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：我们常常依赖三角剖分普查来指导三维流形拓扑中的直觉。然而，如果最小反例过大而无法出现在我们的普查中，这可能导致对猜想产生错误的信任。由于三角剖分的数量随规模呈超指数增长，无法将普查扩展到相对较小的三角剖分之外；当前普查仅涵盖最多10个四面体。在此，我们证明通过使用启发式方法选择性（而非穷举地）枚举三角剖分，可以搜寻大型且难以发现的反例。我们利用这一思路找到了三个猜想的反例，这些猜想针对某些三维流形，询问单顶点三角剖分是否总有一条"标志性"边，使我们能够识别该三维流形。