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Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1353–1361 | Cite as

Achiral 1-Cusped Hyperbolic 3-Manifolds Not Coming from Amphicheiral Null-homologous Knot Complements

  • K. Ichihara
  • I. D. Jong
  • K. Taniyama
Part 2. Special issue “Actual Problems of Algebra and Analysis” Editors: A. M. Elizarov and E. K. Lipachev
  • 6 Downloads

Abstract

It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3- manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.

Keywords and phrases

Amphicheiral knot banding achiral hyperbolic 3-manifold chirally cosmetic filling chirally cosmetic surgery 

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References

  1. 1.
    B. Martelli and C. Petronio, “Dehn filling of the ‘magic’ 3-manifold,” Comm. Anal. Geom. 14, 969–1026 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Weeks, “Hyperbolic structures on three-manifolds,” PhD Thesis (PrincetonUniv. Press, Princeton, 1985).Google Scholar
  3. 3.
    K. Ichihara, I. D. Jong, and H. Masai, “Cosmetic banding on knots and links,” Osaka J. Math. 55, 731–745 (2018).MathSciNetzbMATHGoogle Scholar
  4. 4.
    R. Nikkuni and K. Taniyama, “Symmetries of spatial graphs and Simon invariants,” Fund. Math. 205, 219–236 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    R. Kirby, “Problems in low-dimensional topology,” in Proceedings of the Conference on Geometric Topology, Athens, GA, 1993, Vol. 2. 2 of AMS/IP Studies Adv. Math. (Am. Math. Soc., Providence, RI, 1997), pp. 35–473.Google Scholar
  6. 6.
    S. A. Bleiler, C. D. Hodgson, and J. R. Weeks, “Cosmetic surgery on knots,” in Proceedings of the Kirbyfest, Berkeley, CA, 1998, Vol. 2 of Geom. Topol. Monography (Geom. Topol. Publ., Coventry, 1999), pp. 23–34.Google Scholar
  7. 7.
    K. Ichihara and T. Saito, “Cosmetic surgery and the SL(2,C) Casson invariant for two-bridge knots,” HiroshimaMath. J. 48, 21–37 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    K. Ichihara and Z. Wu, “A note on Jones polynomial and cosmetic surgery,” Comm. Anal. Geom. (in press); arXiv:1606. 03372.Google Scholar
  9. 9.
    Y. Ni and Z. Wu, “Cosmetic surgeries on knots in S3,” J. Reine Angew. Math. 706, 1–17 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. A. Bleiler, “Banding, twisted ribbon knots, and producing reducible manifolds via Dehn surgery,” Math. Ann. 286, 679–696 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J. Hoste, Y. Nakanishi, and K. Taniyama, “Unknotting operations involving trivial tangles,” Osaka J. Math. 27, 555–566 (1990).MathSciNetzbMATHGoogle Scholar
  12. 12.
    J. M. Montesinos, “Surgery on links and double branched covers of S3,” in Knots, Groups, and 3-Manifolds, Papers Dedicated to the Memory of R. H. Fox), Ann. Math. Studies 84, 227–259 (1975).Google Scholar
  13. 13.
    A. Kawauchi, A Survey of Knot Theory (Birkhäuser, Basel, 1996).zbMATHGoogle Scholar
  14. 14.
    A. Zeković, “Computation of Gordian distances and H2-Gordian distances of knots,” Yugosl. J. Oper. Res. 25, 133–152 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    D. Matignon, “On the knot complement problem for non-hyperbolic knots,” TopologyAppl. 157, 1900–1925 (2010).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, College of Humanities and SciencesNihon UniversitySetagaya-ku, TokyoJapan
  2. 2.Department of MathematicsKindai UniversityHigashiosaka City, OsakaJapan
  3. 3.Department of Mathematics, School of EducationWaseda UniversityShinjuku-ku, TokyoJapan

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