Effects connected with the mathematical theory of knots¹ emerge in many areas of science, from physics^2,3 to biology⁴. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems⁵, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹₂. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension^6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems.

与结¹的数学理论相关的影响出现在许多科学领域，从物理学^2,3到生物学⁴。最近的理论工作发现，编织群表征了非厄米周期系统⁵的拓扑结构，其中复能带能量可以在动量空间中编织。然而，这种复杂能带的辫子尚未通过实验实现或控制。在这里，我们介绍了一种紧束缚晶格模型，它可以在两股𝔹 ₂的编织组中实现任意元素。我们通过实验证明了非厄米能带在合成维度^{6,7 中的}这种拓扑复能量编织. 我们的实验利用两个耦合环形谐振器中的频率模式，其中一个同时进行相位和幅度调制。我们观察到构成链接和结的代表性实例的各种双波段编织结构，包括 unlink、unknot、Hopf 链接和三叶草。我们还表明，辫子的手性是可以改变的。我们的结果直接证明了非厄米拓扑的编织群表征，并为在开放的经典和量子系统中设计和实现拓扑稳健相开辟了一条途径。