Floquet engineering, customizing a system using periodic driving, offers a powerful tool to operate topological states of matter and even to create exotic nonequilibrium topological phenomena beyond static scenarios. Here, utilizing the idea of Floquet engineering, we theoretically predict and experimentally observe the fast driving tuned topologically distinct complex-energy braiding, which enables a new topological classification beyond conventional non-Hermitian band topology. Theoretical elucidation of our proposed Floquet engineering scheme is constructed via the Floquet theorem. To experimentally show that, we build a single-photon interferometric network. Distinct braiding topology is directly observed by different knots in the measured eigenspectrum, which is entirely consistent with our measured crossing patterns of the eigenvector loops on the Bloch sphere. We also experimentally show that transitions between Floquet topological phases with distinct knots can be identified by simply counting the total number of emergent exceptional points. Our Letter thus builds the Floquet engineering scheme as a general principle for studying the knot topology.

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• Quantum optics
• Topological phases of matter
• Non-Hermitian systems