Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3‐manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann–Zagier and Moser for ideal triangulations on which HIKMOT is based) showing that there is a redundancy among the edge equations if the edges avoid ‘gimbal lock’. We successfully test the algorithm on known examples such as the orientable closed manifolds in the Hodgson–Weeks census and the bundle census by Bell. We also tackle a previously unsolved problem and determine all knots and links with up to 14 crossings that have a hyperbolic branched double cover.

中文翻译：

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封闭双曲3流形的经过验证的计算

扩展了卡森首先使用的方法，我们展示了如何使用区间算术方法在封闭的3流形的有限三角剖分上验证双曲结构。一个关键因素是新的理论结果（类似于Neumann–Zagier和Moser的定理，它是HIKMOT所基于的理想三角剖分），表明如果边缘避免“万向节锁定”，则边缘方程之间存在冗余。我们在已知示例（例如霍奇森－威克斯普查中的可定向封闭流形和贝尔的普查中）中成功测试了该算法。我们还解决了以前未解决的问题，并确定了多达14个具有双曲线分支双盖的交叉点的所有节点和链接。