Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Topological Imitations and Reni-Mecchia-Zimmermann's Conjecture

  • Received : 2005.03.15
  • Published : 2006.03.23
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Abstract

M. Reni has shown that there are at most nine mutually inequivalent knots in the 3-sphere whose 2-fold branched covering spaces are mutually homeomorphic, hyperbolic 3-manifolds. By observing that the Z-homology sphere version of M. Reni's result still holds, M. Mecchia and B. Zimmermann showed that there are exactly nine mutually inequivalent, knots in Z-homology 3-spheres whose 2-fold branched covering spaces are mutually homeomorphic, hyperbolic 3-manifolds, and conjectured that there exist exactly nine mutually inequivalent, knots in the true 3-sphere whose 2-fold branched covering spaces are mutually homeomorphic, hyperbolic 3-manifolds. Their proof used an argument of AID imitations published in 1992. The main result of this paper is to solve their conjecture affirmatively by combining their argument with a theory of strongly AID imitations published in 1997.

Keywords

References

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