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Hyperbolicity and Volume of Hyperbolic Bongles

Colin Adams, Francisco Gomez-Paz, Jiachen Kang, Lukas Krause, Gregory Li, Reyna Li, Chloe Marple, Ziwei Tan

TL;DR

The paper addresses the problem of classifying hyperbolicity for the infinite family of bongles and obtaining quantitative volume bounds. It uses Adams–Chen type criteria to establish that hyperbolicity occurs exactly for alternating $n$-bongles, and then decomposes the hyperbolic complements into generalized $3$-bipyramids to analyze volumes. For fixed even $n$, balanced alternating $n$-bongles share a common maximal volume $V_n^B$, which provides an upper bound for all hyperbolic $n$-bongles, culminating in the universal bound $ ext{vol} < 5 n v_{tet}$, where $v_{tet}$ is the ideal regular tetrahedron volume; Ushijima's formula and Lagrange multipliers are employed to identify the maximal configuration. The authors also supply explicit volume computations for hyperbolic $3$- through $6$-bongles, illustrating the theory and its computational accessibility. This work advances understanding of volume behavior in a structured class of hyperbolic links with totally geodesic boundary and demonstrates how geometric decompositions and optimization yield sharp global bounds.

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.

Hyperbolicity and Volume of Hyperbolic Bongles

TL;DR

The paper addresses the problem of classifying hyperbolicity for the infinite family of bongles and obtaining quantitative volume bounds. It uses Adams–Chen type criteria to establish that hyperbolicity occurs exactly for alternating -bongles, and then decomposes the hyperbolic complements into generalized -bipyramids to analyze volumes. For fixed even , balanced alternating -bongles share a common maximal volume , which provides an upper bound for all hyperbolic -bongles, culminating in the universal bound , where is the ideal regular tetrahedron volume; Ushijima's formula and Lagrange multipliers are employed to identify the maximal configuration. The authors also supply explicit volume computations for hyperbolic - through -bongles, illustrating the theory and its computational accessibility. This work advances understanding of volume behavior in a structured class of hyperbolic links with totally geodesic boundary and demonstrates how geometric decompositions and optimization yield sharp global bounds.

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 -bongles have the same volume and the corresponding volume is an upper bound on the volume of any hyperbolic -bongle for even. Moreover, all hyperbolic -bongles have volume strictly less than . We also include explicit volume calculations for all hyperbolic 3-bongles through 6-bongles.
Paper Structure (4 sections, 6 theorems, 30 equations, 11 figures)

This paper contains 4 sections, 6 theorems, 30 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Bongles
  3. Volumes of Bongles
  4. Volume Table

Key Result

Theorem 2.1

adamschen Let $F$ be a projection surface with nonempty boundary which is not a disk, and let $L \subset F \times I$ be a link with a connected, reduced alternating projection diagram $\pi(L) \subset F \times \{1/2\}$ with at least one crossing. Let $M = (F \times I)\setminus N(L).$ Then $M$ is tg-h

Figures (11)

  • Figure 1: It is forbidden to push a strand over a stake.
  • Figure 2: Examples of bongles.
  • Figure 3: Essential annuli in the complement of non-alternating bongles. For simplicity, parts of the bongle are contracted into tangles, denoted $T$.
  • Figure 4: Example of a section of an annulus near a pair of stakes for Types II and III. For Type I, both stakes are to the front or back.
  • Figure 5: The core curve of each type of annulus appears in orange.
  • ...and 6 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 3 more