KATAGIRI Minnyou Faculty of Science, Nara Women's University, Associate Professor, 理学部, 助教授 (60263422)
WADA Masaaki Faculty of Science, Nara Women's University, Professor, 理学部, 教授 (80192821)
OCHIAI Mitsuyuki Graduate School of Human Culture, Nara Women's University, Professor, 大学院・人間文化研究科, 教授 (70016179)
NIIDE Naoyuki Faculty of Science, Nara Women's University, Assistant Professor, 理学部, 講師 (40208111)
|Budget Amount *help
¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1998: ¥2,000,000 (Direct Cost: ¥2,000,000)
Recently "low dimensional topology theory" is far going out of the framework of "geometry" and finding out intimate relations between group theory, complex analysis, dynamical system, and even some fields out of mathematics like theoretical physics, and computer science. Within the relations, there are many (very huge, in general) combinatorial structures, for example, train tracks which give coordinates on Teichmuller spaces, canonical decomposition of hyperbolic 3-manifolds by ideal cells (Epstein-Penner), constructions of representations of Hecke algebra by Young diagram (Jones), automatic group theory (Thurston), and normal surface theory by Haken. In connection with these phenomena, it seems that recent development of low dimensional topology and of computer enables us to treat these objects directly and concretely.
In view of these situations, in this research, we intended to study 2 and 3 dimensional manifolds from geometrical can combinatorial viewpoint. Concretely speaking, we studied about the following topics.
・Analyzing 3-manifolds and knots via Heegaard splitting (particularly, with using "graphic" introduced by Rubinstein- Scharlemann), and obtaining useful informations on unknotting tunnels of knots,
・Studying hyperbolic structures on 3-manifolds via triangulations, particularly on hyperbolic structures on 2-bridge knot complements starting from a very simple hyperbolic structure,
・Studying about the relations between moduli spaces of certain kind of Riemannian metrics of 3-manifolds and geometric structures,
・Studying about algorithms (that the computer can handle) to decompose the attaching homeomorphisms of the given Heegaard splittings into canonical Dehn twists.