Imaging nodal knots in momentum space through topolectrical circuits

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 which maps out the topological drumhead region of a Hopf-link.


Supplementary Note 1: Relation between drumhead states and surface topological band structure
Here we briefly illustrate how topological drumhead regions can be read from the surface band structure. Consider for instance a trefoil nodal knot, as shown in the left panels of Fig. 1. Drumhead regions are points in the surface Brillouin zone where topological zero modes exists. As evident in the surface band structures (Right panels) plotted along the dashed paths on the left, these zero modes must necessarily terminate at bulk gap closures, i.e. nodal lines. As such, drumhead states are necessarily demarcated by the surface nodal lines.

Supplementary Note 2: Alexander polynomial from the braid
The Alexander polynomial invariant of a knot can in fact be directly computed from its braid closure. At first sight, this seems tricky, because the closure of a series of braid operations do not uniquely define a knot/link, which can easily be topologically equivalent to a seemingly different braid. That said, there exists a direct means of obtaining the Alexander polynomial A(t) via the (unreduced) Burau representation of a braid: where N is the total number of strands. A generic braid can be expressed as a composition of braid operations σ ± i1 σ ± i2 σ ± i3 ..., with corresponding Burau representation matrix σ(t) = σ ± i1 (t)σ ± i2 (t)σ ± i3 (t).... It turns out that the Alexander polynomial invariant is simply given by where [I N − σ] 11 (t) is the minor matrix of I N − σ(t), which is obtained by omitting its first row and column. Note that when t = 1, σ(1) just gives the permutation matrix for the entire braid, and that each independent permutation cycle gives rise to a separate line node. It is also conventional to normalize A(t) by a power of t, such that it becomes symmetric in t and t −1 .

Supplementary Note 3: Further details of circuit simulation and implementation
Large-scale Xyce simulations provide a platform towards a realistic experimental setting of our circuit design. The compatibility of simulation and experiment for electrical circuits reaches an unprecedented degree of accuracy and agreement in comparison to any other architecture in which topological bands can unfold. A rigorous simulation includes the use of realistic voltage sources supplemented by corresponding shunt resistances, serving as the external excitation, i.e., inhomogeneities to the circuit's differential equations. The simulation further incorporates a realistic measurement process comprising a read-out of voltages at the circuit nodes and of the input current through the shunt resistance, which is analogous to an experimental framework. There, Lock-In amplifiers can be used for the corresponding measurements at fixed frequency. Similar to the experimental sequence of data analysis, the simulation data is subsequently post-processed to reconstruct the admittance band structure from global single-point voltage measurements. In principle, circuit simulations also allow to incorporate component tolerances and serial resistances to study disorder and parasitic effects.

Supplementary Note 4: Explicit forms of Nodal Knots
We next present the explicit forms of the nodal knots used in the above simulations. These may be appropriately truncated for experimental implementations, as is done for the Hopf-link experiment described later. The coefficients in the defining functions z(k) and w(k), which determine the precise nodal line can be modified as long as the winding number, which defines the mapping to the three dimensional Brillouin zone stays invariant. This implies, that small changes of the coefficients will only modify the shape and not the principal structure of the knot. Concerning the simulations of the knot structures, we fine-tuned those coefficients to an optimal value, that mediates between the complexity of Xyce simulations and the resolution of the knot in the three-dimensional momentum grid determined by the implemented system sizes. Supplementary Figure 1: Surface Trefoil knot nodal structure and drumhead states labeled by their multiplicities (Left panels), with corresponding surface band structures along the indicated dashed lines shown on the right. Blue/black curves indicated surface/bulk localization. The top and bottom panels illustrate the same system viewed from different angles.

Hopf-link circuit
For the Hopf-link, we employ a knot function of the form with z(k x , k y , k z ) and w(k x , k y , k z ) as defined in equation 7 of the main text. The real and imaginary part of f (k x , k y , k z ), truncated to admit only nearest-neighbor couplings in each direction with commensurate magnitudes, are given by The corresponding circuit Laplacian is given by , whose form in real space is illustrated in in Fig. 3, with LC normalized to unity.

Trefoil knot circuit
For the Trefoil knot, we employ a knot function of the form   For each slice, the connections are only shown emanating from one reference node (orange) from each sublattice. Also, for the inter-sublattice connections, only connections emanating from the reference node in A, and not B, are shown to avoid clutter. Each connection along x,y or z directions appear in two out of the three diagrams, while each connection in one of the planes appear only once. The black capacitors/inductors have magnitudes C and L, the purple glowing ones have magnitudes 2C and L 2 , and the green glowing ones have magnitudes 4C and L 4 .

Figure-8 knot circuit
For the figure-8 knot, we employ a knot function of the form The real and imaginary part of f (k x , k y , k z ) with additional truncations applied is then given by and

Supplementary Note 5: Future Experimental Enhancements
In this experiment, we have developed a practical framework for constructing tunable circuit arrays that are sufficiently precise for reliably reproducing desired topological band structure features. A central challenge has been the tuning of repeated elements like inductors, which has to be accurately synchronized across all the repeating unit cells. We emphasize that this tunability has rarely been accomplished in the context of circuits with topological band structures.
Moving forward, the most crucial enhancement will be the automation of this tuning with a micro-controller. This will supersede our manual mechanical tuning of the variable inductors, which takes at least a minute for each component. Supplementary Figure 4 describes a solid-state automated tunable circuit with a MOSFET transistor controlling the wire loop around a fixed value inductor. By varying the gate voltage of the MOSFET through a micro-controller, the resistance in the wire loop may be electronically controlled to precisely vary the induced current in the wire loop and hence the impedance/inductance of the coupling unit. Requiring no manual control input i.e. visual reference to an oscilloscope, this approach can achieve drastic speedup for the tuning, and perhaps even allow real-time tuning for the simulation of Floquet Hamiltonians. While additional circuitry is required to connect all inductors to a common impedance measurement circuit, presenting an optimization challenge in dense complicated circuit arrays, this too can be streamlined with intelligent design approaches.
Another important advance will be the realization of nodal knots as a bona-fide spatial 3D circuit arrays. Nodal knot circuits possess more complicated structures than existing realizations of other 3D topolectrical circuits [1,2], and will require more sophisticated PCB designs as well as a more extensive set of component i.e. capacitance values. To realize these capacitances as combinations of commercially available capacitors, it will be advantageous to use an artificial intelligence search-tree algorithm for finding the optimal combinations with fewest components or lowest cost. Related optimization algorithms can also be used to optimize the circuit array configuration in physical space, which is an important consideration when the circuit complexity increases [2,3].   List of physical components that make up the logical components in each unit cell, as labeled in the PCB diagram and photograph in Fig. 9. The fixed value inductor components have an uncertainty of +/-10% and a parasitic resistance of 0.10 Ω, while the fixed value capacitor components have an uncertainty of +/-5%.      Table 7: Simulated data of impedance values for a Hopf Link circuit with N=9. Impedance values in each column correspond to impedance of points k y , k z within a k radius of 0.03 from a particular centered point k y0 , k z0 , in which (k y − k y0 ) 2 + (k z − k z0 ) 2 < 0.03. This table shows only part of the full dataset, the full  dataset is distributed over Tables 4-14. Supplementary Table 8: Simulated data of impedance values for a Hopf Link circuit with N=9. Impedance values in each column correspond to impedance of points k y , k z within a k radius of 0.03 from a particular centered point k y0 , k z0 , in which (k y − k y0 ) 2 + (k z − k z0 ) 2 < 0.03. This table shows only part of the full dataset, the full  dataset is distributed over Tables 4-14.