Knotted optical fields: Seifert fibrations, braided open books and topological constraints
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| Format: | Preprint |
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2024
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| _version_ | 1866916337510514688 |
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| author | Marzuoli, Annalisa Sanna, Nicola |
| author_facet | Marzuoli, Annalisa Sanna, Nicola |
| contents | This paper presents a novel framework for studying knotted and braided configurations of optical fields, moving beyond the conventional Hopfion solution based on the Hopf fibration. By employing the Seifert fibration, a preferred framing is introduced to characterize the ``knottiness" of tubular neighbourhoods of knots embedded in the 3-sphere. This approach yields a specific presentation of Seifert surfaces and facilitates the description of knotted optical fields, drawing on an enriched version of Rañada's formulation. The relation between knots and braids enables one to explore the connections between configurations generated experimentally through the controlled over- and under-crossings of light beams, and concepts from geometric group theory and algebraic topology. This framework, emphasizing the significance of ``braided open books", sheds light on topological constraints inherent in classes of braids suitable for representing interlaced optical fields. These constraints primarily pertain to braids with $n\leq3$ strands -- aligning with existing experimental implementations -- and homogeneous braids with any number of strands. This paper establishes a theoretical foundation for understanding and manipulating knotted optical fields, suggesting potential applications and avenues for further research in this domain. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18855 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Knotted optical fields: Seifert fibrations, braided open books and topological constraints Marzuoli, Annalisa Sanna, Nicola Mathematical Physics Optics This paper presents a novel framework for studying knotted and braided configurations of optical fields, moving beyond the conventional Hopfion solution based on the Hopf fibration. By employing the Seifert fibration, a preferred framing is introduced to characterize the ``knottiness" of tubular neighbourhoods of knots embedded in the 3-sphere. This approach yields a specific presentation of Seifert surfaces and facilitates the description of knotted optical fields, drawing on an enriched version of Rañada's formulation. The relation between knots and braids enables one to explore the connections between configurations generated experimentally through the controlled over- and under-crossings of light beams, and concepts from geometric group theory and algebraic topology. This framework, emphasizing the significance of ``braided open books", sheds light on topological constraints inherent in classes of braids suitable for representing interlaced optical fields. These constraints primarily pertain to braids with $n\leq3$ strands -- aligning with existing experimental implementations -- and homogeneous braids with any number of strands. This paper establishes a theoretical foundation for understanding and manipulating knotted optical fields, suggesting potential applications and avenues for further research in this domain. |
| title | Knotted optical fields: Seifert fibrations, braided open books and topological constraints |
| topic | Mathematical Physics Optics |
| url | https://arxiv.org/abs/2407.18855 |