Hyperbolic Brunnian Theta Curves

Luis Celso Chan Palomo; Scott A. Taylor

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arXiv · 2512.14533

Hyperbolic Brunnian Theta Curves

Luis Celso Chan Palomo·Scott A. Taylor

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Abstract

A nontrivial $θ$-curve in $S^3$ is Brunnian if each of its cycles is the unknot. We show that if the exterior of a Brunnian $θ$-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 $θ$-curve is hyperbolic with totally geodesic boundary. It follows that Brunnian $θ$-curves of low bridge number have exteriors that are hyperbolic with totally geodesic boundary. We also show that two Brunnian $θ$-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.

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BibTeX RIS

Luis Celso Chan Palomo, Scott A. Taylor. 2025-12-16. Hyperbolic Brunnian Theta Curves. https://arxiv.org/abs/2512.14533

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