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Four-dimensional manifolds constructed by lens space surgeries of distinct types

    https://doi.org/10.1142/S0218216517500699Cited by:0
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    Abstract

    A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We study pairs of lens space surgeries along distinct knots whose lens spaces (i.e. the resulting lens spaces of the surgeries) are orientation-preservingly or -reversingly homeomorphic. In the authors’ previous work, we treated with the case both knots are torus knots. In this paper, we focus on the case where one is a torus knot and the other is a Berge’s knot Type VII or VIII, in a genus one fiber surface. We determine the complete list (set) of such pairs of lens space surgeries and study the closed 4-manifolds constructed as above. The list consists of six sequences. All framed links and handle calculus are indexed by integers.

    AMSC: 57M25, 57N35, 57M99

    References

    • 1. K. Baker, Surgery descriptions and volumes of Berge knots. I. Large volume Berge knots, J. Knot Theory Ramifications 17(9) (2008) 1077–1097. Link, ISI, Google Scholar
    • 2. J. Berge, Some knots with surgeries yielding lens spaces, Unpublished manuscript (1990). Google Scholar
    • 3. F. Bonahon, Diffeotopies des espaces lenticulaires, Topology 22(3) (1983) 305–314. Crossref, ISI, Google Scholar
    • 4. S. Boyer, Simply-connected 4-manifolds with a given boundary, Trans. Amer. Math. Soc. 298(1) (1986) 331–357. ISI, Google Scholar
    • 5. R. Gompf and A. Stipsicz, 4-manifolds and Kirby Calculus, Graduate Studies in Mathematics, Vol. 20 (American Mathematical Society, Providence, RT, 1999). Crossref, Google Scholar
    • 6. J. E. Greene, The lens space realization problem, Ann. of Math. (2) 177(2) (2013) 449–511. Crossref, ISI, Google Scholar
    • 7. C. Hodgson and J. H. Rubinstein, Involutions and isotopies of lens spaces, in Knot Theory and Manifolds, Lecture Notes in Mathematics, Vol. 1144 (Springer, Berlin, 1985), pp. 60–96. Crossref, Google Scholar
    • 8. K. Ichihara and T. Saito, Surgical distance between lens spaces, Tokyo J. Math. 34(1) (2011) 153–164. Crossref, ISI, Google Scholar
    • 9. T. Kadokami, Hyperbolicity and identification of Berge knots of types VII and VIII, preprint (2011), arXiv: math.GT/1107.0379. Google Scholar
    • 10. T. Kadokami and Y. Yamada, Lens space surgeries along certain 2-component links related with Park’s rational blow down, and Reidemeister–Turaev torsion, J. Aust. Math. Soc. 96(1) (2014) 78–126. Crossref, ISI, Google Scholar
    • 11. R. Kirby, A calculus for framed links in S3, Invent. Math. 45 (1978) 35–56. Crossref, ISI, Google Scholar
    • 12. R. Kirby, The topology of four manifolds, Lecture Notes in Mathematics, Vol. 1374 (Springer-Verlag, Berlin, 1989). Crossref, Google Scholar
    • 13. P. Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007) 429–472. Crossref, ISI, Google Scholar
    • 14. L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971) 737–745. Crossref, ISI, Google Scholar
    • 15. D. Rolfsen, Knots and Links (Publish or Perish, Berkeley, 1976). Google Scholar
    • 16. T. Saito and M. Teragaito, Knots yielding homeomorphic lens spaces by Dehn surgery, Pacific J. Math. 244(1) (2010) 169–192. Crossref, ISI, Google Scholar
    • 17. H. Sasahira, Instanton Floer homology for lens spaces, Math. Z. 273(1–2) (2013) 237–281, doi:10.1007/s00209-012-1003-2. Crossref, ISI, Google Scholar
    • 18. M. Tange and Y. Yamada, Four-dimensional manifolds constructed by lens space surgeries along torus knots, J. Knot Theory Ramifications 21 (11) (2012) 12501. Link, ISI, Google Scholar
    • 19. M. Ue, On the intersection forms of spin 4-manifolds bounded by spherical 3-manifolds, Algebr. Geom. Topol. 1 (2001) 549–578. Crossref, Google Scholar
    • 20. Y. Yamada, Berge’s knots in the fiber surfaces of genus one, lens spaces and framed links, J. Knot Theory Ramifications 14(2) (2005) 177–188. Link, ISI, Google Scholar
    • 21. Y. Yamada, A family of knots yielding graph manifolds by Dehn surgery, Michigan Math. J. 53(3) (2005) 683–690. Crossref, ISI, Google Scholar
    • 22. Y. Yamada, Generalized rational blow-down, torus knots and Euclidean algorithm, preprint (2007), arXiv:math.GT/0708.2316.. Google Scholar
    • 23. Y. Yamada, Canonical forms of the knots in the genus one fiber surfaces, Bull. Univ. Electro-Commun. 22(1) (2010) 25–31. Google Scholar
    Published: 15 August 2017