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It was believed that modular data are enough to distinguish different modularcategories (and topological orders in 2+1-dimensions). Then counterexamples tothis conjecture were found by Mignard and Schauenburg in 2017. In this work, weshow that the simplest counterexamples can be distinguished by studying therepresentations of mapping class groups of a punctured torus or a genus-2closed manifold.

Distinguish modular categories and 2+1D topological orders beyond modular data: Mapping class group of higher genus manifold