Skip to main content
SpringerLink
Log in

Diffeomorphic vs Isotopic Links in Lens Spaces

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper, we provide a set of moves on disk, band and grid diagrams that connects diffeo-equivalent links: there are up to four isotopy equivalent links in each diffeo-equivalence class. Moreover, we investigate how the diffeo-equivalence relates to the lift of the link in the 3-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in L(p, q) up to diffeo-equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baker, K., Grigsby, J.E.: Grid diagrams and Legendrian lens space links. J. Symplectic Geom. 7, 415–448 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, K., Grigsby, J.E., Hedden, M.: Grid diagrams for lens spaces and combinatorial knot Floer homology. Int. Math. Res. Not. IMRN 10, 39 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Bleiler, S.A., Hodgson, C., Weeks, J. R., Cosmetic surgery on knots. In: Proceedings of the Kirbyfest (Berkeley, 1998), Geom. Topol. Monogr., vol. 2. Geom. Topol. Publ. (1999)

  4. Boileau, M., Flapan, E.: Uniqueness of free actions on \(\mathbf{S}^{3}\) respecting a knot. Can. J. Math. 39, 969–982 (1987)

    Article  MATH  Google Scholar 

  5. Bonahon, F.: Difféotopies des espaces lenticulaires. Topology 22, 305–314 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buck, D., Mauricio, M.: Connect sum of lens spaces surgeries: application to Hin recombination. Math. Proc. Cam. Philos. Soc. 150, 505–525 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cattabriga, A., Manfredi, E., Mulazzani, M.: On knots and links in lens spaces. Topol. Appl. 160, 430–442 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cattabriga, A., Manfredi, E., Rigolli, L.: Equivalence of two diagram representations of links in lens spaces and essential invariants. Acta Math. Hung. 146, 168–201 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cattabriga, A., Nasybullov, T.: Virtual quandle for links in lens spaces. RACSAM (to appear). arXiv:1702.05964

  10. Christensen, A.: A Gordon–Luecke-type argument for knots in lens spaces. Topology 37, 935–944 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cornwell, C.: A polynomial invariant for links in lens spaces. J. Knot Theory Ramif. 21, 125006031 (2012)

    Article  MathSciNet  Google Scholar 

  12. Drobotukhina, Y.V.: An analogue of the Jones polynomial for links in \(\mathbb{R}P^{3}\) and a generalization of the Kauffman-Murasugi theorem. Leningrad Math. J. 2, 613–630 (1991)

    MathSciNet  MATH  Google Scholar 

  13. Gabrovšek, B.: Tabulation of prime knots in lens spaces. Meditter. J. Math. 44, 88 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gabrovšek, B., Manfredi, E.: On the KBSM of links in lens spaces. J. Knot Theory Ramif. 27, 01 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Gabrovšek, B., Manfredi, E.: On the Seifert fibered space link group. Topol. Appl. 206, 255–275 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gabrovšek, B., Mroczkowski, M.: The HOMFLY-PT skein module of the lens space \(L(p,1)\). Topol. Appl. 175, 72–80 (2014)

    Article  MATH  Google Scholar 

  17. Gainullin, F.: Heegaard Floer homology and knots determined by their complements, preprint (2015). arXiv:1504.06180

  18. Hedden, M.: On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Am. Math. Soc. 363, 949–968 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces, in Knot theory and manifold (Vancouver, B.C., 1983), Lecture Notes in Math., vol. 1144. Springer, Berlin (1985)

    Google Scholar 

  20. Hoste, J., Przytycki, J.H.: The \((2,\infty )\)-skein module of lens spaces; a generalization of the Jones polynomial. J. Knot Theory Ramif. 2, 321–333 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Manfredi, E.: Lift in the \(3\)-sphere of knots and links in lens spaces. J. Knot Theory Ramif. 23, 145002221 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mathieu, Y.: Closed 3-manifolds unchanged by Dehn surgery. J. Knot Theory Ramif. 1, 279–296 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Matignon, D.: On the knot complement problem for non-hyperbolic knots. Topol. Appl. 157, 1900–1925 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rolfsen, D.: Knots and links. AMS Chelsea, Madison (1976)

    MATH  Google Scholar 

  25. Sakuma, M.: Uniqueness of symmetries of knots. Math. Z. 192, 225–242 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stevan, S.: Torus knots in lens spaces & topological strings. Ann. Henry Poincaré 16, 1937–1967 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Tietze, H.: Ueber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten. Monatsh. fuer Math. und Phys. 19, 1–118 (1908)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Michele Mulazzani for his useful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna, Italy

    Alessia Cattabriga & Enrico Manfredi

Authors
  1. Alessia Cattabriga
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enrico Manfredi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessia Cattabriga.

Additional information

A. Cattabriga has been supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA-INdAM) and University of Bologna.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cattabriga, A., Manfredi, E. Diffeomorphic vs Isotopic Links in Lens Spaces. Mediterr. J. Math. 15, 172 (2018). https://doi.org/10.1007/s00009-018-1217-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1217-6

Mathematics Subject Classification

Keywords

Search

Navigation