Notes on tiled incompressible tori

Leonid Plachta

Open Mathematics (2012)

  • Volume: 10, Issue: 6, page 2200-2210
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Leonid Plachta. "Notes on tiled incompressible tori." Open Mathematics 10.6 (2012): 2200-2210. <http://eudml.org/doc/269717>.

@article{LeonidPlachta2012,
abstract = {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.},
author = {Leonid Plachta},
journal = {Open Mathematics},
keywords = {Block decomposition; Closed braid complement; Combinatorial pattern; Essential torus; Nontypical tiled torus; Standard tiling of a torus; link; closed braid; incompressible tori; foliation; tiled torus},
language = {eng},
number = {6},
pages = {2200-2210},
title = {Notes on tiled incompressible tori},
url = {http://eudml.org/doc/269717},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Leonid Plachta
TI - Notes on tiled incompressible tori
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 2200
EP - 2210
AB - 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.
LA - eng
KW - Block decomposition; Closed braid complement; Combinatorial pattern; Essential torus; Nontypical tiled torus; Standard tiling of a torus; link; closed braid; incompressible tori; foliation; tiled torus
UR - http://eudml.org/doc/269717
ER -

References

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  1. [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] Zbl0907.57006
  2. [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] Zbl0955.57005
  3. [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] Zbl0833.57004
  4. [4] Burde G., Zieschang H., Knots, 2nd ed., de Gruyter Stud. Math., 5, Walter de Gruyter, Berlin, 1986 
  5. [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] Zbl0901.57010
  6. [6] Jaco W.H., Shalen P.B., Seifert Fibered Spaces in 3-Manifolds, Mem. Amer. Math. Soc., 21(220), American Mathematical Society, Providence, 1979 Zbl0471.57001
  7. [7] Johannson K., Équivalences d’homotopie des variétés de dimension 3, C. R. Acad. Sci. Paris Sér. A-B, 1975, 281(23), A1009–A1010 Zbl0313.57003
  8. [8] Kazantsev A., Essential tori in link complements: detecting the satellite structure by monotonic simplification, preprint available at http://arxiv.org/abs/1005.5263 Zbl1221.57006
  9. [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] Zbl0574.57003
  10. [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] Zbl0898.57004
  11. [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] Zbl1099.57009

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