An algorithmic approach to construct crystallizations of 3-manifolds from presentations of fundamental groups

Abstract

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 + δ 2 n + δ 2 k )-vertex crystallization of the closed connected orientable 3-manifold M m, n, k having fundamental group x 1 , x 2 , x 3 | x m 1 = x n 2 = x k 3 = x 1 x 2 x 3 . 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.MSC 2010 : Primary 57Q15. Secondary 05C15, 57N10, 57Q05.