Computing Fundamental Groups for 3-Manifolds and Their Applications
International Symposium on Visual Computing (ISVC2008)
Junho Kim, Miao Jin, Qianyi Zhou, Feng Luo and Xianfeng Gu
Computing fundamental groups of general 3-manifolds has fundamental
importance since the topologies of prime 3-manifolds are determined by
their fundamental groups. In this work, we provide a series of practical and efficient
algorithms to compute fundamental groups for general 3-manifolds based
on CW cell decomposition. The input is a tetrahedron mesh, while the output
is symbolic representation of its first fundamental group. We further simplify
the fundamental group representation using computational algebraic method. We
present the theoretical arguments of our algorithms and elaborate the algorithms
with a number of examples and applications including ambient isotopic detection
and linkage detection problems.