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The Compressibility of Checkerboard Surfaces of Link Diagrams

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Abstract

Consider the checkerboard surfaces defined by some link diagrams. When they are not orientable, one considers the boundary surfaces of small regular neighborhoods of them. This article studies the compressibility problem of these kinds of surfaces in the link complements. The problem is solved by devising a normalization theory for the compressing discs, which brings up an algorithm to read out compressibility directly from the link diagrams. As an application of the algorithm, the compressibility changes under Reidermeister moves are studied. Diagrams from the knot tables are also studied, and surprisingly, some of them are shown to define completely compressible surfaces of this kind. Infinitely many examples of non-alternating knot diagrams with incompressible surfaces of this kind are also constructed.

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References

  1. Aitchison, I. R., Rubinstein, J. H.: Incompressible surfaces and the topology of 3-dimensional manifolds. J. Austral. Math. Soc. (Series A), 55, 1–22 (1993)

    MATH  MathSciNet  Google Scholar 

  2. Menasco, W. W.: Closed incompressible surfaces in alternating knot and link complements. Topology, 23, 37–44 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aitchison, I. R., Lumsden, E., Rubinstein, J. H.: Cusp structures of alternating links. Invent. Math., 109, 473–494 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Thurston, W. P.: The geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1978

  5. Menasco, W. W., Thistlethwaite, M. B.: The classification of alternating links. Ann. of Math., 138, 113–171 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  6. Menasco, W. W.: Determining incompressibility of surfaces in alternating knot and link complements. Pacific J. Math., 117, 353–370 (1985)

    MATH  MathSciNet  Google Scholar 

  7. Menasco, W. W., Thistlethwaite, M. B.: A geometric proof that alternating knots are nontrivial. Math. Proc. Cambridge Philos. Soc., 109, 425–431 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Menasco, W. W., Thistlethwaite, M. B.: Surfaces with boundary in alternating knot exteriors. J. Reine Angew. Math., 426, 47–65 (1992)

    MATH  MathSciNet  Google Scholar 

  9. Rolfsen, D.: Knots and Links, Publish or Perish, Berkeley, CA, 1976

  10. Bao, Z.: Incompressible surfaces derived from checkerboard coloring of knot diagrams, (unpublished paper), 2002

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Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China

    Zhi Qiang Bao

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  1. Zhi Qiang Bao
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Correspondence to Zhi Qiang Bao.

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Partially supported by NSFC (Grant No. 10201002). Partially supported by JSPS (Research No. P03189)

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Bao, Z.Q. The Compressibility of Checkerboard Surfaces of Link Diagrams. Acta Math Sinica 23, 1845–1858 (2007). https://doi.org/10.1007/s10114-005-0914-9

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  • DOI: https://doi.org/10.1007/s10114-005-0914-9

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