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 $\mathit k (M)$ 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 $$\mathit k (M)\geq 3 (\mathcal{G}(M)+h-1) \geq 3 (\mathcal{G} (\partial M)+h-1).$$ 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 $\mathit k (M)< 3 (\mathcal{G} (\partial M)+1)$ then $M$ is a handlebody. In particular, we prove that $\mathit k (M) =3 \mathcal{G} (\partial M)$ if $M$ is a handlebody and $\mathit k (M) \geq 3 (\mathcal{G} (\partial M)+1)$ if $M$ is not a handlebody. Further, we obtain several combinatorial properties for a crystallization of 3-manifolds with boundary.