Unknotting tunnels in hyperbolic 3-manifolds
Unknotting tunnels in hyperbolic 3-manifolds
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解开双曲 3 流形中的隧道
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
C. Adams
中科院分区：
文献类型：
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作者：
C. Adams
带边界的三维流形中的解结隧道是一个适当嵌入的圆弧，它的一个开邻域的补集是一个手柄。端点在双曲三维流形的尖点边界上并垂直于尖点边界的测地线称为垂直测地线。给出了双曲三维流形M中的一条垂直测地线，得到了它是解结隧道的充分条件。特别是，如果垂直测地线对应于万能盖中的4-手环、5-手环或6-手环，并且具有足够短的长度，则它一定是一个解开的隧道。此外，我们还考虑了一条满足长辈兄弟性质的垂直测地线，这意味着在万能覆盖中，除了中心在无穷远处的那条外，所有的球都通过垂直测地线的升力连接到一个更大的球上。这样一条长度小于ln(2)的垂直测地线被证明是一个解结隧道。
An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic in a hyperbolic 3-manifold M, we find sufficient conditions for it to be an unknotting tunnel. In particular, if the vertical geodesic corresponds to a 4-bracelet, 5-bracelet or 6-bracelet in the universal cover and has short enough length, it must be an unknotting tunnel. Furthermore, we consider a vertical geodesic that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at infinity is connected to a larger horoball by a lift of the vertical geodesic. Such a vertical geodesic with length less than ln(2) is then shown to be an unknotting tunnel.