Modular links: Bunch algorithm and upper volume bounds

TL;DR

This paper introduces the bunch algorithm for modular links, providing new geometric insights and the first upper volume bounds independent of word exponents, with results on volume bounds related to braid index.

Contribution

The paper presents an improved bunch algorithm for modular links and derives novel upper volume bounds that are independent of word exponents.

Findings

First upper volume bound independent of word exponents

Families of modular knot complements with linear volume bounds

Classification of modular links based on word exponents

Abstract

In the 1970s, Williams developed an algorithm that has been used to construct and study modular links in the Lorenz template. We introduce an improved algorithm, which we call the bunch algorithm, to provide more insights into the geometry of modular links and Lorenz links. Using the machinery developed for the bunch algorithm, we provide the first upper volume bound that is independent of word exponents and quadratic in the braid index of the Lorenz link component for all modular link complements. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented.