2014/10/31 by Biplab Basak, BIPLAB BASAK
Computer Science · Mathematics · #Class (philosophy) #Component (thermodynamics) #Computational Geometry and Mesh Generation #Construct (python library) #Geometric and Algebraic Topology #Group (periodic table) #Presentation (obstetrics) #Topological and Geometric Data Analysis #math.GT #msc:05C15 #msc:57N10 #msc:57Q05 #msc:57Q15
paper · pdf · doi:10.1007/s12044-016-0302-7
published as Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 4, November 2016 · 24 pages, 8 figures
arxiv created 2015/09/13 · openalex created_date 2016/06/24 · openalex publication_date 2016/10/03 · arxiv updated 2016/10/28 · openalex updated_date 2026/08/05
We have defined 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 certain class of relations. For m≥ 3 and m ≥ n ≥ k ≥ 2, our generalized algorithm gives a 2(2m+2n+2k-6+δn2 + δk2)-vertex crystallization of the closed connected orientable 3-manifold M⟨ m,n,k ⟩ having fundamental group ⟨ x1,x2,x3 | x1m=x2n=x3k=x1x2x3 ⟩. 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.