Minimal crystallizations of 3-manifolds with boundary

Abstract

Let $(\Gamma ,\gamma )$ be a crystallization of connected compact 3-manifold M with h boundary components. Let $\mathcal {G}(M)$ and $k (M)$ be the regular genus and gem-complexity of M respectively, and let $\mathcal {G}(\partial M)$ be the regular genus of $\partial M$. We prove that

$$\begin{aligned} k (M)\ge 3 (\mathcal {G}(M)+h-1) \ge 3 (\mathcal {G} (\partial M)+h-1). \end{aligned}$$

These bounds for gem-complexity of M are sharp for several 3-manifolds with boundary. Further, we show that if $\partial M$ is connected and $k (M)< 3 (\mathcal {G} (\partial M)+1)$ then M is a handlebody. In particular, we prove that $k (M) =3 \mathcal {G} (\partial M)$ if M is a handlebody and $k (M) \ge 3 (\mathcal {G} (\partial M)+1)$ if M is not a handlebody. Further, we obtain several combinatorial properties for a crystallization of 3-manifolds with boundary.