Combinatorial structures of low dimensional manifolds

Recently "low dimensional topology theory" is far going out of the framework of geometry" andfinding out intimate relations between group theory, complex analysis，动态系统and even some fields out of mathematics like theoretical physics计算机科学：Within the relations， there are many （very huge, very huge）in general) combinatorial结构，for example，tracks which give coordinates on Teichmuller spaceshyperbolic 3-manifolds by ideal cells （Epstein-Penner）constructions of representations of Hecke algebra by Young diagram （Jones）自动组理论（Thurston）and normal surface theory by Haken. In connection with these phenomenait seems that recent development of low dimensional topology and of computer enables us to treatthese objects directly and concretely.In view of these situations, in this research我们有意研究2和3 dimensional manifolds from geometrical can combinatorial viewpoint. 2concretely speaking,分析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 on3-manifolds via triangulations，particularly on hyperbolic structures on 2-bridge knot complements starting from a very simplehyperbolic structure, Studying about the relations between moduli spaces of certain kind ofRiemannian metrics of 3-manifolds and geometric structures，关于算法(that thecomputer can handle) to decompose the attaching homeomorphisms of the given Heegaard splittings intocanonical Dehn twists。

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.