REMARKS ON TILED TORI

摘要

J. S. Birman and W. W. Menasco [Topology, 33, 525-556 (1994)] introduced and studied a class of imbedded tori in closed braid complements that admit a standard tiling. K. Y. Ng [J. Knot Theory Ramifications, 7, 205-216 (1998)] showed that each essential torus in a closed braid complement that admits standard tiling possesses a staircase pattern. The combinatorial and geometric description of the tori from this class was partially given in [Fund. Math., 189, 195-226 (2006)]. In the present paper, we continue the study of the decomposition of tiled tori into building blocks and show that they are similar in a certain sense. In particular, for the class of geometric tiled tori described by Ng in [J. Knot Theory Ramifications, 7, 205-216 (1998)], we show that each torus of this type consists only of elementary blocks.