We often rely on censuses of triangulations to guide our intuition in $3$-manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations; the current census only goes up to $10$ tetrahedra. Here, we show that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations. We use this idea to find counterexamples to three conjectures which ask, for certain $3$-manifolds, whether one-vertex triangulations always have a "distinctive" edge that would allow us to recognise the $3$-manifold.

翻译：我们常常依靠三角测量普查来指导我们的直觉,在3美元比例的地形学上。 但是,如果最小的反例太大,无法出现在我们的人口普查中,这可能导致对猜想的错误信心。 由于三角测量的数量会增加超高的体积,因此无法将人口普查扩大到相对小的三角测量之外;目前的普查只达到10美元四环。 我们在这里表明,通过使用超自然学来选择性地(而不是详尽地)列出三角测量数据,寻找大型和难以找到的反对比样本是可行的。 我们利用这个想法来找到三种预测的反比,这些预测要求,对于某些三元比例的三角测量数据来说,单反比三角测量数据是否总是有一个“清晰”的边缘,让我们能够识别三元比例。</s>