论文标题
有效的三角形和边界坡度
Efficient triangulations and boundary slopes
论文作者
论文摘要
对于紧凑的，不可约的，$ \ partial $ -rirreducible，n-under限制的3个manifold $ m \ ne \ ne \ ne \ mathbb {b}^3 $，那么任何三角元素$ \ mathcal {t} $ $ m $都可以被修改为理想的三角形$ \ mathcal $ \ mathcal {t}^$ $ \ stackrel {\ circ} {m} $。我们使用沿正常表面粉碎三角剖分以及膨胀理想三角剖分以引入和研究边界效率的三角剖分和终端理想的理想三角剖分的逆关系。我们证明，承认环形效率三角剖分的紧凑型3个manifold $ m $所需的拓扑条件足以将$ m $的任何三角剖分修改为也是边界效率的三角剖分，这也是环形效率的。从证据中，我们有任何理想的三角剖分$ t^*$和任何通货膨胀$ \ MATHCAL {t}_λ$，在$ \ Mathcal {T}^*$中封闭的正常表面与$ \ MATHCAL {T}_λ$与Homemoromor的封闭正常表面之间有二个封闭的正常表面与封闭的正常表面。因此，对于理想的三角剖分$ \ MATHCAL {t}^*$，是$ 0 $ - 效率，$ 1 $ - 效率或效率，然后任何通货膨胀$ \ MATHCAL {t}_λ$ of $ \ MATHCAL {有算法可以决定这些有效的三角剖分之一是给定的三角剖分或理想的三角剖分。最后，结果表明，对于环形效率的三角剖分，对于有限的欧拉（Euler）特征的正常表面，只有有限数量的边界斜率。因此，在紧凑的，定向，不可还原，$ \ partial $ - irredcible和An-unnular $ 3 $ manifold中，对于不可压缩和$ \ partial $ compressibal-Incompressibal-Incompressive the Ouler特征的边界斜率只有有限的边界斜率。
For a compact, irreducible, $\partial$-irreducible, an-annular bounded 3-manifold $M\ne\mathbb{B}^3$, then any triangulation $\mathcal{T}$ of $M$ can be modified to an ideal triangulation $\mathcal{T}^*$ of $\stackrel{\circ}{M}$. We use the inverse relationship of crushing a triangulation along a normal surface and that of inflating an ideal triangulation to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. We prove that the topological conditions necessary for a compact 3-manifold $M$ admitting an annular-efficient triangulation are sufficient to modify any triangulation of $M$ to a boundary-efficient triangulation which is also annular-efficient. From the proof we have for any ideal triangulation $T^*$ and any inflation $\mathcal{T}_Λ$, there is a bijective correspondence between the closed normal surfaces in $\mathcal{T}^*$ and the closed normal surfaces in $\mathcal{T}_Λ$ with corresponding normal surfaces being homeomorphic. It follows that for an ideal triangulation $\mathcal{T}^*$ that is $0$-efficient, $1$-efficient, or end-efficient, then any inflation $\mathcal{T}_Λ$ of $\mathcal{T}^*$ is $0$-efficient, $1$-efficient, or $\partial$-efficient, respectively. There are algorithms to decide if a given triangulation or ideal triangulation of a $3$-manifold is one of these efficient triangulations. Finally, it is shown that for an annular-efficient triangulation, there are only a finite number of boundary slopes for normal surfaces of a bounded Euler characteristic; hence, in a compact, orientable, irreducible, $\partial$-irreducible, and an-annular $3$-manifold, there are only finitely many boundary slopes for incompressible and $\partial$-incompressible surfaces of a bounded Euler characteristic.
