. The final definition of a tridiagram of a link in $\mathbb{T}^3$ allows us to then consider additional notions of equivalence relating non-isotopic links in $\mathbb{T}^3$ that possess the same TP tangle preimage. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14254 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diagrammatic representations of 3-periodic entanglements Andriamanalina, Toky Evans, Myfanwy E. Mahmoudi, Sonia Geometric Topology Algebraic Topology 57K10, 57K12 (Primary) 57K35 (Secondary) Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of simple curves in $\mathbb{R}^3$ that are invariant under translations along three non-coplanar axes. As such, these entanglements can be seen as preimages of links embedded in the 3-torus $\mathbb{T}^3 = \mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{S}^1$ in its universal cover $\mathbb{R}^3$, where two non-isotopic links in $\mathbb{T}^3$ may possess the same TP tangle preimage. We consider the equivalence of TP tangles in $\mathbb{R}^3$ through the use of diagrams representing links in $\mathbb{T}^3$. These diagrams require additional moves beyond the classical Reidemeister moves, which we define and show that they preserve ambient isotopies of links in $\mathbb{T}^3$. The final definition of a tridiagram of a link in $\mathbb{T}^3$ allows us to then consider additional notions of equivalence relating non-isotopic links in $\mathbb{T}^3$ that possess the same TP tangle preimage. |
| title | Diagrammatic representations of 3-periodic entanglements |
| topic | Geometric Topology Algebraic Topology 57K10, 57K12 (Primary) 57K35 (Secondary) |
| url | https://arxiv.org/abs/2401.14254 |