Essential tori admitting a standard tiling

Leonid Plachta¹
Fund. Math. 189 (2006), 195-226

Abstract: 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.

MSC (2000): 57M25, 57M15.
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Institute of Applied Problems
of Mechanics and Mathematics
National Academy of Sciences of Ukraine
79060 Naukova 3b, Lviv, Ukraine
and
Institute of Mathematics
University of Gdańsk
Wita Stwosza 57
80-952 Gdańsk, Poland