Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Klein Bottles and Dehn Filling on a Component of Two-component Link Exterior

  • Sayari, Nabil (Departement de Mathematiques et de Statistique, Universite de Moncton)
  • Received : 2019.11.20
  • Accepted : 2020.07.21
  • Published : 2020.12.31
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Abstract

Let M be the exterior of a hyperbolic link K ∪ L in a homology 3-sphere Y, such that the linking number lk(K, L) is non-zero. In this note we prove that if γ is a slope in ∂N(L) such that the manifold ML(γ) obtained by γ-Dehn filling along ∂N(L) contains a Klein bottle, then there is a bound on Δ(μ, γ), depending on the genus of K and on lk(K, L).

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Acknowledgement

We would like to thank the editor and reviewers for careful reading, and constructive suggestions for our manuscript. We would like to thank Professor Rebhi Salem for helpful comments.

References

  1. M. Culler, C. McA. Gordon, J. Luecke and P. Shalen, Dehn surgery on knots, Ann. Math., 125(1987), 237-300. https://doi.org/10.2307/1971311
  2. C. McA. Gordon and J. Luecke, Dehn surgeries on knots creating essential tori, I, Comm. Anal. Geom., 3(3-4)(1995), 597-644. https://doi.org/10.4310/CAG.1995.v3.n4.a3
  3. C. Hayashi and K. Motegi, Only single twists on unknots can produce composite knots, Trans. Amer. Math. Soc., 349(11)(1997), 4465-4479. https://doi.org/10.1090/S0002-9947-97-01628-0
  4. K. Ichihara and M. Teragaito, Klein bottle surgery and genera of knots, Pacific J. Math., 210(2003), 317-333. https://doi.org/10.2140/pjm.2003.210.317
  5. D. Matignon and N. Sayari, Non-orientable surfaces and Dehn surgeries, Canad. J. Math., 56(5)(2004), 1022-1033. https://doi.org/10.4153/CJM-2004-046-9
  6. W. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6(1982), 357-381. https://doi.org/10.1090/S0273-0979-1982-15003-0