Creation of electrical knots and observation of DNA topology

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. The braided function is expressed with a formal variable u and has N distinct zeros at positions parameterized by the height h in equation (A1) as

$$\begin{equation}{p}_{h}(u)=\prod\limits _{j=0}^{N-1}(u-(\mathrm{cos}\enspace {h}_{j}+\mathrm{i}\mathrm{sin}\enspace \beta {h}_{j})).\end{equation} \tag{ A1 }$$

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 2πぱい, so after h changed by 2πぱい, the zeroes in the complex u plane are in the same positions as where they started. Moreover, the term e^ih 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., p_h (u) = u² − v² = 0 represents the standard hopf link and p_h (u) = u² − v³ = 0 denotes the standard trefoil knot.

The formal variables in the expression above can be mapped to the R³ space through the Milnor map. We use Cartesian coordinates x, y, and z to express the formal variables u and v in the following equations:

$$\begin{equation}\begin{aligned}u& =\frac{{x}^{2}+{y}^{2}+{z}^{2}-1+2\mathrm{i}z}{{x}^{2}+{y}^{2}+{z}^{2}+1}=\frac{{R}^{2}+{z}^{2}-1+2\mathrm{i}z}{{R}^{2}+{z}^{2}+1}\\ v& =\frac{2\left(x+\mathrm{i}y\right)}{{x}^{2}+{y}^{2}+{z}^{2}+1}=\frac{2R\enspace {\text{e}}^{\text{i}\phi }}{{R}^{2}+{z}^{2}+1}.\end{aligned}\end{equation} \tag{ A2 }$$

Given in both Cartesian and cylindrical coordinates, R² = x² + y² and R e^iϕ = x + iy, the expression for the standard trefoil knot is written as

$$\begin{align}{f}_{\text{tre}}\left( \overrightarrow {r}\right)& =\left[({R}^{4}-1)\left({R}^{2}-1\right)-8{R}^{3}\enspace {\mathrm{e}}^{\mathrm{i}3\phi }\right]+4\mathrm{i}z\left({R}^{4}-1\right)\\ & \quad +{z}^{2}\left(3{R}^{4}-6{R}^{2}-5\right)+8\mathrm{i}{z}^{3}{R}^{2}+{z}^{4}\left(3{R}^{2}-5\right)+4\mathrm{i}{z}^{5}+{z}^{6}=0\cdot \end{align} \tag{ A3 }$$

In figure 5(a), we provide the standard trefoil knot in the R³ space.

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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 figure 5(b), we provide one structure obtained from the deformation of the structure in figure 5(a). 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 figure 5(b) is shown as

$$\begin{align}{f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{t}\mathrm{r}\mathrm{e}}& =-1+{a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}{x}^{2}+{a}_{5}xy+{a}_{6}xz+{a}_{7}{y}^{2}+{a}_{8}yz+{a}_{9}{z}^{2}+{a}_{10}{x}^{3}\\ & \quad +{a}_{11}{x}^{2}y+{a}_{12}{x}^{2}z+{a}_{13}x{y}^{2}+{a}_{14}x{z}^{2}+{a}_{15}xyz+{a}_{16}{y}^{3}+{a}_{17}{y}^{2}z\\ & \quad +{a}_{18}y{z}^{2}+{a}_{19}{z}^{3}+{a}_{20}{x}^{4}+{a}_{21}{x}^{3}y+{a}_{22}{x}^{3}z+{a}_{23}{x}^{2}{y}^{2}\\ & \quad +{a}_{24}{x}^{2}yz+{a}_{25}{x}^{2}{z}^{2}+{a}_{26}x{y}^{3}+{a}_{27}x{y}^{2}z+{a}_{28}xy{z}^{2}+{a}_{29}x{z}^{3}+{a}_{30}{y}^{4}\\ & \quad +{a}_{31}{y}^{3}z+{a}_{32}{y}^{2}{z}^{2}+{a}_{33}y{z}^{3}+{a}_{34}{z}^{4}+{a}_{35}{x}^{6}+{a}_{36}{x}^{4}{y}^{2}+{a}_{37}{x}^{2}{y}^{4}\\ & \quad +{a}_{38}{y}^{6}+{a}_{39}z{x}^{4}+{a}_{40}z{x}^{2}{y}^{2}+{a}_{41}z{y}^{4}+{a}_{42}{z}^{2}{x}^{4}+{a}_{43}{x}^{2}{y}^{2}{z}^{2}\\ & \quad +{a}_{44}{z}^{2}{y}^{4}+{a}_{45}{z}^{3}{x}^{2}+{a}_{46}{z}^{3}{y}^{2}+{a}_{47}{z}^{4}{x}^{2}+{a}_{48}{z}^{4}{y}^{2}+{a}_{49}{z}^{5}\\ & \quad +{a}_{50}{z}^{6}+{a}_{51}{x}^{5}+{a}_{52}{y}^{5}.\end{align} \tag{ A4 }$$

The deformed trefoil knot can be composed of two fitting curves when we set $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{t}\mathrm{r}\mathrm{e}}\right\vert {\leqslant}5{\times}1{0}^{-9}$. The ranges of coordinates are, x ∈ [2, 7] and y ∈ [2, 8], z ∈ [1, 2]. The values from a₁ to a₅₂ for these two fitting curves are listed below (table 1).

The previous step has shown how to obtain one knot or link in space. Figures 5(a) and (b) show the trefoil knot. Here we provide the construction of trefoil knot in the lattice. We construct the lattice as

$$\begin{equation}{H}_{\mathrm{d}\mathrm{e}\mathrm{f}-\mathrm{t}\mathrm{r}\mathrm{e}}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ A5 }$$

The site in the lattice is often described by the integral coordinate. In figure 5(c), the ranges of x, y, z coordinates in the lattice are x ∈ [1, 7], y ∈ [1, 8], z ∈ [0, 2], and the values of x, y, z are all integers. Therefore, we seek all integral coordinates satisfying $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{t}\mathrm{r}\mathrm{e}}\right\vert {\leqslant}5{\times}1{0}^{-9}$ and the sites described by these coordinates can comprise the deformed 'discrete' trefoil knot. We list all these coordinates in sequence below.

In the Hamiltonian, we set the coordinates shown in table 2 to connect with the six adjacent sites through the coupling strength t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1. We provide the construction of this Hamiltonian in the three-dimensional lattice in figure 5(c). Red (blue) cylinders represent the coupling strength t_i,j (s_k,l ). We also show the distribution of localized eigenstates of H in figure 5(c). 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 figure 5(c) to connect these sites and find that they comprise the trefoil knot. This trefoil knot is the 'discrete' deformed trefoil knot in figure 5(b).

Since the lattice contains three layers along the z direction, we use three PCB layers in the electric circuit. The detailed structure is provided in figure 5(d). Every two nearest neighboring nodes in the circuit is connected through the capacitor C₁ (C₂), which corresponds to the coupling strength t_i,j (s_k,l ) in the lattice. In the electric design, the dynamics of this circuit can be described by one Laplacian J which bridges the current and voltages in the circuit. By choosing appropriate grounding inductors, we can make the Laplacian J similar to the Hamiltonian H_def-tre. For three nodes in the circuit (contained in the orange dotted triangle) in figure 5(d), the Laplacian is addressed as

$$\begin{equation}J=\left(\begin{matrix}\hfill \dots \hfill & \hfill \hfill & \hfill \dots \hfill & \hfill \hfill & \hfill \dots \hfill \\ \hfill \hfill & \hfill 6\mathrm{i}\omega {C}_{2}+6{(\mathrm{i}\omega {L}_{2})}^{-1}\hfill & \hfill \mathrm{i}\omega {C}_{2}\hfill & \hfill 0\hfill & \hfill \hfill \\ \hfill \dots \hfill & \hfill \mathrm{i}\omega {C}_{2}\hfill & \hfill 5\mathrm{i}\omega {C}_{2}+5{(\mathrm{i}\omega {L}_{2})}^{-1}+\mathrm{i}\omega {C}_{1}+{(\mathrm{i}\omega {L}_{1})}^{-1}\hfill & \hfill \mathrm{i}\omega {C}_{1}\hfill & \hfill \dots \hfill \\ \hfill \hfill & \hfill 0\hfill & \hfill \mathrm{i}\omega {C}_{1}\hfill & \hfill 6\mathrm{i}\omega {C}_{1}+6{(\mathrm{i}\omega {L}_{1})}^{-1}\hfill & \hfill \hfill \\ \hfill \dots \hfill & \hfill \hfill & \hfill \dots \hfill & \hfill \hfill & \hfill \dots \hfill \end{matrix}\right)\end{equation} \tag{ A6 }$$

In our design, when we set $\omega =\frac{\mathrm{1}}{\sqrt{{L}_{1}{C}_{1}}}=\frac{1}{\sqrt{{L}_{2}{C}_{2}}}$, the diagonal elements in the matrix J are eliminated, and the Laplacian changes to the expression shown in the inset of figure 5(d).

In our experiment, we use the impedance as the measured quantity. The impedance between ath node and bth node is ${Z}_{a,b}=\frac{{V}_{a}-{V}_{b}}{{I}_{a,b}}={\sum }_{{j}_{n}\ne 0}\frac{\vert {\varphi }_{n,a}-{\varphi }_{n,b}{\vert }^{2}}{{j}_{n}}$. Here, V_a (V_b ) is the electric potential at a (b)th node and I_a,b is the current between ath node and bth node. The symbols j_n and φふぁい_n represent the nth admittance eigenvalue and the corresponding eigenstate of J. For convenience, we set the bth node as the ground. When the node is connected by the capacitor C₁ (C₂), the corresponding term in Laplacian J is iωおめがC₁ (iωおめがC₂). When the electric circuit shown in figure 5(d) is designed, we can obtain the distribution of impedance which is similar to that of localized eigenstates in figure 5(c). The experimental distributions of impedance have been provided in the main text.

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 figure 1(a) and large spheres in figure 1(f) 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 figure 1(a) and (f) 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.

When we change one crossing in the trefoil knot, the trefoil knot can change to the unknot. Here we provide the realizations of structures during the change process from figures 7(a)–(e).

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Constructions of these structures are similar to the description in appendix A. These five structures from figures 7(a)–(e) are constructed in the lattices as equation (D1),

$$\begin{equation}{H}_{un}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ D1 }$$

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 figures 7(b)–(e) can be obtained in a similar way as the deformed trefoil knot (equation (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 t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1.

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 (figure 7(a)) → second (figure 7(b)), (4,5,2) → (4,5,1); second (figure 7(b)) → third (figure 7(c)), (4,5,1) → (3,5,1); third (figure 7(c)) → fourth (figure 7(d)), (3,5,1) → (3,4,1), (4,6,1) → (4,5,1); and fourth (figure 7(d)) → fifth (figure 7(e)), (3,4,1) → (3,4,2), (4,5,1) → (4,5,2). In the change process, only one or two sites coupled to neighboring sites by strength t_i,j = 0.01 are changed and these sites are 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 some coupling strengths at each step in the change process. Red spheres from figures 7(a)–(e) are the occupations of localized eigenstates. We can find that for the first trefoil knot (figure 7(a)), the third structure (figure 7(c)) and the fifth unknot (figure 7(e)), the coordinates of sites occupied by the localized eigenstates are exactly those presented in tables 2 and 4; but for the second (figure 7(b)) and fourth (figure 7(d)) 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 figure 6. Moreover, if we change the structures continuously following the sequence as, first (figure 7(a)), second (figure 7(b)), third (figure 7(c)), second (figure 7(b)) and first (figure 7(a)) 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 figure 7(f). 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 C₁ = 100 pF. Other nodes are connected by the large capacitor C₂ = 10 nF. 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 figure 7(g).

In figure 8, we provide the measurement outcomes of distributions of impedances for the corresponding electric designs in figure 7(f1)–(f5).

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From the distributions of impedance in figure 8, we can find that the nodes connected by small electric capacitors C₁ are exactly having a large impedance for the first, third and fifth cases (figures 8(a), (c) and (e)). But for the second and fourth cases (figures 8(b) and (d)), some nodes connected by the capacitors C₁ are not possessing large impedances. So these experimental results are consistent with those in the simulations.

Not only the change of crossing once in the trefoil knot can be revealed in the electric circuit, but changes of crossings in other knots can also be provided. The construction details in the figure-8 knot and 8₃ knot are the same as the electrical trefoil knot above. We firstly provide the constructions of figure-8 knot and 8₃ knot in lattices, then map to the electric circuits. The Hamiltonian to realize the change from figure-8 knot to unknot in the lattice is shown as,

$$\begin{equation}{H}_{\mathrm{fi}\mathrm{g}\mathrm{u}\mathrm{r}\mathrm{e}-\mathrm{8}}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ D2 }$$

Here, we do not provide the details of functions for these structures from figure-8 knot to unknot. Consider the coordinates in the lattices are integers, we only provide the integral coordinates satisfying the corresponding functions in table 5. To realize the construction from the figure-8 knot to unknot, we show the coordinates in table 5 that connect with the six adjacent sites through the coupling strength t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1.

When compared with the chosen coordinates to realize the five structures in the lattices, in table 5, we find that the coordinates from the first to the fifth structure are changed as: first (figure 9(a)) → second (figure 9(b)), (3,2,1) → (3,2,2); second (figure 9(b)) → third (figure 9(c)), (3,2,2) → (3,1,2); third (figure 9(c)) → fourth (figure 9(d)), (3,2,3) → (3,2,2); and fourth (figure 9(d)) → fifth (figure 9(e)), (3,2,2) → (3,2,1), (3,1,2) → (3,2,1). In the change process, only one or two sites coupled to neighboring sites by strength t_i,j = 0.01 are changed and these sites are 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 some coupling strengths at each step in the change process. The corresponding realizations in lattices are presented in figure 9.

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Red spheres in figure 9 represent the sites occupied by the localized eigenstates. We can find that for the first figure-8 knot (figure 9(a)), the third structure (figure 9(c)) and the fifth unknot (figure 9(e)), the coordinates of sites occupied by the localized eigenstates are exactly those presented in table 5; but for the second (figure 9(b)) and fourth (figure 9(d)) structures, some sites with coordinates shown in table 5 are not occupied by localized eigenstates. This phenomenon corresponds to the description of changing one crossing in the text around figure 6.

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 5 are connected by small capacitors C₁ = 100 pF. Other nodes are connected by large capacitors C₂ = 10 nF. 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 figure 10, we provide the distributions of impedances. We can find that for the first figure-8 knot (figure 10(a)), the third structure (figure 10(c)) and the fifth unknot (figure 10(e)), the coordinates of nodes possessing large impedance are exactly those presented in table 5; but for the second (figure 10(b)) and fourth (figure 10(d)) 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 figure 9.

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To show the change process that changes from 8₃ knot to unknot, we need to change crossing in 8₃ knot twice. The Hamiltonian to realize the change from the 8₃ knot to unknot in lattice is shown as

$$\begin{equation}{H}_{{\mathrm{8}}_{3}\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{t}}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ D3 }$$

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 tables 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 t_i,j = 0.01 for the first five structures in figure 11, and the coupling strengths between other sites are s_k,l = 1. 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.

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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 (figure 11(a)) → second (figure 11(b)), (5,14,1) → (5,14,2), (6,15,1) → (6,15,2), (7,16,1) → (7,16,2); second (figure 11(b)) → third (figure 11(c)), (5,14,2) → (5,14,3), (6,15,2) → (6,15,3), (7,16,2) → (7,16,3); third (figure 11(c)) → fourth (figure 11(d)), (5,16,3) → (5,16,2), (6,15,3) → (6,15,2), (7,14,3) → (7,14,2); and fourth (figure 11(d)) → fifth (figure 11(e)), (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 t_i,j = 0.01 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 figure 11 represent the sites occupied by the localized eigenstates. We can find that for the first 8₃ knot (figure 11(a)), the third structure (figure 11(c)) and the fifth structure (figure 11(e)), the coordinates of sites occupied by the localized eigenstates are exactly those presented in table 6; but for the second (figure 11(b)) and fourth (figure 11(d)) structures, some sites with coordinates shown in table 6 are not occupied by localized eigenstates. The change from figures 11(a)–(e) corresponds to the description of changing one crossing in the text around figure 6.

Then we show the coordinates in table 7 that connect with the six adjacent sites through the coupling strength t_i,j = 0.01 for the remaining four structures in figure 11 (figures 11(f)–(i)), and the coupling strengths between other sites are s_k,l = 1.

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 changed as: fifth (figure 11(e)) → sixth (figure 11(f)), (5,8,3) → (5,8,2), (6,7,3) → (6,7,2), (7,6,3) → (7,6,2); sixth (figure 11(f)) → seventh (figure 11(g)), (5,8,2) → (5,8,1), (6,7,2) → (6,7,1), (7,6,2) → (7,6,1); seventh (figure 11(g)) → eighth (figure 11(h)), (5,6,1) → (5,6,2), (6,7,1) → (6,7,2), (7,8,1) → (7,8,2); and eighth (figure 11(h)) → ninth (figure 11(i)), (5,6,2) → (5,6,3), (6,7,2) → (6,7,3), (7,8,2) → (7,8,3). In the change process, only three sites coupled to neighboring sites by strength t_i,j = 0.01 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. We can find that for the seventh structure (figure 11(g)) and the ninth structure (figure 11(i)), the coordinates of sites occupied by the localized eigenstates are exactly those presented in table 7; but for the sixth (figure 11(f)) and eighth (figure 11(h)) structures, some sites with coordinates shown in table 7 are not occupied by localized eigenstates. The change from figures 11(e)–(i) corresponds to the description of changing one crossing in the text around figure 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 tables 6 and 7 are connected by small capacitors C₁ = 100 pF. Other nodes are connected by large capacitors C₂ = 10 nF. 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 figure 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 figure 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 figure 6. So we need to change the crossings twice to make the 8₃ knot become unknot.

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Here, we describe how to construct knots and links during the action of topoisomerase on the DNA molecules. In figures 3(a) and (d) of the main text, the DNA molecules display the structures of unknot and hopf link. Here, we provide the expression for the structure in figure 3(a) of the main text as

$$\begin{align}{f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{e}\mathrm{n}\mathrm{z}}& =-1+{a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}{x}^{2}+{a}_{5}xy+{a}_{6}xz+{a}_{7}{y}^{2}+{a}_{8}yz+{a}_{9}{z}^{2}+{a}_{10}{x}^{3}\\ & \quad +{a}_{11}{x}^{2}y+{a}_{12}{x}^{2}z+{a}_{13}x{y}^{2}+{a}_{14}x{z}^{2}+{a}_{15}xyz+{a}_{16}{y}^{3}+{a}_{17}{y}^{2}z\\ & \quad +{a}_{18}y{z}^{2}+{a}_{19}{z}^{3}+{a}_{20}{x}^{4}+{a}_{21}{x}^{3}y+{a}_{22}{x}^{3}z+{a}_{23}{x}^{2}{y}^{2}+{a}_{24}{x}^{2}yz+{a}_{25}{x}^{2}{z}^{2}\\ & \quad +{a}_{26}x{y}^{3}+{a}_{27}x{y}^{2}z+{a}_{28}xy{z}^{2}+{a}_{29}x{z}^{3}+{a}_{30}{y}^{4}\\ & \quad +{a}_{31}{y}^{3}z+{a}_{32}{y}^{2}{z}^{2}+{a}_{33}y{z}^{3}+{a}_{34}{z}^{4}+{a}_{35}{x}^{6}+{a}_{36}{x}^{4}{y}^{2}+{a}_{37}{x}^{2}{y}^{4}+{a}_{38}{y}^{6}\\ & \quad +{a}_{39}z{x}^{4}+{a}_{40}z{x}^{2}{y}^{2}+{a}_{41}z{y}^{4}+{a}_{42}{z}^{2}{x}^{4}+{a}_{43}{x}^{2}{y}^{2}{z}^{2}\\ & \quad +{a}_{44}{z}^{2}{y}^{4}+{a}_{45}{z}^{3}{x}^{2}+{a}_{46}{z}^{3}{y}^{2}+{a}_{47}{z}^{4}{x}^{2}+{a}_{48}{z}^{4}{y}^{2}+{a}_{49}{z}^{5}+{a}_{50}{z}^{6}+{a}_{51}{x}^{5}+{a}_{52}{y}^{5}\end{align} \tag{ E1 }$$

The structure in figure 3(a) is composed of four fitting curves when we set $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{e}\mathrm{n}\mathrm{z}}\right\vert {\leqslant}1{\times}1{0}^{-7}$. The ranges of coordinates are x ∈ [2, 8], y ∈ [2, 16] and z ∈ [1, 3]. The values of a₁ to a₅₂ for these four fitting curves are listed below in tables 8 and 9.

The other structure shown in figure 3(d) 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 figures 3(a) and (d) of the main text are constructed in the lattices as,

$$\begin{equation}{H}_{\text{enzyme}}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ E2 }$$

The site in the lattice is often described by the integral coordinate. Since we have provided the function describing the structure in figure 3(a) as $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{e}\mathrm{n}\mathrm{z}}\right\vert {\leqslant}1{\times}1{0}^{-7}$. The ranges of x, y, z coordinates in the lattice are x ∈ [2, 8], y ∈ [2, 16], z ∈ [1, 3], and the values of x, y, z are all integers. Therefore, to construct this structure in the lattice, we seek all integral coordinates satisfying $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{e}\mathrm{n}\mathrm{z}}\right\vert {\leqslant}1{\times}1{0}^{-7}$. We list all these coordinates in sequence in table 10 below. In the realization of this structure, the coordinates in table 10 are those connected with the six adjacent sites through the coupling strength t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1. Moreover, for the structure in figure 3(d), we also provide the coordinates that connect with the six adjacent sites through the coupling strength t_i,j = 0.01 in table 10, and other sites are connected by s_k,l = 1.

In figures 13(a) and (b), we provide the realizations of such unknot (figure 3(a) of the main text) and hopf link (figure 3(d) of the main text) in the lattices. Red (blue) cylinders in the lattices represent the coupling strengths t_i,j = 0.01 (s_k,l = 1). Red spheres in the lattices represent the sites occupied by localized eigenstates. As shown in figure 13, these sites are exactly those presented in table 10. To illustrate the distribution clearly, we use the purple and orange tubes in figure 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.

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The corresponding electric circuits are presented in figure 13(c). 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 C₁ = 100 pF, and other nodes are connected by large capacitors C₂ = 10 nF. Experimentally measured impedances have been shown in figure 3 of the main text. Although we provide only one experiment setup in figure 13(c), we can only modulate some electric capacitors and associated grounding inductors to accomplish the constructions of these two structures in the circuits.

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 figure 3(d) 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,

$$\begin{equation}{H}_{\text{enzyme}}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ E3 }$$

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 t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1.

In figure 14, we provide the realizations of these two structures in the lattices. Red (blue) cylinders in the lattices represent the coupling strengths t_i,j = 0.01 (s_k,l = 1).

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As shown in figure 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 figure 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 C₁ = 100 pF, and other nodes are connected by large capacitors C₂ = 10 nF. 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 figures 4(a), (d) and (g) of the main text, we provide three topologically equivalent structures. The first structure (figure 4(a)) has the twist Tw = 0 and the writhe Wr = 1. The third structure (figure 4(g)) has the twist Tw = 1 and the writhe Wr = 0. The electric realizations of these three structures have been provided in figures 4(b), (e) and (h), respectively. We provide the expression (equation (F1)) for the structure shown in figure 4(a) below,

$$\begin{align}{f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}& =-1+{a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}{x}^{2}+{a}_{5}xy+{a}_{6}xz+{a}_{7}{y}^{2}+{a}_{8}yz+{a}_{9}{z}^{2}+{a}_{10}{x}^{3}\\ & \quad +{a}_{11}{x}^{2}y+{a}_{12}{x}^{2}z+{a}_{13}x{y}^{2}+{a}_{14}x{z}^{2}+{a}_{15}xyz+{a}_{16}{y}^{3}+{a}_{17}{y}^{2}z\\ & \quad +{a}_{18}y{z}^{2}+{a}_{19}{z}^{3}+{a}_{20}{x}^{4}+{a}_{21}{x}^{3}y+{a}_{22}{x}^{3}z+{a}_{23}{x}^{2}{y}^{2}+{a}_{24}{x}^{2}yz\\ & \quad +{a}_{25}{x}^{2}{z}^{2}+{a}_{26}x{y}^{3}+{a}_{27}x{y}^{2}z+{a}_{28}xy{z}^{2}+{a}_{29}x{z}^{3}+{a}_{30}{y}^{4}\\ & \quad +{a}_{31}{y}^{3}z+{a}_{32}{y}^{2}{z}^{2}+{a}_{33}y{z}^{3}+{a}_{34}{z}^{4}+{a}_{35}{x}^{6}+{a}_{36}{x}^{4}{y}^{2}+{a}_{37}{x}^{2}{y}^{4}+{a}_{38}{y}^{6}\\ & \quad +{a}_{39}z{x}^{4}+{a}_{40}z{x}^{2}{y}^{2}+{a}_{41}z{y}^{4}+{a}_{42}{z}^{2}{x}^{4}+{a}_{43}{x}^{2}{y}^{2}{z}^{2}+{a}_{44}{z}^{2}{y}^{4}\\ & \quad +{a}_{45}{z}^{3}{x}^{2}+{a}_{46}{z}^{3}{y}^{2}+{a}_{47}{z}^{4}{x}^{2}+{a}_{48}{z}^{4}{y}^{2}+{a}_{49}{z}^{5}\\ & \quad +{a}_{50}{z}^{6}+{a}_{51}{x}^{5}+{a}_{52}{y}^{5}.\end{align} \tag{ F1 }$$

The structure in figure 4(a) is composed of two fitting curves when we set $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}\right\vert {\leqslant}1{\times}1{0}^{-8}$. The ranges of coordinates are, x ∈ [3, 11], y ∈ [2, 14], z ∈ [1, 3]. The values of a₁ to a₅₂ for these two fitting curves are listed below in table 12.

Other structures shown in figures 4(d) and (g) 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 figure 4 of the main text are constructed in the lattices as,

$$\begin{equation}{H}_{\text{dna}}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ F2 }$$

The site in the lattice is often described by the integral coordinate. Since we have provided the function describing the structure in figure 4(a) as $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}\right\vert {\leqslant}1{\times}1{0}^{-8}$. The ranges of x, y, z coordinates in the lattice are x ∈ [3, 11], y ∈ [2, 14], z ∈ [1, 3], and the values of x, y, z are all integers. Therefore, to construct this structure in the lattice, we seek all integral coordinates satisfying $\left\vert {f}_{\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}}\right\vert {\leqslant}1{\times}1{0}^{-8}$. We list all these coordinates in sequence in table 13 below. In the realization of this structure, these coordinates connecting with the six adjacent sites through the coupling strength t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1. Moreover, for the structures in figures 4(d) and (g), we also provide the coordinates connecting with the six adjacent sites through the coupling strength t_i,j = 0.01 in table 13.

In figures 15(a)–(c), we provide the realizations of these three structures in the lattices. Red (blue) cylinders in the lattices represent the coupling strengths t_i,j = 0.01 (s_k,l = 1). Red spheres in the lattices represent the sites occupied by localized eigenstates. As shown in figures 15(a)–(c), these sites are exactly those presented in table 13. To illustrate the distribution clearly, we use the purple and orange tubes in figures 15(a)–(c) to connect these sites. We find that they comprise the first structure with the twist Tw = 0 and the writhe Wr = 1 (figure 15(a)), the second structure (figure 15(b)) and the third structure with the twist Tw = 1 and the writhe Wr = 0 (figure 15(c)), respectively. So we realize 'discrete' versions of three structures in figures 4(a), (d) and (g) of the main text.

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The corresponding electric circuits are also presented in figure 15(d). Similar to the electric realization above, we connect every two nodes in the electric circuit through the capacitors. The nodes with the coordinates presented in table 13 are connected by C₁ = 100 pF, and other nodes are connected by C₂ = 10 nF. The corresponding experimental designs have been provided in figures 4(b), (e) and (h) of the main text, respectively. Experimentally measured impedances in these circuits have been shown in figures 4(c), (f) and (i) of the main text, respectively.

To demonstrate these two structures (Tw = 0 and Wr = 1, Tw = 1 and Wr = 0) are topological equivalent, we need to start from the structure with Tw = 0 and Wr = 1, and finally reach the structure with Tw = 1 and Wr = 0 continuously. Constructions of these structures are similar to the description in appendix A. That is shown as

$$\begin{equation}{H}_{\text{dna}}=\sum\limits _{\left\langle i,j\right\rangle }{t}_{i,j}{a}_{i}^{{\dagger}}{a}_{j}+\sum\limits _{\left\langle k,l\right\rangle }{s}_{k,l}{a}_{k}^{{\dagger}}{a}_{l}+\mathrm{H}.\mathrm{C}.\end{equation} \tag{ F3 }$$

During this change process, we need to modulate the corresponding designs continuously, and do not change the crossing at any structures. In the change process, only some sites coupled to neighboring sites by the strength t_i,j = 0.01 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. In the following, we provide the details of remaining structures in this continuous change process in the lattices. In the realizations of different structures between the first structure in figure 4(a) and the second structure in figure 4(d), the coordinates in table 14 connect with the six adjacent sites through the coupling strength t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1.

In figure 16, we provide the realizations of these five structures in the lattices.

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Red (blue) cylinders in the lattices represent the coupling strengths t_i,j = 0.01 (s_k,l = 1). Red spheres in the lattices represent the sites occupied by localized eigenstates. As shown in figures 16(a), (c), (e), (g) and (i), these sites are exactly those presented in table 14. To illustrate the distribution clearly, we use the purple and orange tubes in figure 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 C₁ = 100 pF, and other nodes are connected by the large capacitor C₂ = 10 nF. The distributions of large impedances are consistent with those of localized eigenstates in figures 16(b), (d), (f), (h) and (j). 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 figure 4(d) and the structure in figure 4(f). 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 t_i,j = 0.01, and the coupling strengths between other sites are s_k,l = 1.

In figure 17, we provide the realizations of these five structures in the lattices. Red (blue) cylinders in the lattices represent the coupling strengths t_i,j = 0.01 (s_k,l = 1). Red spheres in the lattices represent the sites occupied by localized eigenstates. As shown in figures 17(a), (c), (e), (g) and (i), these sites are exactly those presented in table 15. To illustrate the distribution clearly, we use the purple and orange tubes in figure 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 C₁ = 100 pF, and other nodes are connected by the large electric capacitor C₂ = 10 nF. The distributions of large impedances are consistent with those of localized eigenstates in figures 17(b), (d), (f), (h) and (j). We can find that the nodes having large impedances in the circuits are consistent with those sites occupied by localized eigenstates in the lattices.

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