2025/12/16 by Luis Celso Chan Palomo, Scott A. Taylor, Palomo, Luis Celso Chan +1
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2512.14533
openalex publication_date 2025/12/16 · openalex created_date 2025/12/18 · openalex updated_date 2026/07/28
A nontrivial θ-curve in S3 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.