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：在三流形拓扑中，我们常依赖三角剖分的普查来指导直觉。然而，若最小反例过大而无法出现在普查中，这种依赖可能导致对猜想产生错误信心。由于三角剖分数量随尺寸超指数增长，将普查范围扩展到相对较小的三角剖分之外是不可能的；当前普查仅覆盖最多10个四面体。本文证明，通过启发式方法有选择性地（而非穷举地）枚举三角剖分，可以搜索到大型且难以发现的反例。利用这一思想，我们找到了三个猜想的反例，这些猜想询问：对于特定三流形，单顶点三角剖分是否总存在一条"标志性"边，使我们能够识别该三流形。