On a computer recognition of 3-manifolds

Algorithmic construction and recognition of hyperbolic 3-manifolds, links, and graphs

This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological "objects" of various types, and to classify several classes of such objects using such structures.

Construction and Recognition of Hyperbolic 3-Manifolds with Geodesic Boundary

We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical triangulation, to an algorithm that computes the Kojima triangulation. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boun…

2-manifold recognition is in logspace

We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.

不可压缩曲面、层次结构与平凡纽结识别

Incompressible surfaces, hierarchies and unknot recognition

提出用层次结构技术判定三维流形中曲面是否不可压缩的新算法，无需计算复杂不变量和冗长搜索，还给出检测平凡纽结新算法，为判定紧致可定向三维流形是否有不可压缩边界属于NP提供新证明。

动态系统的算法与拓扑不变量. III. 二维曲面的识别与分类算法

Algorithms and topological invariants for dynamic systems. III. Algorithms for Recognition and Classification of 2-Dimensional Surfaces

该研究构造了可区分曲面拓扑类型的算法和拓扑不变量，能识别单纯复形及正则CW复形中的2-流形与3-流形，并确定2-流形的连通性、可定向性等拓扑类型，是动态系统拓扑理论的第三部分成果。

Uniqueness of Walkup's 9-vertex 3-dimensional Klein bottle

Via a computer search, Altshuler and Steinberg found that there are 1296 +1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold $K^{3}_{9}$ triangulates the twisted $S^{2}$-bundle over $S^{1}$. It was first constructed by Walkup. In this paper, we present a computer-free proof of the uniqueness of this non-sphere combinatorial 3-manifold. As opposed to the computer-generated proof, ours does not require wading through all the 9-vertex 3-spheres. As a preliminary result, we also show that any 9-vertex combinatorial 3-manifold is equivalent by…