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

Abstract

We have defined weight of the pair $(⟨S\mid R⟩,R)$ for a given presentation $⟨S\mid R⟩$ 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 $(⟨S\mid R⟩,R)$ is $n$ then our algorithm constructs all the $n$-vertex crystallizations which yield $(⟨S\mid R⟩,R)$. 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 $m\ge 3$ and $m\ge n\ge k\ge 2$, 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 $⟨x_1,x_2,x_3\mid x_1^m=x_2^n=x_3^k=x_1{x}_2{x}_3⟩$. These crystallizations are minimal and unique with respect to the given presentations. If `$n=2$' or `$k\ge 3$ and $m\ge 4$' then our crystallization of $M⟨m,n,k⟩$ is vertex-minimal for all the known cases.

24 pages, 8 figures

References in corpus (2)

• A note about complexity of lens spaces
• A census of genus two 3-manifolds up to 42 coloured tetrahedra

Cited by in corpus (1)

• Handle decompositions for a class of closed orientable PL 4-manifolds