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arXiv · 2407.18855

Knotted optical fields: Seifert fibrations, braided open books and topological constraints

Annalisa Marzuoli·Nicola Sanna
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Abstract

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 Ra\~{n}ada'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.

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Annalisa Marzuoli, Nicola Sanna. 2024-07-26. Knotted optical fields: Seifert fibrations, braided open books and topological constraints. https://arxiv.org/abs/2407.18855

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