Notes on tiled incompressible tori

• Autorzy: Leonid Plachta
• 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.

Słowa kluczowe

EN

Block decomposition; Closed braid complement; Combinatorial pattern; Essential torus; Nontypical tiled torus; Standard tiling of a torus

Kategorie tematyczne

• 57M25: Knots and links in S 3
• 57M15: Relations with graph theory

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 6
• Strony: 2200-2210

Daty

wydano

2012-12-01

online

2012-10-12

Twórcy

autor

Leonid Plachta

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]
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• [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]
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• [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]

Identyfikatory

DOI

10.2478/s11533-012-0117-4