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Exceptional and cosmetic surgeries on knots

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Abstract

We show that the bridge distance of a knot determines a lower bound on the genera of essential surfaces and Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries on the knot. In particular, knots with high bridge distance do not admit non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries. We further show that if a knot has bridge distance at least 3 then its bridge number is bounded above by a function of Seifert genus, or indeed by the genus of (almost) any essential surface or Heegaard surface in the surgered manifold.

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Acknowledgments

The authors are grateful to the American Institute of Mathematics for its support through the SQuaREs program. The third author was also supported by NSF Grant DMS-1006369. The fifth author was supported by NSF Grant DMS-1054450. We are also grateful to a referee for numerous helpful comments and for suggesting Corollary 2.

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  1. Jesse Johnson

    Present address: Google, Cambridge, USA

Authors and Affiliations

  1. Department of Mathematics, California State University, Long Beach, Long Beach, CA, 90840, USA

    Ryan Blair

  2. Department of Mathematics, San Jose State University, San Jose, CA, 95192, USA

    Marion Campisi

  3. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Jesse Johnson

  4. Department of Mathematics and Statistics, Colby College, Waterville, ME, 04901, USA

    Scott A. Taylor

  5. Department of Mathematics, The University of Iowa, Iowa City, IA, 52240, USA

    Maggy Tomova

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  1. Ryan Blair
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  2. Marion Campisi
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  5. Maggy Tomova
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Correspondence to Maggy Tomova.

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Blair, R., Campisi, M., Johnson, J. et al. Exceptional and cosmetic surgeries on knots. Math. Ann. 367, 581–622 (2017). https://doi.org/10.1007/s00208-016-1392-3

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