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

TL;DR

This paper introduces a new topological framework using Seifert fibrations and braided open books to analyze and characterize knotted optical fields, expanding beyond traditional Hopfion solutions and linking to experimental and mathematical concepts.

Contribution

It develops a novel theoretical approach employing Seifert fibrations and braided open books to describe knotted optical fields, connecting experimental configurations with advanced topological methods.

Findings

Framework based on Seifert fibrations characterizes knotted optical fields.

Connections established between optical configurations and algebraic topology.

Topological constraints identified for braids with up to 3 strands.

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.