Generating Infinitely Many Hyperbolic Knots with Plats

TL;DR

This paper explores the connections between plat-position links, braid group dynamics, and Heegaard splittings, leading to a method for constructing infinitely many distinct hyperbolic knots with bounded genus and volume.

Contribution

It introduces a novel approach linking plat position, Hempel distance, and hyperbolic knot construction, producing infinitely many prime hyperbolic knots with specific properties.

Findings

Constructed infinitely many distinct hyperbolic n-bridge knots for n ≥ 3.

Established a lower bound for Hempel distance of the plat from the Heegaard splitting.

Proved that knot genus and hyperbolic volume are bounded below by a linear function.

Abstract

In this paper we study the relationships between links in plat position, the dynamics of the braid group, and Heegaard splittings of double branched covers of $S^3$ over a link. These relationships offer new ways to view links in plat position and a new tool kit for analyzing links. In particular, we show that the Hempel distance of the Heegaard splitting of the double branched cover obtained from a plat is a lower bound for the Hempel distance of that plat. Using the Hempel distance of a knot in bridge position and pseudo-Anosov braids we obtain our main result: a construction of infinitely many sequences of prime hyperbolic $n$-bridge knots for $n \geq 3$, infinitely many of which are distinct. We consider known results to show that the knot genus and hyperbolic volume of these knots are bounded below by a linear function.