Hyperbolicity and Volume of Hyperbolic Bongles

TL;DR

This paper characterizes hyperbolic properties of a class of links called bongles, establishes volume bounds for hyperbolic bongles, and provides explicit volume calculations for small cases.

Contribution

It introduces necessary and sufficient conditions for hyperbolicity of bongles and determines volume bounds, including explicit calculations for small hyperbolic bongles.

Findings

Balanced hyperbolic n-bongles have the same volume.

Volume of hyperbolic n-bongles is bounded above by 5n(1.01494...).

Explicit volume calculations for hyperbolic 3- to 6-bongles.

Abstract

We consider a simple but infinite class of staked links known as bongles. We provide necessary and sufficient conditions for these bongles to be hyperbolic. Then, we prove that all balanced hyperbolic $n$-bongles have the same volume and the corresponding volume is an upper bound on the volume of any hyperbolic $n$-bongle for $n$ even. Moreover, all hyperbolic $n$-bongles have volume strictly less than $5n(1.01494\dots)$. We also include explicit volume calculations for all hyperbolic 3-bongles through 6-bongles.