Bridge Positions and Plat Presentations of Links

TL;DR

This paper explores the connection between bridge positions and plat presentations of links, establishing a correspondence that simplifies algebraic questions and yields new insights into specific classes of links.

Contribution

It introduces a correspondence between bridge positions and plat presentations, enabling new proofs and perspectives on link classifications.

Findings

Unique Hilden double coset class for the n-bridge unknot

Single double coset class for torus knots in plat position

Potential for new research on plat closures of knots

Abstract

In this paper we investigate the relationship between links in bridge position and plat presentations. We will show that the Hilden double coset classes of plat presentations of a link are equivalent to bridge positions of the link up to bridge isotopy. This correspondence allows us to reframe algebraic questions about plat presentations in terms of bridge positions. We demonstrate some results about both plat presentations and links in bridge position using this correspondence. For instance, we reprove that there is only one Hilden double coset class of the n-bridge unknot in $\mathbb{S}^3.$ We also show that there is only a single double coset class for torus knots in plat position. Finally, we discuss how this correspondence may be used to investigate plat closures of knots, which is the subject of ongoing research.