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are three closely related ideas, all of which signify structures in a physical system that are stable against perturbations. Solitons won't decay, dissipate, disperse or evaporate in the way that ordinary waves (or solutions or structures) might. The stability arises from an obstruction to the decay,
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provides a natural setting for description and classification of defects in ordered systems. Topological methods have been used in several problems of condensed matter theory. Poénaru and
Toulouse used topological methods to obtain a condition for line (string) defects in liquid crystals that can
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Topological defects, of the cosmological type, are extremely high-energy phenomena which are deemed impractical to produce in Earth-bound physics experiments. Topological defects created during the universe's formation could theoretically be observed without significant energy expenditure.
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Topological defects have not been identified by astronomers; however, certain types are not compatible with current observations. In particular, if domain walls and monopoles were present in the observable universe, they would result in significant deviations from what astronomers can see.
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examples of solitons: there is no obstruction to their decay; they will dissipate after a time. The mathematical solution describing a tornado can be continuously transformed, by weakening the rotation, until there is no rotation left. The details, however, are context-dependent: the
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of the lattice that spirals around. It can be moved from one location to another by pushing it around, but it cannot be removed by simple continuous deformations of the lattice. (Some screw dislocations manifest so that they are directly visible to the naked eye: these are the
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the system with a soliton in it, to one without it. The mathematics behind topological stability is both deep and broad, and a vast variety of systems possessing topological stability have been described. This makes categorization somewhat difficult.
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in the material containing it. One common manifestation is the repeated bending of a metal wire: this introduces more and more screw dislocations (as dislocation-anti-dislocation pairs), making the bent region increasingly
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into another; to get from one to the other would require "cutting" (as with scissors), but "cutting" is not a defined operation for solving PDE's. The cutting analogy arises because some solitons are described as mappings
94:(PDE) having distinct classes of solutions, with each solution class belonging to a distinct homotopy class. In many cases, this arises because the base space -- 3D space, or 4D spacetime, can be thought of as having the
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The authenticity of a topological defect depends on the nature of the vacuum in which the system will tend towards if infinite time elapses; false and true topological defects can be distinguished if the defect is in a
1341:(with integer Hopf index). The definition can be extended to include dislocations of the helimagnetic order, such as edge dislocations and screw dislocations (that have an integer value of the Burgers vector)
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is deeply related to the stability of topological defects. In the case of line defect, if the closed path can be continuously deformed into one point, the defect is not stable, and otherwise, it is stable.
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Various types of defects in the medium can be characterized by elements of various homotopy groups of the order parameter space. For example, (in three dimensions), line defects correspond to elements of
525:, allowing them to interact with one-another and "connect up", and thus disconnect (fracture) the whole. The idea that critical densities of solitons can lead to phase transitions is a recurring theme.
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Unlike in cosmology and field theory, topological defects in condensed matter have been experimentally observed. Ferromagnetic materials have regions of magnetic alignment separated by domain walls.
695:; much like the Skyrmion, it owes its stability to belonging to a non-trivial homotopy class for maps of 3-spheres. For the monopole, the target is the magnetic field direction, instead of the
118:. As a result, the mapping from space(time) to the variables in the PDE is describable as a mapping from a sphere to a (different) sphere; the classes of such mappings are given by the
807:, but cannot decay or be undone or be de-tangled, precisely because there is no continuous transformation that will map them (homotopically) to a uniform or "trivial" solution.
423:
1804:
Dussaux, A.; Schoenherr, P.; Koumpouras, K.; Chico, J.; Chang, K.; Lorenzelli, L.; Kanazawa, N.; Tokura, Y.; Garst, M.; Bergman, A.; Degen, C. L.; Meier, D. (18 August 2016).
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543:; these include both vortex-like solutions to the Einstein field equations, and vortex-like solutions in more complex systems, coupling to matter and wave fields.
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106:: adding a point at infinity. This is reasonable, as one is generally interested in solutions that vanish at infinity, and so are single-valued at that point. The
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F. A. Bais, Topological excitations in gauge theories; An introduction from the physical point of view. Springer
Lecture Notes in Mathematics, vol. 926 (1982)
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cross each other without entanglement. It was a non-trivial application of topology that first led to the discovery of peculiar hydrodynamic behavior in the
1175:, cube-like defects that form when a spherical symmetry is broken, are predicted to have magnetic charge, either north or south (and so are commonly called "
759:. Although homotopic considerations prevent the classical field from being deformed into the ground state, it is possible for such a transition to occur via
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liquid crystals display a variety of defects including monopoles, strings, textures etc. In crystalline solids, the most common topological defects are
1680:
Cruz, M.; Turok, N.; Vielva, P.; Martínez-González, E.; Hobson, M. (2007). "A Cosmic
Microwave Background Feature Consistent with a Cosmic Texture".
1447:
1165:, two-dimensional membranes that form when a discrete symmetry is broken at a phase transition. These walls resemble the walls of a closed-cell
309:, its direction plus length, can be thought of as specifying a point on a 3-sphere. The orientation of the vector specifies a subgroup of the
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652:, which were previously the exclusive domain of mathematics. Further development identified the pervasiveness of the idea: for example, the
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form when larger, more complicated symmetry groups are completely broken. They are not as localized as the other defects, and are unstable.
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767:. If quantum tunneling erases the distinction between this and the ground state, then the next higher group of homotopies is given by the
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of the order parameter space of a medium to discuss the existence, stability and classifications of topological defects in that medium.
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a string around a stick: the string cannot be removed without cutting it. The most common extension of this winding analogy is to maps
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Poénaru and
Toulouse showed that crossing defects get entangled if and only if they are members of separate conjugacy classes of π
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was observed in the 19th century, as a solitary water wave in a barge canal. It was eventually explained by noting that the
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775:. These are theoretical hypotheses; demonstrating such concepts in actual lab experiments is a different matter entirely.
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As the universe expanded and cooled, symmetries in the laws of physics began breaking down in regions that spread at the
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distinct solutions. Typically, this occurs because the boundary on which the conditions are specified has a non-trivial
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973:) can be deformed continuously to each other, and hence, distinct defects correspond to distinct conjugacy classes.
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Schoenherr, P.; Müller, J.; Köhler, L.; Rosch, A.; Kanazawa, N.; Tokura, Y.; Garst, M.; Meier, D. (May 2018).
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Poénaru, V.; Toulouse, G. (1977). "The crossing of defects in ordered media and the topology of 3-manifolds".
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direction. Monopoles are usually called "solitons" rather than "defects". Solitions are associated with
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is a cyclone, for which soliton-type ideas have been offered up to explain its multi-century stability.
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1435:{\displaystyle {\mathcal {L}}=\partial _{\mu }\phi \partial ^{\mu }\phi -\left(\phi ^{2}-1\right)^{2}}
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provide examples of circle-map type topological solitons in fluids. More abstract examples include
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is perhaps the simplest way of understanding the general idea: it is a soliton that occurs in a
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for the medium. Then, the order parameter space can be written as the Lie group quotient
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Topological defects were studied as early as the 1940's. More abstract examples arose in
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much like what happens in condensed-matter systems such as superconductors. Certain
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A soliton and an antisoliton colliding with velocities ±sinh(0.05) and annihilating.
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take in a field that was otherwise dominated by perturbative calculations done with
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To restate more plainly: solitons are found when one solution of the PDE cannot be
1806:"Local dynamics of topological magnetic defects in the itinerant helimagnet FeGe"
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have been suggested as providing the initial 'seed'-gravity around which the
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Mermin, N. D. (1979). "The topological theory of defects in ordered media".
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which is explained by having the soliton belong to a different topological
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of matter has condensed. Textures are similarly benign. In late 2007, a
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http://demonstrations.wolfram.com/SeparationOfTopologicalSingularities/
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The existence of a topological defect can be demonstrated whenever the
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Azhar, Maria; Kravchuk, Volodymyr P.; Garst, Markus (12 April 2022).
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In magnetic systems, topological defects include 2D defects such as
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Other more complex hybrids of these defect types are also possible.
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than the base physical system. More simply: it is not possible to
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predict the formation of stable topological defects in the early
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Topologically stable solution of a partial differential equation.
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is a physical example of the abstract mathematical setting of a
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stands for compactified 3D space, while the second stands for a
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is highly constrained, requiring special circumstances (see
636:, rose to considerable popularity because these offered a
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Because of these observations, the formation of defects
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is defined as a region of space described by a function
803:. Topological defects are not only stable against small
771:. If the process repeats, this results in a walk up the
1337:(with integer skyrmion charge), or 3D defects such as
457:.) The mathematical stability comes from the non-zero
425:.) Such maps occur in PDE's describing vector fields.
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822:) that assigns to every point in the region an
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87:provided the needed topological understanding.
1900:Cosmic Strings & other Topological Defects
961:). However, defects which belong to the same
571:was proposed in the 1960's as a model of the
517:of the material. This can be thought of as a
8:
1169:, dividing the universe into discrete cells.
945:), point defects correspond to elements of π
1278:In condensed matter physics, the theory of
1219:is completed for the surrounding regions.
1152:. The well-known topological defects are:
832:. The homotopy theory of defects uses the
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1329:Topological defects in magnetic systems
1148:in the early universe according to the
953:), textures correspond to elements of π
341:; the length fixes a point. This has a
1857:"Screw Dislocations in Chiral Magnets"
1077:universality class systems including:
436:, typically studied in the context of
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1925:Large-scale structure of the cosmos
1245:large-scale structure of the cosmos
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444:. The prototypical example is the
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1442:in (1 + 1)-dimensional spacetime.
1028:or solitary wave which occurs in
418:{\displaystyle SU(2)\simeq S^{3}}
227:. Such maps can be thought of as
1510:Topological quantum field theory
1255:provided evidence of a possible
1132:during these phase transitions.
212:{\displaystyle U(1)\simeq S^{1}}
81:Korteweg-De Vries (KdV) equation
1659:10.1051/jphys:01977003808088700
1088:Magnetic flux "tubes" known as
732:. In abstract settings such as
497:{\displaystyle S^{1}\to S^{1};}
1883:10.1103/PhysRevLett.128.157204
1608:Geometry, Topology and Physics
1495:Topological entropy in physics
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894:then, it can be shown that π
859:be the symmetry subgroup of
755:, and thus will be called a
223:; the mappings arise in the
170:{\displaystyle U(1)\to U(1)}
1253:cosmic microwave background
1140:Depending on the nature of
1081:Screw/edge-dislocations in
674:Belinski–Zakharov transform
535:and pinned vortex tubes in
1946:
1515:Topological quantum number
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996:and are believed to drive
120:homotopy groups of spheres
104:one-point compactification
1783:10.1038/s41567-018-0056-5
1590:10.1103/RevModPhys.51.591
1570:Reviews of Modern Physics
1520:Topological string theory
1039:in crystalline materials,
634:Wess–Zumino–Witten models
116:compact topological space
1606:Nakahara, Mikio (2003).
1056:Wess–Zumino–Witten model
1045:in quantum field theory,
787:entail the existence of
713:is used in the sense of
662:Einstein field equations
1861:Physical Review Letters
1712:10.1126/science.1148694
1073:Topological defects in
1030:exactly solvable models
537:type-II superconductors
1744:. Cambridge cosmology.
1637:Le Journal de Physique
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1239:). On the other hand,
1150:Kibble-Zurek mechanism
1126:grand unified theories
851:of transformations on
797:differential equations
795:which is preserved in
757:topological excitation
701:topological invariants
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1930:Inflation (cosmology)
1810:Nature Communications
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1353:A static solution to
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1237:Inflation (cosmology)
1024:Examples include the
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679:The terminology of a
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685:topological soliton
438:solid state physics
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430:topological defect
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1621:978-0-7503-0606-5
1505:Topological order
1480:Quantum mechanics
1142:symmetry breaking
1136:Symmetry breaking
1122:phase transitions
1075:lambda transition
1069:Lambda transition
1049:Magnetic skyrmion
998:phase transitions
834:fundamental group
761:quantum tunneling
715:charge in physics
689:magnetic monopole
642:Feynmann diagrams
511:stiff and brittle
446:screw dislocation
442:materials science
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1761:Nature Physics
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605:
602:
597:
593:
554:Great Red Spot
541:cosmic strings
493:
488:
484:
480:
475:
471:
459:winding number
412:
408:
404:
401:
398:
395:
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389:
369:
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288:
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102:, obtained by
72:
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53:homotopy class
26:
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14:
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1234:
1229:
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1213:ordered phase
1210:
1205:
1199:
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1187:
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805:perturbations
802:
798:
794:
790:
789:homotopically
786:
778:
776:
774:
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766:
762:
758:
754:
750:
745:
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739:
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734:string theory
731:
727:
723:
722:fiber bundles
718:
716:
712:
708:
707:
702:
698:
697:isotopic spin
694:
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658:Kerr solution
655:
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582:
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347:unitary group
344:
325:
319:
312:
308:
304:
286:
282:
274:
256:
252:
243:
239:
230:
226:
225:circle bundle
222:
204:
200:
196:
190:
184:
161:
155:
146:
140:
131:
128:
123:
121:
117:
113:
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105:
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93:
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86:
82:
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75:The original
70:
68:
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49:
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1332:
1319:dislocations
1308:
1301:
1284:
1277:
1232:
1230:
1226:
1206:
1203:
1163:Domain walls
1139:
1115:
1111:
1099:Vortices in
1072:
1023:
1010:false vacuum
1006:
991:
981:
975:
970:
958:
950:
942:
935:
926:
925:denotes the
921:
916:
910:
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896:
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887:
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877:
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860:
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852:
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840:
838:
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811:
809:
782:
769:string group
756:
753:vacuum state
749:ground state
746:
719:
710:
704:
684:
680:
678:
562:
548:
527:
429:
427:
307:three-vector
303:vector field
273:three-sphere
127:continuously
124:
89:
74:
61:continuously
47:
43:
29:
1475:Dislocation
1223:Observation
1196:and higher
1101:superfluids
1014:true vacuum
742:knot theory
709:. The word
666:black holes
533:superfluids
450:dislocation
130:transformed
32:mathematics
1914:Categories
1874:2109.04338
1774:1704.06288
1537:References
1323:plasticity
1289:superfluid
1287:-phase of
1198:dimensions
1032:, such as
919:), where π
765:spin group
650:cohomology
448:; it is a
1832:2041-1723
1791:1745-2481
1695:0710.5737
1645:CiteSeerX
1416:−
1407:ϕ
1397:−
1394:ϕ
1389:μ
1385:∂
1381:ϕ
1376:μ
1372:∂
1335:skyrmions
1249:cold spot
1189:Skyrmions
1173:Monopoles
1004:physics.
849:Lie group
730:monodromy
630:instanton
601:→
529:Vorticies
506:stiffness
479:→
403:≃
249:→
197:≃
153:→
85:Lax pairs
64:transform
1920:Solitons
1728:12735226
1720:17962521
1667:93172461
1525:Topology
1458:See also
1339:Hopfions
1183:Textures
1130:universe
1043:Skyrmion
988:Examples
839:Suppose
740:, as in
728:and the
726:holonomy
693:monopole
646:homotopy
569:Skyrmion
545:Tornados
177:, where
112:codomain
96:topology
71:Overview
40:solitons
1841:4992142
1700:Bibcode
1682:Science
1578:Bibcode
1311:Nematic
1257:texture
1251:in the
1116:In the
1090:fluxons
1026:soliton
660:to the
577:neutron
573:nucleon
558:Jupiter
549:are not
345:by the
229:winding
219:is the
77:soliton
36:physics
1838:
1830:
1789:
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1345:Images
1292:helium
1012:and a
855:. Let
711:charge
683:vs. a
581:proton
567:. The
380:, and
221:circle
100:sphere
1869:arXiv
1769:arXiv
1724:S2CID
1690:arXiv
1663:S2CID
1096:, and
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882:is a
847:be a
738:knots
305:. (A
108:range
98:of a
1828:ISSN
1787:ISSN
1716:PMID
1616:ISBN
1313:and
1294:-3.
1167:foam
965:of π
929:-th
886:for
656:and
648:and
440:and
46:and
34:and
1879:doi
1865:128
1836:PMC
1818:doi
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1092:in
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556:of
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