Essential tori admitting a standard tiling

Volume 189 / 2006

Leonid Plachta Fundamenta Mathematicae 189 (2006), 195-226 MSC: 57M25, 57M15. DOI: 10.4064/fm189-3-1


Birman and Menasco (1994) introduced and studied a class of embedded tori in closed braid complements which admit a standard tiling. The geometric description of the tori from this class was not complete. Ng showed (1988) that each essential torus in a closed braid complement which admits a standard tiling possesses a staircase tiling pattern.

In this paper, we introduce and study the so-called longitude-meridional patterns for essential tori admitting a standard tiling. A longitude-meridional pattern of an essential torus can be derived from the corresponding tiled torus and carries a portion of geometric information about the embedded torus. We also study the interplay between the geometry of essential embedded tori and combinatorics of the corresponding tiled tori.


  • Leonid PlachtaInstitute of Applied Problems
    of Mechanics and Mathematics
    National Academy of Sciences of Ukraine
    79060 Naukova 3b, Lviv, Ukraine
    Institute of Mathematics
    University of Gdańsk
    Wita Stwosza 57
    80-952 Gdańsk, Poland

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