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.
@article{bwmeta1.element.doi-10_2478_s11533-012-0117-4,
author = {Leonid Plachta},
title = {Notes on tiled incompressible tori},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {2200-2210},
zbl = {1261.57010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0117-4}
}
Leonid Plachta. Notes on tiled incompressible tori. Open Mathematics, Tome 10 (2012) pp. 2200-2210. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0117-4/
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