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

TL;DR

This paper introduces an algorithm to construct minimal crystallizations of 3-manifolds from fundamental group presentations, enabling the explicit realization of these manifolds with optimized vertex counts.

Contribution

The authors develop a novel algorithm for constructing crystallizations of 3-manifolds from presentations with two or three generators, including a generalization for specific relations, producing minimal and unique realizations.

Findings

Constructed new crystallizations of 3-manifolds.

Provided an algorithm for presentations with two generators and two relations.

Generalized the algorithm for three generators, yielding minimal crystallizations.

Abstract

We have defined weight of the pair $(\langle S \mid R \rangle, R)$ for a given presentation $\langle S \mid R \rangle$ 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 $(\langle S \mid R \rangle, R)$ is $n$ then our algorithm constructs all the $n$-vertex crystallizations which yield $(\langle S \mid R \rangle, R)$. 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 $m\geq 3$ and $m \geq n \geq k \geq 2$, 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 $M\langle m,n,k \rangle$ having fundamental group $\langle x_1,x_2,x_3 \mid x_1^m=x_2^n=x_3^k=x_1x_2x_3 \rangle$. These crystallizations are minimal and unique with respect to the given presentations. If `$n=2$' or `$k\geq 3$ and $m \geq 4$' then our crystallization of $M\langle m,n,k \rangle$ is vertex-minimal for all the known cases.