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2012 | 10 | 6 | 2200-2210
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Notes on tiled incompressible tori

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
Let Θ denote the class of essential tori in a closed braid complement which admit a standard tiling in the sense of Birman and Menasco [Birman J.S., Menasco W.W., Special positions for essential tori in link complements, Topology, 1994, 33(3), 525–556]. Moreover, let R denote the class of thin tiled tori in the sense of Ng [Ng K.Y., Essential tori in link complements, J. Knot Theory Ramifications, 1998, 7(2), 205–216]. We define the subclass B ⊂ Θ of typical tiled tori and show that R ⊂ B. We also describe a method allowing to construct new examples of tiled essential tori T which are outside the class B in the strong sense. In [Kazantsev A., Essential tori in link complements: detecting the satellite structure by monotonic simplification, preprint available at http://arxiv.org/abs/1005.5263], Kazantsev showed that the inclusion R ⊂ Θ is proper by giving the corresponding example of a nonthin tiled torus T. It turns out this torus T is inside the class B. We show that the inclusion B ⊂ Θ is proper. It follows that the tori from the class B do not provide the complete geometric description of the class Θ. The main results of the paper are Theorems 2.1 and 2.2 which give a constructive procedure for obtaining examples of nontypical tiled essential tori.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
6
Strony
2200-2210
Opis fizyczny
Daty
wydano
2012-12-01
online
2012-10-12
Bibliografia
  • [1] Birman J.S., Finkelstein E., Studying surfaces via closed braids, J. Knot Theory Ramifications, 1998, 7(3), 267–334 http://dx.doi.org/10.1142/S0218216598000176[Crossref]
  • [2] Birman J.S., Hirsch M.D., A new algorithm for recognizing the unknot, Geom. Topol., 1998, 2, 175–220 http://dx.doi.org/10.2140/gt.1998.2.175[Crossref]
  • [3] Birman J.S., Menasco W.W., Special positions for essential tori in link complements, Topology, 1994, 33(3), 525–556 http://dx.doi.org/10.1016/0040-9383(94)90027-2[Crossref]
  • [4] Burde G., Zieschang H., Knots, 2nd ed., de Gruyter Stud. Math., 5, Walter de Gruyter, Berlin, 1986
  • [5] Finkelstein E., Closed incompressible surfaces in closed braid complements, J. Knot Theory Ramifications, 1998, 7(3), 335–379 http://dx.doi.org/10.1142/S0218216598000188[Crossref]
  • [6] Jaco W.H., Shalen P.B., Seifert Fibered Spaces in 3-Manifolds, Mem. Amer. Math. Soc., 21(220), American Mathematical Society, Providence, 1979
  • [7] Johannson K., Équivalences d’homotopie des variétés de dimension 3, C. R. Acad. Sci. Paris Sér. A-B, 1975, 281(23), A1009–A1010
  • [8] Kazantsev A., Essential tori in link complements: detecting the satellite structure by monotonic simplification, preprint available at http://arxiv.org/abs/1005.5263
  • [9] Lozano M.T., Przytycki J.H., Incompressible surfaces in the exterior of a closed 3-braid. I. Surfaces with horizontal boundary components, Math. Proc. Cambridge Philos. Soc., 1985, 98(2), 275–299 http://dx.doi.org/10.1017/S0305004100063465[Crossref]
  • [10] Ng K.Y., Essential tori in link complements, J. Knot Theory Ramifications, 1998, 7(2), 205–216 http://dx.doi.org/10.1142/S0218216598000139[Crossref]
  • [11] Plachta L., Essential tori admitting a standard tiling, Fund. Math., 2006, 189(3), 195–226 http://dx.doi.org/10.4064/fm189-3-1[Crossref]
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-012-0117-4
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