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Theoretical analysis demonstrates an exact equivalence between Hilden double cosets and link bridge positions, indicating a geometric bridge for algebraic knot problems.
This paper investigates the relationship between links in bridge position and plat presentations. We prove that Hilden double coset classes of plat presentations correspond exactly to bridge positions of a link up to bridge isotopy. This correspondence allows us to translate algebraic questions about plat presentations into geometric questions about bridge positions. Using this perspective, we re-establish several results from both frameworks. In particular, we show that there is a unique Hilden double coset class for the n-bridge unknot in S 3 , and that torus knots admit a single double coset class in plat position. Finally, we discuss how this correspondence can be used to study plat closures of braids, which is the subject of ongoing research.
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Seth Hovland (2026) studied this question.
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