An algorithmic approach to construct crystallizations of 3-manifolds from presentations of fundamental groups
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- 作者：BIPLAB BASAK
- 关键词：Pseudotriangulations of manifolds ; crystallizations of manifolds ; spherical and hyperbolic 3 ; manifolds ; presentations of groups.
- 刊名：Proceedings Mathematical Sciences
- 出版年：2016
- 出版时间：November 2016
- 年：2016
- 卷：126
- 期：4
- 页码：629-653
- 全文大小：1,098 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics - 出版者：Springer India
- ISSN：0973-7685
- 卷排序：126
文摘
We have defined the weight of the pair (〈S∣R〉,R) for a given presentation 〈S∣R〉 of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of (〈S∣R〉,R) is n, then our algorithm constructs all the n-vertex crystallizations which yield (〈S∣R〉,R). As an application, we have constructed some new crystallizations of 3-manifolds. We have generalized our algorithm for presentations with three generators and a certain class of relations. For m≥3 and m≥n≥k≥2, our generalized algorithm gives a \(2(2m+2n+2k-6+{\delta _{n}^{2}} + {\delta _{k}^{2}})\)-vertex crystallization of the closed connected orientable 3-manifold M〈m,n,k〉 having fundamental group \(\langle x_{1},x_{2},x_{3} \mid {x_{1}^{m}}={x_{2}^{n}}={x_{3}^{k}}=x_{1}x_{2}x_{3} \rangle \). These crystallizations are minimal and unique with respect to the given presentations. If ‘ n=2’ or ‘ k≥3 and m≥4’ then our crystallization of M〈m,n,k〉 is vertex-minimal for all the known cases.