Creation of electrical knots and observation of DNA topology

Knots are fascinating topological structures that have been observed in various contexts, ranging from micro-worlds to macro-systems, and are conjectured to play a fundamental role in their respective fields. In order to characterize their physical properties, some topological invariants have been introduced, such as unknotting number, bridge number, Jones Polynomial and so on. While these invariants have been proven to theoretically describe the topological properties of knots, they have remained unexplored experimentally because of the difficulty associated with control. Herein, we report the creation of isolated electrical knots based on discrete distributions of impedances in electric circuits and observation of the unknotting number for the first time. Furthermore, DNA structure transitions under the action of enzymes were studied experimentally using electrical circuits, and the topological equivalence of DNA double strands was demonstrated. As the first experiment on the creation of electrical knots in real space, our work opens up the exciting possibility of exploring topological properties of DNA and some molecular strands using electric circuits.


I. INTRODUCTION
The exploration of knot physics has become one of the most fascinating frontiers in recent years, due to its complex topology that plays an important role in physical and life sciences [1][2][3][4][5]. At present, various knots have been constructed based on different physical implementations.
"Discrete" knots are usually found in the microscopic world, particularly on the cellular level. Different molecular knots can be synthesized through the accurate control of chemical reactions [1,[33][34][35][36][37][38]. After acting on duplex cyclic DNA molecules with direct repeats, enzymes called topoisomerases are able to generate different nontrivial DNA knots. Topoisomerases have shown to be tendentious in knotting or unknotting DNA molecules, which have completely different functions in living systems. For instance, type II topoisomerase manifests a preference to unknot DNA molecules. Therefore, understanding the possible topologies of DNA molecules is helpful to explore life mechanisms. In recent years, many theoretical investigations, including lattice-based simulations, equilateral chain model simulations, grid diagrams and so on, have proven to generate the topology of different DNA knots [35][36][37][38]. However, the direct experimental evidence of DNA topology, especially the topological equivalences between two different DNA structures, is still lacking.
In this work, we experimentally constructed the "discrete" knots based on circuits [39][40][41][42][43][44][45][46][47][48], which differ from other realizations of "continuous" knots for macroscopic objects. The advantage of constructing such an experimental platform is that it can be used to study various phenomena corresponding to DNA topology. For example, through the change of impendence in the electric circuits, we could for the first time have been able to render one important topological invariant in knot theory experimentally observable: namely, the unknotting number.
The topological change of DNA molecules under the action of enzyme Tn3 resolvase was also observed. In addition, the topological equivalence between two different structures of DNA molecules was experimentally demonstrated using the circuit platforms.

II. ELECTRICAL REALIZATION OF KNOTS AND OBSERVATION OF UNKNOTTING NUMBER
To implement electric knots in an electric circuit, the following steps were performed. At first, we established a connection between the knot theory and the corresponding construction in the lattice. The coupling strengths between sites in the lattice were designed, and the sites in the lattice occupied by the localized eigenstates comprised the knot structure. Then, we designed the electric circuit and correlated the Laplacian describing the circuit to the Hamiltonian in the lattice by choosing the appropriate electric capacitors and grounding elements. Finally, the distributions of impedance in the circuit were measured, where nodes possessing large impedances correspond to the sites occupied by localized eigenstates in the lattice and form the knot. The detailed construction method is described in Appendix A. Following such a method, various electrical knots were created. Figs. 1a and 1b display the designs of three-layer electric circuits, which were integrated with a trefoil knot and unknot structures, respectively. The red spheres represent the nodes possessing large impedances (>1 k ), and other nodes have an impedance smaller than 0.1 k . The purple tubes indicate the connections among the red spheres through the small electric capacitor. In order to observe such a phenomenon experimentally, we designed the corresponding circuits (the circuit corresponding to Fig. 1a is shown in Fig. 1c). For the convenience of experiment, the total cube was cut into three layers, which were then positioned on three printed circuit boards (PCBs). Capacitors were connected to every node on adjacent layers. A cyclic boundary condition was applied to avoid the edge effect. The experimental setup of the fabricated circuit is shown in Fig. 1d. In one PCB layer, the capacitors are arranged on the front and grounding inductors on the back. The enlarged image of the front and back of a plaquette is given in Fig. 1e. The fabricated sample has exactly the same construction as that shown in The dots in Figs. 1f and 1g represent the nodes with impedance larger than 1 k . Accordingly, the experimental results were found to agree well with the results presented in Figs. 1a and 1b.
Because circuit networks possess remarkable advantages of being tunable and reconfigurable, many interesting problems associated with the knot theory using electric knots can be explored. In mathematics, some invariants are usually introduced to describe the topological properties of knots, such as unknotting number, bridge number, Jones Polynomial, and Alexander Polynomial [1,4,5]. Although these concepts and theories have existed for around 100 years, they have never been proven by experiments. For the first time, this work provides the observation of these topological invariants in the designed circuit platforms.
Particularly, one invariant, the unknotting number, is defined as the least times in the change of crossing to make the knot become an unknot. Thus, the key to observe the unknotting number is to realize the change of crossing in the electric circuit.  We further studied the change of crossing in the knot structures, considering the transition from the electrical trefoil knot in Fig. 1a to unknot in Fig. 1b  The above discussions present only one phase transition of DNA molecules from the unknotted structure to the hopf link under the action of topoisomerases. In fact, multiple phase transitions can occur when the topoisomerases react repeatedly. For example, the structure of DNA molecules in Fig. 3d transitioned into the figure-8 knot structure when enzyme Tn3 resolvase was continuously applied. Such multiple phase transitions can also be well demonstrated in designing circuits by modulating the capacitors and associated grounding elements (see Appendix E). That is to say, all kinds of phase transitions of DNA molecules can be well observed by designing the circuits, further demonstrating the powerful ability to study related problems in this way.

STRANDS IN THE ELECTRIC CIRCUIT
DNA double strands play a dominant role in various biological functions that are necessary for life, such as replication, transcription and recombination [1,2]. Cyclic duplex DNA exhibits different geometric structures under different ambient environments. Figure 4a shows the cyclic duplex DNA structure in the relaxed state, which becomes an intermediate state ( Fig.1d) with the change of ambient environment, and then transitions into an unrelaxed state (Fig. 4g). In the unrelaxed state, the DNA structure is more tightly twisted. To describe the different geometric structures of cyclic duplex DNA, the twist ( Tw ) is defined as how much the DNA structure twists around its axis, and the writhe ( Wr ) refers to how much the axis of the DNA structure is contorted in space [1,49,50]. When the cyclic duplex DNA structure is in the relaxed state ( Fig. 4a), it does not twist around its axis; instead, its axis is contorted once in space and, thus, the Tw 0  and Wr 1  . In Fig. 4g, the axis of the DNA structure lies flat in the plane, but the structure twists around its axis once, and thereby, Tw 1  and Wr 0  .
The topology of the DNA geometric configuration determines the functions of living mechanisms. The linking number Lk Tw Wr  is usually introduced to characterize the topological properties of different geometric configurations. It has been proven that, although the geometric configuration is different, the linking number is the same, and the function of DNA is the same. This is called the topological equivalence. For example, the two geometric configurations shown in Fig. 4a and 4g seem different, yet their linking numbers are the same.
The demonstration of topological equivalence between two different geometric configurations is very meaningful, but is very difficult to experimentally demonstrate in living cells. Using the designed circuit platforms, we were able to observe the phenomena very well.
By integrating the cyclic duplex DNA structure function in the relaxed state into the circuit design, we constructed a circuit that displays the DNA configuration (Fig. 4b). Similar to the above figures, red spheres denote the nodes with impedances larger than 1 k , and other nodes have impedances smaller than 0.1 k . It is seen clearly that the red spheres in Fig. 4b exactly constitute the connection representing the structure shown in Fig. 4a. By tuning the capacitors and associated grounding elements, the corresponding circuits for the intermediate state (Fig. 4d) and unrelaxed state (Fig. 4g)  Therefore, to demonstrate the topological equivalence between the structures, a continuous change from one structure to another needed to occur. That is, there is no change of crossing in the process. In fact, in the designed circuits, it is convenient to study the continuous change between two structures by modulating the electric capacitors and related grounding inductors step-by-step. Details of the change process for such a case are given in Appendix F. Results reveal that there is no change of crossing occurred in the process of continuous change between the two structures in Fig.4c and 4i, thereby verifying their topological equivalence.

V. CONCLUSION
This work theoretically and experimentally proposes schemes to create isolated electrical knots based on discrete distributions of impedances in electric circuits. Using the designed circuit platforms, we experimentally observed unknotting numbers for the first time, and the corresponding DNA structure transitions under the action of enzymes using electrical circuits.
The topological equivalence of DNA double strands was further demonstrated experimentally.
Our results provide convincing evidence that the electric circuits offer reliable platforms to study the various topological properties of knots and links, especially microscopic "discrete" knots in the molecular strands. Moreover, such circuit platforms revealed new phenomena in this work and also provide a basis for future exploration of unsolvable problems.

Appendia A: The general scheme to create electrical knots
Here, we describe how to construct knots and links in electric circuits. We start from the standard procedure of obtaining the knots and links, and then provide the design on the circuit.

Construction of standard knots
According to the theory [1,2], since the abstract group definitions of braids have remarkable geometric and topological interpretation, the knots and links can be obtained from the braids. Since the knots or links are composed by closed braids, the braid function needs to be periodic in h. It is convenient that the range of h is taken from 0 to 2 , so after h changed by 2 , the zeroes in the complex u plane are in the same positions as where they started. Moreover, the term ih e can be substituted by a second formal variable v. After this process, we can obtain the expression for different knots and links with these two formal variables u and v. E.g., 22 ( )= 0 ( 1) 1 8 4 1 In Fig. 5a, we provide the standard trefoil knot in the 3 R space. The mathematic theory tells us that the knots and links can be deformed freely in space, but not allowed to be cut off or glued. After such a deformation process, the geometric structure representing the knot (link) can be changed to another structure. Although these two structures seem different at the first sight, they are actually the same knots (links). It is often called these two knots (links) are isotopic. It is hard to find an ambient isotopy function in general, the researchers often project the knot into two-dimension and manipulate this projection in three formal ways, i.e., the Reidemeister moves. These manipulations are equivalent to an ambient isotopy in 3D. In Fig. 5b, we provide one structure obtained from the deformation of the structure in Fig. 5a. These two are isotopic since they can be deformed into each other without being cut off or glued. The expression for the deformed trefoil knot in Fig. 5b  1 deformed tre f a x a y a z a x a xy a xz a y a yz a z a x a x y a x z a xy a xz a xyz a y a y z a yz a z a x a x y a x z a x y a x yz a x z a xy a xy z a xyz a xz ay a y z a y z a yz a z a x a x y a x y a y a zx a zx y a zy a z x a x y z a z y a z x a z y a z x a z y a z a z a x a y The deformed trefoil knot can be composed of two fitting curves when we set 9    The site in the lattice is often described by the integral coordinate. In Fig. 5c, the ranges of , , x y z coordinates in the lattice are [1,7], [1,8] In the Hamiltonian, we set the coordinates shown in Table 2 to connect with the six adjacent sites through the coupling strength We also show the distribution of localized eigenstates of H in Fig. 5c. Red spheres label the sites occupied by the localized eigenstates. We find that the coordinates of these sites are exactly those shown in Table 2. To illustrate the distribution clearly, we use the purple tube in Fig. 5c to connect these sites and find that they comprise the trefoil knot. This trefoil knot is the "discrete" deformed trefoil knot in Fig. 5b.

Electric realization
Since the lattice contains three layers along the z direction, we use three printed circuit board (PCB) layers in the electric circuit. The detailed structure is provided in Fig. 5d. Every two nearest neighboring nodes in the circuit is connected through the capacitor  Fig. 5d, the Laplacian is addressed as In our design, when we set , the diagonal elements in the matrix J are eliminated, and the Laplacian changes to the expression shown in the inset of Fig. 5d.
In our experiment, we use the impedance as the measured quantity. The impedance between ath node and bth node is After completing these three steps, we realize the "discrete" trefoil knot in the electric circuit. To make this realization clear, we connect these "special" nodes having large impedances together. Since these nodes distribute at the diagonal nodes of each plaquette in the circuit (red spheres in Fig. 1a and large spheres in Fig. 1f of the main text), we connect these diagonal nodes of each plaquette in the circuit in sequence. The obtained connection is exactly the trefoil knot (see Fig. 1a and Fig. 1f in the main text). Compared with other experimental realizations of knots, the realization of trefoil knot in the electric circuit is easy and controllable.
Moreover, not only the trefoil knot is implemented electrically, other knots and links can also be realized in the electric circuit in the same way.

Appendia B: The eaperimental details in the electric circuit
Here, we describe the experimental details in the electric circuit. In our experiment, we choose the electric capacitors CC41-0603-CG-50V-100pF-F(N) and CC41-0603-CG-50V-10nF-F(N), which are described as the small capacitors in red ( 1 C ) and the large capacitors in blue ( 2 C ) in the main text, respectively. The types of inductors are NLV32T-3R3J-EF and NLV32T-033J-EF. We measure the impedance at each node by connecting the impedance analyzer to the measurement connectors. The cooper pillars are connected to the grounding inductors, which also sustain the three PCB layers.
We use the WK6500B impedance analyzer to measure the impedance. In principle, we need to measure the impedance between the nodes and the ground at the resonant frequency Due to the existence of various errors in the experiments, the measured frequency is a little smaller than the resonant frequency. To obtain the appropriate results, we use the impedance analyzer to sweep the frequencies around the resonant frequency. We choose the certain frequency where the peak of impedance appears and we use this peak value as the impedance of this node.
In our experiment, the values of electric inductors and capacitors, of course, are not ideal, and there exists spurious inductive coupling in the experimental setup, but the dominant error is from the connecting wires. The type of these wires is DB9. These wires are used to realize cyclic boundary condition and connect different PCB layers. The parasitic inductance from long connecting wires cannot be neglected. When we measure the impedance of nodes connecting with adjacent nodes through the small electric capacitor 1 C , the impedance of these nodes are very large in the ideal circuit simulation. However, in fact, due to the non-negligible parasitic inductance of the long wires in the experiment, we find that the impedances for these nodes at the edge of the PCB layer are often a little bit smaller than those at the inner part of the layer, but are still much larger than the nodes with small impedances. So this error does not ruin our experimental results.

Appendia C: The realization of change process in the lattice and electric circuit
Here, we describe how to realize the change process in the electric circuit. Similar to the construction in Appendix A, we firstly provide the functions describing the curves from Figs. (C5) Here we provide the constructions of these five structures in the lattices. The ranges of , , x y z coordinates in the lattices are [1,4], [1,3], [1,4] x y z    , and the values of , , In the Hamiltonian, we set the coordinates shown in Table 3 Fig. 6a to Fig. 6e. Red spheres label the sites occupied by the localized eigenstates. We find that for the cases ,, i a c e  , the coordinates of these sites are exactly those shown in Table 3. But for the case ib  , the sites with coordinates (2,2,2) in  The first structure is to form the trefoil knot and the fifth structure is to form the unknot.
The construction for the first structure has been provided in Appendix A. The functions for the four structures from Fig. 7b to Fig. 7e can be obtained in a similar way as the deformed trefoil knot (Eq. (A4)). Consider the lengthy expressions of these functions for these four structures, we do not provide the details of functions here. Since the coordinates in the lattice are integers, we provide the coordinates satisfying the corresponding functions in Table 4. In our design, the coordinates in Table 4 connect with the six adjacent sites through the coupling strength  The fifth structure (the unknot in Fig. 1b  Comparing the chosen coordinates in Table 4 to realize the five structures in the lattices, we find that the coordinates from the first to the fifth structure are changed as: first (Fig.   7a)→second (Fig. 7b), (4,5,2)→(4,5,1); second (Fig. 7b)→third (Fig. 7c) to each other at any two adjacent steps. In this sense, we can view this change process continuously. It means that we need only continuously to modulate some coupling strengths at each step in the change process. Red spheres from Fig. 7a to Fig. 7e are the occupations of localized eigenstates. We can find that for the first trefoil knot (Fig. 7a), the third structure ( Fig.   7c) and the fifth unknot (Fig. 7e), the coordinates of sites occupied by the localized eigenstates are exactly those presented in Table 2 and 4; but for the second (Fig. 7b) and fourth (Fig. 7d) structures, some sites with coordinates shown in Table 4 are not occupied by localized eigenstates. This phenomenon corresponds to the description of changing one crossing in the text around Fig. 6. Moreover, if we change the structures continuously following the sequence as, first (Fig. 7a), second (Fig. 7b), third (Fig. 7c), second (Fig. 7b) and first (Fig. 7a) structures, there are also two cases where the connections formed by localized eigenstates cannot recover the geometric structures. During this change, the trefoil return back to itself finally. Fortunately, the definition of unknotting number is the least time of crossing change necessary to change the knot into an unknot [1]. So the change making trefoil return back to itself is meaningless for the invariant unknotting number.
The corresponding electric realizations are presented in Fig. 7f. Similar to the electric realization in Appendix A, we connect every two nearest neighboring nodes in the electric circuit through the capacitors. The nodes with the coordinates presented in Table 4 are connected by the small electric capacitor 1 C =100pF. Other nodes are connected by the large capacitor 2 C =10nF. Consider the correspondence between the lattice and the electric circuit, we need to change some capacitors and their associated grounding inductors at each step. The corresponding electric setup is presented in Fig. 7g.
In Fig. 10, we provide the distributions of impedances. We can find that for the first figure-8 knot (Fig. 10a), the third structure (Fig. 10c) and the fifth unknot (Fig. 10e), the coordinates of nodes possessing large impedance are exactly those presented in Table 5; but for the second (Fig. 10b) and fourth (Fig. 10d) structures, some sites with coordinates shown in Table 5 do not have large impedances, and the connections are broken apart. This corresponds to the observation of localization of eigenstates in the lattices in Fig. 9.

c. 83 knot
To show the change process that changes from 83 knot to unknot, we need to change Here, we do not provide the details of functions for these structures from the 83 knot to unknot.
Consider the coordinates in the lattices are integers, we only provide the integral coordinates satisfying the corresponding functions in Table 6 and 7. To realize the construction of 83 knot to unknot, we show the coordinates in Table 6 that connect with the six adjacent sites through the coupling strength , 0.01 ij t  for the first five structures in Fig. 11, and the coupling strengths between other sites are , 1 kl s  . When obtaining the fifth structure, we have changed the crossing in the 83 knot once. At this time, the obtained fifth structure is still knotted. The second structure in Fig. 11b ( The third structure in Fig. 11c ( The fourth structure in Fig. 11d ( The fifth structure in Fig. 11e ( When compared with the chosen coordinates to realize the five structures in the lattices, in Table 6, we find that the coordinates from the first to the fifth structure are changed as: first ( Fig. 11a)→second (Fig. 11b) and fourth ( Fig. 11d)→fifth (Fig. 11e), (5,16,2)→(5,16,1), (6,15,2)→(6,15,1), (7,14,2)→(7,14,1). In the change process, only three sites coupled to neighboring sites by strength , 0.01 ij t  are changed and these sites are the nearest neighboring to each other at any two adjacent steps. In this sense, we can view this change process continuously. It means that we need only continuously to modulate three coupling strengths at each step in the change process.
Red spheres in Fig. 11 represent the sites occupied by the localized eigenstates. We can find that for the first 83 knot (Fig. 11a), the third structure (Fig. 11c) and the fifth structure (Fig.   11e), the coordinates of sites occupied by the localized eigenstates are exactly those presented in Table 6; but for the second (Fig. 11b) and fourth (Fig. 11d) structures, some sites with coordinates shown in Table 6 are not occupied by localized eigenstates. The change from Fig.   11a to 11e corresponds to the description of changing one crossing in the text around Fig. 6. Then we show the coordinates in Table 7 that connect with the six adjacent sites through the coupling strength , 0.01 ij t  for the remaining four structures in Fig. 11 (Fig. 11f, 11g, 11h and 11i), and the coupling strengths between other sites are , 1 kl s  . When compared with the chosen coordinates to realize from the fifth to the ninth structures in the lattices, in Table 7, we find that the coordinates from the fifth to the ninth structures are We can find that for the seventh structure (Fig. 11g) and the ninth structure (Fig. 11i), the coordinates of sites occupied by the localized eigenstates are exactly those presented in Table   7; but for the sixth (Fig. 11f) and eighth (Fig. 11h) structures, some sites with coordinates shown in Table 7 are not occupied by localized eigenstates. The change from Fig. 11e to 11i corresponds to the description of changing one crossing in the text around Fig. 6. Considering the change process making the 83 knot to unknot, we have gone through the change of crossings twice. So the unknotting number for the 83 knot is two. Similar to the electric realization above, we connect every two nearest neighboring nodes in the electric circuit through the capacitors, and the nodes with the coordinates presented in Table 6 and 7 are connected by small capacitors 1 C =100pF. Other nodes are connected by large capacitors 2 C =10nF. Consider the correspondence between the lattice and the electric circuit, we only need to change some capacitors and their associated grounding inductors at each step.
In Fig. 12, we provide the distributions of impedances. We can find that for the first five structures, the coordinates of nodes possessing large impedance in the 83 knot, the third structure and the fifth unknot are exactly those presented in Table 6; but in the second and fourth structures, some sites with coordinates shown in Table 6 do not have large impedances. This phenomenon corresponds to the description of changing one crossing in the text around Fig. 6.
For the remaining four structures, the coordinates of nodes possessing large impedance in the seventh structure and the ninth unknot are exactly those presented in Table 7; but in the sixth and eighth structures, some sites with coordinates shown in Table 7 do not have large impedances. This phenomenon also corresponds to the description of changing one crossing in the text around Fig. 6. So we need to change the crossings twice to make the 83 knot become unknot.
Appendia E: The construction details to create electrical knots and links under the action of topoisomerase.

a. The change from an unknot to one hopf link
Here, we describe how to construct knots and links during the action of topoisomerase on the DNA molecules. In Fig. 3a and 3d of the main text, the DNA molecules display the structures of unknot and hopf link. Here, we provide the expression for the structure in Fig. 3a of the main text as 2  2  2  3  1  2  3  4  5  6  7  8  9  10   2  2  2  2  3  2  2  3  4  11  12  13  14  15  16  17  18  19  20   3  3  2 2  2  2 2  3  2  2  3  21  22  23  24  25  26  27  28  29 1 deformed enz f a x a y a z a x a xy a xz a y a yz a z a x   a x y a x z a xy a xz a xyz a y a y z a yz a z a x   a x y a x z a x y a x yz a x z a xy a xy z a 4  30   3  2 2  3  4  6  4 2  2 4  6  4  2 2  31  32  33  34  35  36  37  38  39  40   4  2 4  2 2 2  2 4  3 2  3 2  4 2  4 2  5  41  42  43  44  45  46  47  48  49   6  5  5  50 51 52 ay a y z a y z a yz a z a x a x y a x y a y a zx a zx y a zy a z x a x y z a z y a z x a z y a z x a z y a z a z a x a y

xyz a xz
The structure in Fig. 3a     The other structure shown in Fig. 3d can be obtained in functions in the same way, we do not provide functions here for the lengthy expressions. Here we show how to construct these structures in the lattices. These two structures in Fig. 3a  . We list all these coordinates in sequence in Table 10 below. In the realization of this structure, the coordinates in Table 10   In Figs. 13a and 13b, we provide the realizations of such unknot (Fig. 3a Table 10. To illustrate the distribution clearly, we use the purple and orange tubes in Fig. 13 to connect these sites. We find that they comprise the unknot and hopf link, respectively. So we realize "discrete" versions of unknot and hopf link. The electrically experimental setup to realize these two structures.
The corresponding electric circuits are presented in Fig. 13c. Similar to the electric realization above, we connect every two nearest neighboring nodes in the electric circuit through the capacitors. The nodes with the coordinates presented in Table 10 are connected by small capacitors 1 C =100pF, and other nodes are connected by large capacitors 2 C =10nF.
Experimentally measured impedances have been shown in Fig. 3 of the main text. Although we provide only one experiment setup in Fig. 13c, we can only modulate some electric capacitors and associated grounding inductors to accomplish the constructions of these two structures in the circuits.

b. The change from the hopf link to the figure-8 knot
Actually, the DNA molecule changes to a new topology ( figure-8 knot) if the Tn3 resolvase acts on the DNA molecule with the topology in Fig. 3d of the main text again [1]. Here we will show how to realize these two structures in the lattices, the corresponding electric designs can be realized following the way above. These two structures are constructed in the lattices as, † † ,, ,, H.C. Here, we do not provide the functions of these two structures for their lengthy expressions.
Consider the sites in the lattices are described by the integral coordinates, we provide the coordinates satisfying those functions in Table 11. In the realizations of these two structures, the coordinates in Table 11 connect with the six adjacent sites through the coupling strength  In Fig. 14, we provide the realizations of these two structures in the lattices. Red (blue) cylinders in the lattices represent the coupling strengths  ( 1 s  ). Red spheres in the lattices are the sites occupied by the localized eigenstates. We connect these sites through red and orange tubes. In (c) and (d), the value of impedance at each node has been normalized to the maximum value.
As shown in Fig. 14, red spheres in the lattices represent the sites occupied by localized eigenstates. These sites are exactly those presented in Table 11. To illustrate the distribution clearly, we use the purple and orange tubes in Fig. 14 to connect these sites. We find that they comprise the "discrete" hopf link and figure-8 knot, respectively. The corresponding electric realizations are similar to the above. We connect every two nearest neighboring nodes in the electric circuit through the capacitors. The nodes with the coordinates presented in Table 11 are connected by small capacitors 1 C =100pF, and other nodes are connected by large capacitors 2 C =10nF. The large impedances will emerge at those nodes exactly presented in Table 11. Here, we describe how to construct DNA structures with different twists and writhes. In Figs.
4a, 4d and 4g of the main text, we provide three topologically equivalent structures. The first structure (Fig. 4a) has the twist Tw 0  and the writhe Wr 1  . The third structure (Fig. 4g) has the twist Tw 1  and the writhe Wr =0 . The electric realizations of these three structures have been provided in Fig. 4b, 4e and 4h, respectively. We provide the expression (Eq. (F1)) for the structure shown in Fig. 4a y a z a x a xy a xz a y a yz a z a x   a x y a x z a xy a xz a xyz a y a y z a yz a z a x   a x y a x z a x y a x yz a x z a xy a xy z a xyz a y z a yz a z a x a x y a x y a y a zx a zx y a zy a z x a x y z a z y a z x a z y a z x a z y a z a z a x a y The structure in Fig. 4a is composed of two fitting curves when we set 8 1*10 . The ranges of coordinates are, [3,11], [2,14], The values of 1 a to 52 a for these two fitting curves are listed below in Table 12.  Other structures shown in Fig. 4d and Fig. 4g can be obtained in functions in the same way, and we do not provide the detailed lengthy expressions here. Then we show how to construct these structures in the lattices. These three structures in Fig. 4 of the main text are constructed in the lattices as, Red spheres in the lattices represent the sites occupied by localized eigenstates. As shown in Figs. 16a, 16c, 16e, 16g and 16i, these sites are exactly those presented in Table 14. To illustrate the distribution clearly, we use the purple and orange tubes in Fig. 16 to connect these sites. We find that they comprise the five structures. So we realize "discrete" versions of these five structures. The corresponding electric realizations are similar to those above. We connect every two nodes in the electric circuit through the electric capacitors. The nodes with the coordinates presented in Table 14 are connected by the small capacitor 1 C =100pF, and other nodes are connected by the large capacitor 2 C =10nF. The distributions of large impedances are consistent with those of localized eigenstates in Figs. 16b, 16d, 16f, 16h and 16j. We can find that the nodes having large impedances in the circuits are consistent with those sites occupied by localized eigenstates in the lattices.
We also provide the continuous change between the structure in Fig. 4d and the structure in Fig. 4f. To realize such continuous change, we need to construct another five structures. The coordinates in Table 15 connect with the six adjacent sites through the coupling strength  The structure in Fig. 17a In Fig. 17, we provide the realizations of these five structures in the lattices. Red (blue) cylinders in the lattices represent the coupling strengths , 0.01 ij t  ( , 1 kl s  ). Red spheres in the lattices represent the sites occupied by localized eigenstates. As shown in Fig. 17a, 17c, 17e, 17g and 17i, these sites are exactly those presented in Table 15. To illustrate the distribution clearly, we use the purple and orange tubes in Fig. 17 to connect these sites. We find that they comprise the five structures. So we realize "discrete" versions of these five structures. The corresponding electric realizations are similar to those above. We connect every two nodes in the electric circuit through the electric capacitors. The nodes with the coordinates presented in Table 15 are connected by the small electric capacitor 1 C =100pF, and other nodes are connected by the large electric capacitor 2 C =10nF. The distributions of large impedances are consistent with those of localized eigenstates in Figs. 17b, 17d, 17f, 17h and 17j. We can find that the nodes having large impedances in the circuits are consistent with those sites occupied by localized eigenstates in the lattices. ( 1 s  ) in the lattices. Red spheres denote the sites occupied by localized eigenstates. We connect these sites in purple and orange tubes. In (b), (d), (f), (h) and (j), we provide the simulated distributions of impedance for electric realizations of (a), (c), (e), (g) and (i), respectively. The value of impedance at each node has been normalized to the maximum impedance.