Notes on tiled incompressible tori

作者：Plachta Leonid^*

来源：Central European Journal of Mathematics, 2012, 10(6): 2200-2210.

DOI：10.2478/s11533-012-0117-4

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

Let Theta 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 subset of Theta of typical tiled tori and show that R subset of 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. %26lt;br%26gt;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 subset of Theta 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 subset of Theta is proper. It follows that the tori from the class B do not provide the complete geometric description of the class Theta. 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.

• 出版日期2012-12