Topological solitonic macromolecules

Being ubiquitous, solitons have particle-like properties, exhibiting behaviour often associated with atoms. Bound solitons emulate dynamics of molecules, though solitonic analogues of polymeric materials have not been considered yet. Here we experimentally create and model soliton polymers, which we call “polyskyrmionomers”, built of atom-like individual solitons characterized by the topological invariant representing the skyrmion number. With the help of nonlinear optical imaging and numerical modelling based on minimizing the free energy, we reveal how topological point defects bind the solitonic quasi-atoms into polyskyrmionomers, featuring linear, branched, and other macromolecule-resembling architectures, as well as allowing for encoding data by spatial distributions of the skyrmion number. Application of oscillating electric fields activates diverse modes of locomotion and internal vibrations of these self-assembled soliton structures, which depend on symmetry of the solitonic macromolecules. Our findings suggest new designs of soliton meta matter, with a potential for the use in fundamental research and technology.

designs of soliton meta matter, with a potential for the use in fundamental research and technology.

Introduction
Recent advances in understanding light-matter interactions and topology allow for designing various artificial media, called metamaterials, assembled of pre-engineered atom-like building blocks [1][2][3][4][5] .Metamaterials often help to overcome limitations of properties of the natural matter [1][2][3][4] ; however, known designs target only very specific physical properties, typically in a narrow range of the parameter space.On the other hand, in many cases solitons feature classical atom-like (alias particle-like) properties, with their stability and particle-like behaviour enabled by nonlinear effects [6][7][8][9][10][11] .Proposals for the creation of bound states of solitons, in the form of "molecules" 12,13 , have attracted a great deal of interest and produced experimental demonstrations in various settings.Known examples include the Newton's cradle in non-uniform soliton chains in optical fibres, which appear as an intermediate stage in the generation of the supercontinuum 9 , and "supersolitons", i.e., self-trapped collective excitations in chains of fluxons trapped by a periodic lattice of defects in a long Josephson junction 6 , or in chains of matter-wave solitons 10 .
However, the possibility of forming soliton macromolecules or metamaterial analogues of polymers with inter-soliton binding resembling strong covalent-like chemical bonds has not been considered, while they can potentially become a basis for new forms of meta matter and promise exhibiting complex dynamic behaviours.Here we show that chirality of liquid-crystal (LC) host media helps to stabilize not only individual atom-like 2D skyrmions (or small molecule-like assemblies dubbed "twistions" 27,28 ) but also their macromolecule-like complex bound states.
Using 3D nonlinear optical imaging 21,24,29,30 we map spatial structures of the director field n(r), which describes spatial variations of the local average direction of rod-like molecular orientations, the nonpolar director n-n, revealing how individual topological solitons are inter-bonded into macromolecules with the help of topological point defects 31,32 .Results of numerical modelling based on the Frank-Oseen free energy are consistent with these experimental findings 21,24,30 , providing insights into the spontaneous self-assembly of polyskyrmionomers.Laser tweezers allow us to robustly control linear, branched and other macromolecule-resembling architectures, allowing, in particular, to encode and store data in the spatial distribution of the skyrmion number, which is a topological invariant characterizing the topology of n(r) in terms of the second homotopy group 17,18,33 .Various types of locomotion and internal vibrations of polyskyrmionomer structures arise, caused by asymmetric responses of n(r) to pulses of the oscillating electric field.
The emergent physical behaviour of polyskyrmionomers suggests new designs of solitonic meta matter [1][2][3][4] , including soliton counterparts of active matter 34 .Beyond LCs, due to the similarity of Hamiltonians, we foresee that similar structures may emerge in solid-state chiral magnets 17,18 , where they can be used in the race-track-memory data storage 34,35 and other spintronics applications 36 .

Results
Polymer-like assembly of skyrmions by defect structures.2D skyrmions are topological solitons of order parameter fields, such as magnetization field of magnets or n(r) of LCs, lowdimensional condensed matter analogues of Skyrme solitons in high energy and nuclear physics (thus often called "baby skyrmions") 37 .Within the simplest 2D skyrmion configuration (Fig. 1a), the director  rotates by 180 degrees from the centre to the peripheral far-field background, while overall different 2D skyrmion structures are labelled as elements of the second homotopy group  2 ( 2 /ℤ 2 ).The topological structures do not allow one to smoothly transform the 2D skyrmion into a uniform state.In 3D, a 2D skyrmion tube can terminate on two  2 ( 2 /ℤ 2 ) point defects with opposite hedgehog charges (±1 ) 38 , when embedded in a uniform background setting with vertical (homeotropic) boundary conditions on confining surfaces (Fig. 1b).The so constructed 3D structures are called torons 39 .Both 2D skyrmions and torons can be stable or meta-stable in chiral LCs, playing the role of macroscopic "atoms".For perpendicular boundary conditions in the unwound uniform background, their interaction is repulsive, therefore they cannot be bound into "molecules".However, additional opposite-charged point defects in this chiral LC system can bind the soliton "atoms" into a stable structure, effectively playing the role of shared electrons in analogy to covalent bonds (Fig. 1c).Furthermore, the diversity of possible binding scenarios includes the asymmetric structures within which the defects close to the bottom and top surfaces differ, breaking the mirror symmetry with respect to the cell midplane (Fig. 1d,e).Due to this asymmetry, the 2D skyrmion number  sk () = 1 8 ∫    2  computed for 2D planar cross-sections (Fig. 1e) varies at different sample depths along the z direction, where   = 1 8    • (   ×   ) is the skyrmion number density along i direction, i,j and k are spatial coordinates and   is the totally antisymmetric tensor 24,40 .Therefore, Nsk(z) may be different near top and bottom confining surfaces, though it is equal to unity at the midplanes.To classify the structural diversity of such solitonic assemblies, a structure exhibiting Nsk(z)=Nt skyrmions at the top and Nsk(z)=Nb skyrmions at the bottom is denoted as S  b  t , where "S" stands for the "skyrmion".Individual skyrmions can be assembled into a higher-order polymer-like structure (Fig. 1f), which we name "polyskyrmionomer", as said above.Much like mapping director structures from 2D planes to the two-sphere order parameter space of elementary and other skyrmions covers the two-sphere Nsk times, S  b  t describes the more complex distribution of this topological invariant within 2D planes intersecting our polyskyrmionomers.We define the polyskyrmionomer's order N as the larger number of Nt and Nb, where polyskyrmionomer of a given order can have multiple isomers.For example, there are three types of dimers (Fig. 1g and Fig. 2): S 2 1 , S 2 2 and S 1 2 .In the experiment, these polyskyrmionomers (Fig. 1g-j and Fig. 2) can be generated by laser tweezers (Methods) under an applied voltage within a certain range.
Data encoding in polyskyrmionomers.The 2D skyrmion number of a polyskyrmionomer not only varies along the sample depth, the z direction, but also along the polyskyrmionomer's chain extension direction, the x direction (Fig. 1e and Fig. 3).For the S 1 3 trimer (Fig. 3a), Nsk(z)=3 in the 2D top horizontal planes parallel confining surfaces (Fig. 1e and Fig. 3b), while in the planes beneath the two point defects (Fig. 3a) the skyrmion number is reduced to unity (Fig. 1e and Fig. 3c).When computing the skyrmion number in the vertical 2D planes (parallel z) at different locations along the x direction (Fig. 1e and Fig. 3d-f), we find that the skyrmion number density is concentrated in specific structural regions.The integration of the skyrmion number density in these localized regions yields -1/2 or +1/2, which implies the presence of merons (half-skyrmions, also referred to as twist-escaped disclinations or lambda lines) 33,41,42 .In 3D, they form meron tubes which can terminate at point defects and fade at lateral sides (Fig. 3g-i).Different combinations of positive-and negative-charge half-integer merons yield three possible values of the total skyrmion number in each (y,z) cross-section (along the x direction): -1, 0 and +1 (Fig. 3d-f).By changing the number and positions of point defects, multiple meta-stable states of polyskyrmionomers can be created (Fig. 3g-q), which feature different skyrmion number series along the x axis.For example, S 1 3 configurations produce three states for different positions of the bottom negative point defect (Fig. 3g-i), whose skyrmion number series are (+1,-1,-1), (+1,-1) and (+1,+1,-1), respectively (we ignore regions with Nsk(x) = 0).These skyrmion number alternations suggest the possibility to design a scheme encoding data into the high-order polyskyrmionomers (Fig. 3r): if we treat Nsk(x) = -1 as (0)2 and Nsk(x) = +1 as (1)2 in the binary data presentation form, with Nsk(x) = 0 as a space separating different bits, we can use a polyskyrmionomer of order 9 to store an 8bit binary string by arranging the specific sequence of defect positions (Fig. 3r and Supplementary Fig. 1).The standard English alphabet can be encoded by 26 different order-9 polyskyrmionomers (Supplementary Fig. 1).As an example, we show a polyskyrmionomer-based encoding of "CU" in Fig. 3r.The polyskyrmionomer-stored information remains robust over long time because the energetic barriers associated with reconfiguring different meta-stable polyskyrmionomer structures are hundreds-to-thousands times higher than thermal energy.

Detailed structures of polyskyrmionomers.
To gain insights into the 3D n(r) structures of polyskyrmionomers, we utilize the three-photon excitation fluorescence polarizing microscopy (Supplementary Fig. 2, Methods and Supplementary Information), which we then compare with results of simulations via minimizing the Frank-Oseen free energy.Based on 3PEF-PM imaging with different polarizations of excitation laser light 29,30 , the reconstructed n(r) (see Fig. 4a,b and Supplementary Movie 1) reveals point defects and localized configurations of n(r) around them (Methods).To obtain the polyskyrmionomers in computer simulations, we place the S 1 1 toron configurations, obtained as described previously 39 , in a linear series with overlapped edges.The subsequent free energy minimization gives rise to self-compensating point defects.Within the numerical relaxation routine based on a variational method, the structure of the polyskyrmionomer relaxes to a stable state with a continuous n(r) except at the singular point defects.Asymmetric polyskyrmionomers, such as S 1 2 and S 1 3 , can be obtained from the symmetric configurations, S 2 2 and S 3 3 , mediated by directed annihilation of pairs of positive and negative defects (see Fig. 4a,b).
The agreement of computer-simulated and experimental 3PEF-PM images in different crosssections (see Fig. 4c,d) validates the numerical analysis of studied structures and the distribution of skyrmion numbers in our configurations.The net charge of all  2 ( 2 /ℤ 2 ) point defects within the polyskyrmionomers is zero (see Fig. 4e-j), which is consistent with the uniform far-field background and boundary conditions.Spatial changes of the skyrmion numbers  2 ( 2 /ℤ 2 ) at different 2D cross-sections orthogonal to z and x axes are mediated by point defects  2 ( 2 /ℤ 2 ) .
For example, while there is only one linearly stretched skyrmion in the midplane of Fig. 4f,i and Supplementary Fig. 3, this stretched skyrmion splits when the plane crosses the point defects (Supplementary Fig. 3).Imaging and modelling reveal n(r) structures comprising nonsingular configurations of the so-called lambda-lines line (see Fig. 4g-k) 33 , fractional skyrmions that are pervasive in chiral nematic LCs due to their energy-costs-minimizing nature.
Self-propelling rotation and intrinsic dynamics of the topological "molecules" .The structure and symmetry of polyskyrmionomers determine properties of their driven dynamics, which can be illustrated with examples of dimers S 1 2 , S 2 2 and S 2 1 (Fig. 1 and Fig. 2).When subjected to an oscillating electric field, these structures exhibit squirming dynamical behaviour (Fig. 5a-c) arising from their nonreciprocal asymmetric temporal evolution of n(r) 14 : under the action of oscillating voltage U (amplitude U0~3.9 V), S 1 2 and S 2 1 configurations rotate clockwise and counter-clockwise, respectively.This observation is consistent with the symmetry of their 3D structures, which have broken mirror symmetry planes passing through their middles orthogonally to both x and z axes.
As one can obtain S 1 2 and S 2 1 from each other by upside-down flipping, as anticipated, their rotation directions are opposite.The S 2 2 structures remain static, except for Brownian motion, consistent with their mirror symmetry: flipping S 2 2 transforms it into itself.Directional rotation of asymmetric polyskyrmionomers depends on frequency of the oscillating electric field (see Fig. 5d,e): the rotation velocities of S 2 1 and S 1 2 decay exponentially and exhibit the same frequency dependence in opposite directions; S 1 3 and higher-order structures rotate slower than the lower-order ones, though yielding a similar exponential decay coefficient.These high-order polyskyrmionomers also exhibit more complex dynamics, like bending, self-folding and waving, as also established for their semi-flexible chemical macromolecular analogues (see Supplementary Movie 2).
Since skyrmions are fundamental elements of our topological molecule entities, similar to atoms, thermal fluctuations or external fields can cause various vibrations reflecting interactions between the bound quasi-atoms.The observed over-damped vibrations of polyskyrmionomer are consistent with low Reynold number in our system (see Fig. 5f,g).The initial length L of the polyskyrmionomer depends on an equilibrium-defining voltage U.As the amplitude of voltage U0 is suddenly changed by  , the polyskyrmionomers shrink or extend so as to reach a new equilibrium length (see Fig. 5f-h), according to an equation of the over-damped vibrations describing the change of the length : where β is the effective damping coefficient and ω0 is the eigenfrequency of the macrosoliton's vibrations.The best fits of solutions of Eq. (1) to the observed data for dimers, trimers, tetramers and pentamers (Fig. 5h) demonstrates that β and ω0 scale inversely to the order N.This finding is reasonable, similar to how β and ω0 would scale for N strings connected in a series, where the respective constants also decay as 1/N.When switching U, the polyskyrmionomer oscillates between two equilibrium lengths (see Supplementary Movie 3) and displays breathing-like behavior (Fig. 6).Non-contact laser tweezers also prompt linear over-damped vibrations in the polyskyrmionomers of different modes (see Supplementary Movie 4, Supplementary Fig. 4 and Supplementary Information).
Polyskyrmionomers exhibit out-of-equilibrium inter-solitonic interactions when placed close to each other (see Fig. 5i and Supplementary Movie 5), altering directional rotations of each other (Fig. 5j) and forming an array or gas (Fig. 7a,b and Supplementary Movie 5).The gas of solitonic molecules exhibits an interesting dynamic behavior different from that of passive soliton lattices 21,24,39,43 and other collective-motion systems 44,45 .The polyskyrmionomers can also interact with other spatially localized topological director configurations, like torons (Fig. 7c,d), hopfions (Fig. 7e,f) and möbiusons 46 (Fig. 8).When embedded in a gas-like disordered states of polyskyrmionomers (see Supplementary Movie 5), the solitonic dimers can collide and interact with torons or hopfions 33 , yielding motion trajectories dependent on the size and respective viscous drag resistance to motions of the solitons experiencing these interactions.If we embed a small möbiuson 46 near a dimer, the interaction leads to their co-locomotion (Fig. 8a) while the dimer rotates with a time-varying angular velocity (Fig. 8b).This co-propagation reveals the polyskyrmionomer's potential in implementing transportation of topological cargo.
Polyskyrmionomer architectures.In addition to the linearly shaped polyskyrmionomers, we have found star-and ring-shaped ones (Fig. 9), analogues of corresponding macromolecules.The star-shaped polyskyrmionomers exhibit both directional motion and rotation (see Fig. 9f-h and Supplementary Movie 6), with peaks of the angular velocity of a trimer at f ~ 100 Hz.Ringshaped polyskyrmionomers can adopt different forms (Fig. 9i,j) at applied stabilizing voltage, with the voltage stability range of which being larger than for the linearly and star-shaped polyskyrmionomers.Polyskyrmionomers with different shapes can inter-transform outside of the range of stabilizing voltage.In particular, linearly shaped polyskyrmionomers transform from higher-order to lower-order ones with the increase of voltage (ΔU > 0.2 V), which is accompanied by a progressive decrease of the length and discrete decrease of the skyrmion number measured in specifically selected planes (Fig. 10 and Supplementary Movie 7), in agreement with the simulation results.Furthermore, the ring-shaped polyskyrmionomer can shrink and transform into linear ones (Supplementary Movie 8), and the star-shaped polyskyrmionomers can transform into linear ones or monomers (S 1 1 ).Such processes of bringing the applied voltage outside the stability range to cause splitting of polyskyrmionomers could be considered being analogues of electrolytic splitting of conventional molecules 47 .

Discussion
We have reported creation of polyskyrmionomers, solitonic analogues of polymers, which are composed of point-defect-bound LC skyrmions.Much like in molecules comprising atoms bound by covalent bonds, while sharing electrons, polyskyrmionomers form via individual skyrmions sharing singular point defects.Such solitonic configurations are observed in highly technological chiral LCs, exhibiting facile reconfiguration and emergent dynamics activation with weak external stimuli.For example, high-order polyskyrmionomers exhibit self-interaction and folding in response to low-frequency voltage and laser tweezers (Supplementary Movie 9), with a striking resemblance of conventional polymer chain folding.This observation may help establish solitonic structures in chiral LCs as model systems to study not only the topological solitons in various experimentally inaccessible physical systems, but also to use them to model behavior of polymers and both equilibrium and active matter systems made from semi-flexible chain-like building blocks.The fact that the LC's director is also the optical axis in these birefringent soft matter systems will allow designing such spatially localized structures as versatile reconfigurable beam deflectors, steerers 5 and lasing elements 48 and microcargo transporters 49 .Since the director field of all observed configurations can be smoothly vectorized, they can also emerge as stable or metastable configurations in the solid-state chiral magnetic systems with similar Hamiltonians 50 .Thus, our polyskyrmionomer architectures have potential for data storage in their magnetic nanostructured analogues 51 .In fact, junctions of skyrmion tubes with point defects, which are pervasive within the polyskyrmionomers, are also found in magnetic solids, albeit in different structures 18 .For example, similar to skyrmion bags 50 , polyskyrmionomers could allow for new breeds of high-density multibit magnetic racetrack memories.From a fundamental standpoint, interesting open questions include how polyskyrmionomer structures behave in presence of flows 52 or during collective motions 44,45 .

Generating and imaging localized structures
We utilize an ytterbium-doped fibre laser (YLR-10-1064, IPG Photonics, operating at 1064 nm) to generate and non-contact manipulate polyskyrmionomers.The low-order polyskyrmionomers (linearly, star-and ring-shaped ones) are laser-generated in a uniform unwound background under the action of sinusoidal electric field (we tune the voltage to the balance value), with the laser power in the range of 40-50 mW.For higher-order linearly shaped polyskyrmionomers, we first turn on the laser, move the sample stage linearly and melt a strip region.When the LCs quench back, a high-order linear polyskyrmionomers emerge simultaneously.Torons, hopfions and möbiusons are also generated by laser tweezers, following the previously reported procedures 14,21,46 .
POMs and brightfield transmission-mode optical images are obtained with a multi-modal imaging setup built around an IX-81 Olympus inverted microscope (which is also integrated with the 3PEF-PM imaging setup described below) and charge-coupled-device cameras (Grasshopper and Flea FMVU-13S2C-CS, Point Grey Research).High-numerical-aperture (NA) Olympus objectives 100×, 40× and 20× with NA =1.4,0.75 and 0.4, respectively, are used.The angular velocities and trajectories of solitons are analysed using freeware (ImageJ) from National Institutes of Health.

Numerical modeling
For chiral nematic LCs, the energy cost of spatial deformations of n(r) can be expressed by the Frank-Oseen free energy functional: Here the Frank elastic constant  11 ,  22 and  33 determine the energy cost of splay, twist and bend deformations, respectively.Further, the surface energy is where W is the surface anchoring strength and   the preferred orientation of which is perpendicular to the surface.When external electric field  is applied, the dielectric properties of LCs induce an additional dielectric coupling term in the free energy, so that the free energy is supplemented by the following electric coupling term: where  0 is the vacuum permittivity, and Δ is the dielectric anisotropy of the LC.The total free energy F is the sum of Felastic, Felectric and Fsurface.Polyskyrmionomers of different orders emerge as local or global minima of F, and a relaxation routine based on the variational method is used to identify an energy-minimizing configuration n(r).Applying this method, at each iteration of the numerical simulation n(r) is updated based on a formula derived from the Euler-Lagrange equation,   new =   old − MSTS 2 []   , where subscript i denotes spatial coordinates, []   denotes the functional derivative of F with respect to ni, and MSTS is the maximum stable time step of the minimization routine, determined by the elastic constants and the spacing of the computational grid.To scale the time step in the real system, we assume that the director produced images separately for three different wavelengths spanning the entire visible spectrum (450, 550, and 650 nm) and then superimposed them, according to light source intensities at the corresponding wavelengths.

Three-dimensional nonlinear optical imaging
3D nonlinear optical imaging of polyskyrmionomer structures is performed using the 3PEF-PM setup built around the IX-81 Olympus inverted optical microscope (see Supplementary Information).To reduce material birefringence that leads to the 3PEF-PM imaging artifacts, before imaging we applied the weak ultraviolet light to photo-polymerize the polyskyrmionomers which are stabilized in the partially polymerizable chiral LC mixtures; then Using these parameters, we calculate the fields () and (): where  is the amplitude of the normalized 3PEF-PM signal.The ensuing iso-surfaces are visualized in perspective views with the help of an open-source freeware, PARAVIEW.
Computer simulations of the 3PEF-PM images are also based on the signal intensity.The 3D structures, produced from both experimental and numerical data, for different linear polarizations of excitation light are coloured as shown in Fig. 4a,b, and 2D cross-sections are colourd by red and green for  0 and  /2 , respectively (Fig. 4c,d).   .o-q, The skyrmion number density in different cross-sections of the S 3 6 along z.The locations of the cross-sections corresponding to panels of (o-q) are marked in (n).The colour codes used in (k-m,o-q) are presented on their right.r, Encoding of word "CU" using the ASCII binary table.The selected n(r)-orientations corresponding to preimages are marked on the right-top, respectively.

Chiral LCs are prepared by mixing 4 -
Cyano-4'-pentylbiphenyl (5CB, EM Chemicals) or a nematic mixture E7 (Slichem) with left-handed chiral additive cholesterol pelargonate (Sigma-Aldrich) at a weight fraction  dopant = 1/(ℎ htp ) to define the equilibrium pitch (p) of the ensuing chiral LCs, where the helical twisting power ℎ htp = 6.25 μm −1 for cholesterol pelargonate.In addition, to provide high-resolution, aberration-free 3D nonlinear optical imaging, a partially polymerizable chiral LC mixture is prepared by mixing E7-based chiral LC (at fraction of 69%), reactive mesogens RM-82 and RM-257 (at fractions of 12% and 18%, respectively, Merck), and ultraviolet-sensitive photoinitiator Irgacure 369 (at fraction of 1%, Sigma-Aldrich).The samples are prepared by sandwiching LCs between indium-tin-oxide (ITO)-coated glass slides or coverslips treated with 25% solution of polyimide SE5661 (Nissan Chemicals) to obtain perpendicular boundary conditions.The polyimide is applied to the substrates by spin-coating at 2,700 rpm for 30 s followed by a 5 min prebaking at 110 °C and 1 hour baking at 185 °C.The cell gap thickness d = 10 μm is defined by silica spacers and the cell gap to pitch ratio is d/p ≈ 2. Metal wires are attached to ITO glasses and connected to an external voltage supply (GFG-8216A, GW Instek) or a data acquisition board (NIDAQ-6363, National Instruments) for electric control.Additionally, we use an in-house MATLAB code controlling the data acquisition board, connected to a computer for fast modulation of the voltage output.
we washed away and replaced the unpolymerized LCs by infiltrating an index-matching immersion oil.We use a Ti-Sapphire oscillator (Chameleon Ultra II; Coherent) operating at 870 nm with 140-fs pulses at an 80 MHz repetition rate, as the source of the laser excitation light.An oil-immersion 100× objective (NA = 1.4) is used to collect the fluorescence signals, which are detected by a photomultiplier tube (H5784-20, Hamamatsu) after a 417/60-nm bandpass filter.With a third-order nonlinear process, the LC molecules are excited via the three-photon absorption process and the signal intensity scales ∝ cos 6  0 , where  0 is the angle between the polarization of the excitation light and the long axis of the LC molecule.Different polarization states of the excitation are controlled by a half-wave plate.When n(r) is nearly parallel to the polarization of the laser beam, large cos  0 corresponds to the strong 3PEF-PM signal intensity.Then, the experimental 3D fluorescence images are processed by applying contrast enhancement and depth-dependent intensity correction.To reconstruct the polyskyrminomer structures experimentally, we define the director with respect to polar angle  and azimuthal angle , where () = [sin  cos  , sin  sin  , cos ].We calculate the Stokes parameters using four 3PEF-PM images with the excitation laser beam linearly polarized at four different angles with respect to the x axis ( 0 ,  /4 ,  /2 ,  3/4 ), the angle being zero for component  0 .Then, the appropriately defined Stokes parameters are  0 = ( 0 +  /4 +  /2 +  3/4 )/2 ,  1 =  0 −  /2 ,  2 =  /4 −  3/4 .(5) Figures Captions

Fig. 3|
Fig. 3| Spatial distribution of 2D skyrmion number of polyskyrmionomers.a-f, The preimage (a) and skyrmion number density in different cross-sections (b-f) of an S 1 3 .Locations of the cross-

Fig. 4|
Fig. 4| Experimental and numerical analysis of structures of    and    .a,b, Experimentally reconstructed (left) and numerically simulated (right) iso-surfaces that depicts n(r) of S 1 2 (a) and

Fig. 5|
Fig. 5| Locomotion, vibration and dynamic interactions of polyskyrmionomers.a, Rotation angles of S 1 2 (red), S 2 2 (green) and S 2 1 (blue) versus time with their preimages.The clockwise direction is defined as positive, and angles are measured with an error of ±3°.b,c, Snapshots of self-propelling rotation of an S 1 2 (b) and S 2 1 (c).d, Angular velocities () of S 1 2 (red), S 2 1 (blue) and S 1 3 (green) versus time.The rotation direction of S 2 1 is counter-clockwise, and we plot values

Fig. 6|
Fig. 6| Breathing dynamics of a trimer.Red circles represent the length of a trimer, under the

Fig. 7|
Fig. 7| The interaction of polyskyrmionomers with other solitons.a, A dimer array.b, A

Fig. 8|
Fig. 8| The co-propulsion of polyskyrmionomer and möbiuson.a, POM of the interacting