论文标题
具有边界的3个manifolds的最小结晶
Minimal crystallizations of 3-manifolds with boundary
论文作者
论文摘要
令$（γ，γ）$是连接的紧凑型3-manifold $ m $，$ h $边界组件的结晶。令$ \ MATHCAL {G}（M）$和$ \ Mathit K（M）$分别为$ M $的常规属和宝石 - 复杂性，让$ \ Mathcal {G}（\ partial m）$为$ \ partial m $的常规属。我们证明了这一点 $$ \ MATHIT K（M）\ GEQ 3（\ MATHCAL {G}（M）+H-1）\ GEQ 3（\ MATHCAL {G}（\ partial M）+H-1）。$$对于几个具有边界的3-Manifolds的宝石复杂性的$ M $的bounds bounts bounds bungs bounts bounts。此外，我们表明，如果连接了$ \ partial m $，并且$ \ mathit k（m）<3（\ mathcal {g}（\ partial m）+1）$，则$ m $是一个句柄。特别是，如果$ \ Mathit k（m）= 3 \ Mathcal {g}（\ partial m）$如果$ m $是一个handlebody，而$ \ Mathit k（m）\ geq 3（\ Mathcal {g}（\ partial m）+1）此外，我们获得了几种组合特性，以结晶具有边界。
Let $(Γ,γ)$ 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.
