Bridge Positions and Plat Presentations of Links

Seth Hovland

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arXiv · 2410.22556

Bridge Positions and Plat Presentations of Links

Seth Hovland

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Abstract

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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BibTeX RIS

Seth Hovland. 2024-10-29. Bridge Positions and Plat Presentations of Links. https://arxiv.org/abs/2410.22556

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