Imaging nodal knots in momentum space through topolectrical circuits

Abstract

Knots are intricate structures that cannot be unambiguously distinguished with any single topological invariant. Momentum space knots, in particular, have been elusive due to their requisite finely tuned long-ranged hoppings. Even if constructed, probing their intricate linkages and topological "drumhead" surface states will be challenging due to the high precision needed. In this work, we overcome these practical and technical challenges with RLC circuits, transcending existing theoretical constructions which necessarily break reciprocity, by pairing nodal knots with their mirror image partners in a fully reciprocal setting. Our nodal knot circuits can be characterized with impedance measurements that resolve their drumhead states and image their 3D nodal structure. Doing so allows for reconstruction of the Seifert surface and hence knot topological invariants like the Alexander polynomial. We illustrate our approach with large-scale simulations of various nodal knots and an experiment that maps out the topological drumhead region of a Hopf-link.

arXiv comment37 pages, 14 figures, 14 tables

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• Dimensional transmutation from non-Hermiticity
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• Non-Abelian Braiding of Topological Edge Bands
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• Braiding Topology of Non-Hermitian Open-Boundary Bands
• Quench dynamics and bulk-edge correspondence in nonlinear mechanical systems
• Anisotropic non-linear optical response of nodal loop materials
• Observation of non-local impedance response in a passive electrical circuit
• Braiding topology of symmetry-protected degeneracy points in non-Hermitian systems
• Observing the nodal-line conversion determined by the relative homotopy
• Creation of electrical knots and observation of DNA topology
• Berry curvature inside parity-time-symmetry protected exceptional surface
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• Entanglement manifestation of knot topology in a non-Hermitian lattice
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