Abstract
L. Paoluzzi and B. Zimmermann constructed a family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary, and classified them up to homeomorphism. Our main purpose is to determine the canonical decompositions of these manifolds. Using the result, we can obtain an alternative proof of the classification theorem of these manifolds and determine their isometry groups. We also determine their unknotting tunnels. Some of these manifolds are related to certain spatial graphs, so-called Suzuki′s Brunnian graphs. The properties of these manifolds enable us to obtain those of the graphs. Moreover, we give an affirmative answer to Kinoshita′s problem concerning these graphs. In the Appendix, we calculate the volume of these manifolds.
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Ushijima, A. The Canonical Decompositions of Some Family of Compact Orientable Hyperbolic 3-Manifolds with Totally Geodesic Boundary. Geometriae Dedicata 78, 21–47 (1999). https://doi.org/10.1023/A:1005181102798
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DOI: https://doi.org/10.1023/A:1005181102798