Hyperbolic Brunnian Theta Curves

TL;DR

This paper investigates the geometric and topological properties of Brunnian theta curves, proving conditions under which their exteriors are hyperbolic with totally geodesic boundary and classifying their isotopy types.

Contribution

It establishes that atoroidal Brunnian theta curve exteriors are hyperbolic with totally geodesic boundary and classifies Brunnian spines of genus 2 handlebody knots.

Findings

Exteriors of certain Brunnian theta curves are hyperbolic.

Two Brunnian theta curves are isotopic iff neighborhood isotopic.

Classification of Brunnian spines of genus 2 handlebody knots.

Abstract

A nontrivial $\theta$-curve in $S^3$ is Brunnian if each of its cycles is the unknot. We show that if the exterior of a Brunnian $\theta$-curve is atoroidal, then it does not contain an essential annulus. Previously, Ozawa-Tsutsumi showed that there is no essential disc. Consequently, by Thurston's work, the exterior of an atoroidal Brunnian $\theta$-curve is hyperbolic with totally geodesic boundary. It follows that Brunnian $\theta$-curves of low bridge number have exteriors that are hyperbolic with totally geodesic boundary. We also show that two Brunnian $\theta$-curves are isotopic if and only if they are neighborhood isotopic and classify Brunnian spines of genus 2 handlebody knots. We rely heavily on a classification of annuli in the exteriors of genus two handlebody knots by Koda-Ozawa and further developed by Wang in conjunction with sutured manifold theory results of Taylor.