Modular links: Bunch algorithm and upper volume bounds

TL;DR

This paper introduces the bunch algorithm, a refined method for constructing modular and Lorenz links that yields geometric and combinatorial insights beyond Williams' original approach. It derives a new upper volume bound for all modular link complements that is independent of word exponents and quadratic in the braid index, using annular Dehn filling and a Seifert-fibred-space framework with parent links. The authors also classify modular link complements by base orders, establishing that classes share a parent manifold and a common bound, and they identify families with volume bounds linear in the word period. Together, these results advance our ability to bound hyperbolic volumes in modular/Lorenz settings and raise questions about uniform bounds and possible extensions to other hyperbolic lifts.

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.

Figures (14)

• Figure 1: Left: The Lorenz template. Right: The split template, i.e., the Lorenz template cut along the branch line. If the $y$-split line is glued to the right half of the branch line and the $x$-split line is glued to the left half of the branch line (without twisting of strips), we obtain the Lorenz template.
• Figure 2: Illustration of Williams' algorithm
• Figure 3: Colour the letters in the code word $w$ and then colour the corresponding strands in Figure \ref{['Fig:WilliamsAlg2']} with the corresponding colour.
• Figure 4: Top: Split template with the ${x_1}^{10}$-arc and the ${y_3}^6$-arc for the labelled code word ${x_1}^{10}{y_1}^2{x_2}^5{y_2}^2{x_3}^7{y_3}^6{x_4}^2{y_4}^2{x_5}^5{y_5}^3$. Note that the ${x_1}^{10}$-arc has $10$$x$-turns and the ${y_3}^6$-arc has $6$$y$-turns. Bottom: Lorenz template with the same arcs, obtained after gluing the split template.
• Figure 5: The $(x_1,1)$-turn and $(y_1,1)$-turn of the labelled code word $w={x_1}^{10}{y_1}^2{x_2}^5{y_2}^2{x_3}^7{y_3}^6{x_4}^2{y_4}^2{x_5}^5{y_5}^3$. These turns correspond to the $1^{\textup{st}}$ and $11^{\textup{th}}$ letters in $w$ respectively.

Theorems & Definitions (37)

• Remark 1.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Remark 3.5
• Definition 3.6: Bunches in a modular link
• Theorem 3.7
• proof
• Proposition 3.8