Adams, C. C. The Knot Book (American Mathematical Society, 2004).
Yang, Y. et al. Non-abelian physics in light and sound. Science 383, 844–858 (2024).
Chen, Z. G., Zhang, R. Y., Chan, C. T. & Ma, G. Classical non-abelian braiding of acoustic modes. Nat. Phys. 18, 179–184 (2022).
Ashida, Y., Gong, Z. & Ueda, M. Non-Hermitian physics. Adv. Phys. 69, 249–435 (2020).
Özdemir, S., Rotter, S., Nori, F. & Yang, L. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).
Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).
El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).
Miri, M. A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).
Li, A. et al. Exceptional points and non-Hermitian photonics at the nanoscale. Nat. Nanotechnol. 18, 706–720 (2023).
Wang, C. et al. Non-Hermitian optics and photonics: from classical to quantum. Adv. Opt. Photonics 15, 442–523 (2023).
Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity-time symmetry. Nat. Photonics 11, 752–762 (2017).
Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76–79 (2016).
Yoon, J. W. et al. Time-asymmetric loop around an exceptional point over the full optical communications band. Nature 562, 86–90 (2018).
Song, Q., Odeh, M. & Genevet, P. Plasmonic topological metasurface by encircling an exceptional point. Science 373, 1133–1137 (2021).
Schumer, A. et al. Topological modes in a laser cavity through exceptional state transfer. Science 375, 884–888 (2022).
Ergoktas, M. S. et al. Topological engineering of terahertz light using electrically tunable exceptional point singularities. Science 376, 184–188 (2022).
Ding, K., Fang, C. & Ma, G. Non-Hermitian topology and exceptional-point geometries. Nat. Rev. Phys. 4, 745–760 (2022).
Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).
Wang, K., Dutt, A., Wojcik, C. C. & Fan, S. Topological complex-energy braiding of non-Hermitian bands. Nature 598, 59–64 (2021).
Zhang, Q. et al. Observation of acoustic non-Hermitian Bloch braids and associated topological phase transitions. Phys. Rev. Lett. 130, 017201 (2023).
Tong, S. et al. Observation of Floquet–Bloch braids in non-Hermitian spatiotemporal lattices. Phys. Rev. Lett. 134, 126603 (2025).
Patil, Y. S. S. et al. Measuring the knot of non-Hermitian degeneracies and non-commuting braids. Nature 607, 271–275 (2022).
Guria, C. et al. Resolving the topology of encircling multiple exceptional points. Nat. Commun. 15, 1369 (2024).
Cao, M. M. et al. Probing complex-energy topology via non-Hermitian absorption spectroscopy in a trapped ion simulator. Phys. Rev. Lett. 13, 163001 (2023).
Rao, Z. et al. Braiding reflectionless states in non-Hermitian magnonics. Nat. Phys. 20, 1904–1911 (2024).
Hu, Y. et al. Ultrafast control of braiding topology in non-Hermitian metasurfaces. Preprint at https://arxiv.org/abs/2410.16756 (2024).
Gao, J. et al. Topology-imprinting in nonlinear metasurfaces. Sci. Adv. 11, eadv5190 (2025).
Harari, G. et al. Topological insulator laser: theory. Science 359, eaar4003 (2018).
Milburn, T. J. et al. General description of quasiadiabatic dynamical phenomena near exceptional points. Phys. Rev. A 92, 052124 (2015).
Zhu, B., Wang, Q., Wang, Y., Wang, Q. J. & Chong, Y. D. Versatile braiding of non-Hermitian topological edge states. Phys. Rev. B 110, 134317 (2024).
Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time–symmetric microring lasers. Science 346, 975–978 (2014).
Gu, Z. et al. Experimental demonstration of PT-symmetric stripe lasers. Laser Photonics Rev. 10, 588–594 (2016).
Lai, Y. H., Lu, Y. K., Suh, M. G., Yuan, Z. & Vahala, K. Observation of the exceptional-point-enhanced Sagnac effect. Nature 576, 65–69 (2019).
Brandstetter, M. et al. Reversing the pump dependence of a laser at an exceptional point. Nat. Commun. 5, 4034 (2014).
Zhu, B. et al. Anomalous single-mode lasing induced by nonlinearity and the non-Hermitian skin effect. Phys. Rev. Lett. 129, 013903 (2022).
Corato-Zanarella, M. et al. Widely tunable and narrow-linewidth chip-scale lasers from near-ultraviolet to near-infrared wavelengths. Nat. Photonics 17, 157–164 (2023).
Su, Q. et al. A self-injection locked laser based on high-Q micro-ring resonator with adjustable feedback. J. Lightwave Technol. 41, 6756–6763 (2023).
Kondratiev, N. M. et al. Recent advances in laser self-injection locking to high-Q microresonators. Front. Phys. 18, 21305 (2023).
Xiang, C. et al. 3D integration enables ultralow-noise isolator-free lasers in silicon photonics. Nature 620, 78–85 (2023).
Snigirev, V. et al. Ultrafast tunable lasers using lithium niobate integrated photonics. Nature 615, 411–417 (2023).
Kondratiev, N. M. et al. Self-injection locking of a laser diode to a high-Q WGM microresonator. Opt. Express 25, 28167–28178 (2017).
Zhang, X. L. et al. Non-abelian braiding on photonic chips. Nat. Photonics 16, 390–395 (2022).
Wang, Y. et al. Non-Hermitian non-abelian topological transition in the S = 1 electron spin system of a nitrogen vacancy centre in diamond. Nat. Nanotechnol. 20, 873–880 (2025).
Wang, K. et al. Observation of braid-protected unpaired exceptional points. Phys. Rev. Lett. 136, 056602 (2026).
Chen, Y. et al. High-dimensional non-abelian holonomy in integrated photonics. Nat. Commun. 16, 3650 (2025).
Sun, S. et al. Reconfigurable non-abelian integrated photonics. Nat. Commun. 16, 7089 (2025).
Mao, W. Supporting data for ‘Laser mode braiding on a chip’. figshare https://doi.org/10.6084/m9.figshare.31883344 (2026).