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Two-dimensional topological solitons, commonly called Skyrmions, are extensively studied in solid-state magnetic nanostructures and promise many spintronics applications. However, three-dimensional topological solitons dubbed hopfions have not been demonstrated as stable spatially localized structures in solid-state magnetic materials. Here we model the existence of such static solitons with different Hopf index values in noncentrosymmetric solid magnetic nanostructures with a perpendicular interfacial magnetic anisotropy. We show how this surface anisotropy, along with the Dzyaloshinskii-Moriya interactions and the geometry of nanostructures, stabilize hopfions. We demonstrate knots in emergent field lines and computer simulate Lorentz transmission electron microscopy images of such solitonic configurations to guide their experimental discovery in magnetic solids.Topological solitons exist in the effective field theories of many physical systems [1][2][3][4][5][6][7][8][9]. For example, two-dimensional (2D) particlelike Skyrmions are solitonic field configurations classified by elements of the second homotopy group ! " ($ " ) = ℤ and indexed by a topological charge, the Skyrmions number. Such Skyrmions are widely studied in solid magnets due to the wealth of new

Starting from Gauss and Kelvin, knots in fields were postulated behaving like particles, but experimentally they were found only as transient features or required complex boundary conditions to exist and couldn't self-assemble into three-dimensional crystals. We introduce energetically stable micrometer-sized knots in helical fields of chiral liquid crystals. While spatially localized and freely diffusing in all directions, they resemble colloidal particles and atoms, self-assembling into crystalline lattices with open and closed structures. These knots are robust and topologically distinct from the host medium, though they can be morphed and reconfigured by weak stimuli under conditions like in displays. A combination of energyminimizing numerical modeling and optical imaging uncovers the internal structure and topology of individual helical field knots and various hierarchical crystalline organizations they form.One Sentence Summary: Stable solitonic and vortex knots in molecular alignment fields behave like particles and form triclinic crystals. Main Text:Topological order and phases represent an exciting frontier of modern research (1), but topologyrelated ideas have a long history in physics (2). Gauss postulated that knots in fields could behave like particles whereas Kelvin, Tait and Maxwell believed that the matter, including crystals, could be made of real-space free-standing knots of vortices (2-4). These early physics models, introduced long before even the very existence of atoms was widely accepted, gave origins to modern mathematical knot theory (2-4). Expanding this topological paradigm, Skyrme and others modeled subatomic particles with different baryon numbers as nonsingular topological solitons and their clusters (3-5). Knotted fields emerged in classical and quantum field theories (3-7) and in scientific branches ranging from fluid mechanics to particle physics and cosmology (2-11). In condensed matter, arrays of singular vortex lines and low-dimensional analogs of Skyrme solitons were found as topologically nontrivial building blocks of exotic thermodynamic phases in superconductors, magnets and liquid crystals (LCs) (12-14). Could they be knotted, and could these knots self-organize into three-dimensional (3D) crystals? Knotted fields in condensed matter found many experimental and theoretical embodiments, including both nonsingular solitons and knotted vortices (7)(8)(9)(15)(16)(17)(18)(19)(20)(21)(22)(23). However, they were metastable and decayed with time (7-9,15-17) or could not be stabilized without colloidal Shnir, H. Sohn, R. Voinescu and Y. Yuan for discussions.

Liquid crystals are widely known for their facile responses to external fields, which forms a basis of the modern information display technology. However, switching of molecular alignment field configurations typically involves topologically trivial structures, although singular line and point defects often appear as short-lived transient states. Here, we demonstrate electric and magnetic switching of nonsingular solitonic structures in chiral nematic and ferromagnetic liquid crystals. These topological soliton structures are characterized by Hopf indices, integers corresponding to the numbers of times that closed-loop-like spatial regions (dubbed "preimages") of two different single orientations of rod-like molecules or magnetization are linked with each other. We show that both dielectric and ferromagnetic response of the studied material systems allow for stabilizing a host of topological solitons with different Hopf indices. The field transformations during such switching are continuous when Hopf indices remain unchanged, even when involving transformations of preimages, but discontinuous otherwise.

Chiral condensed matter systems, such as liquid crystals and magnets, exhibit a host of spatially localized topological structures that emerge from the medium's tendency to twist and its competition with confinement and field coupling effects. We show that the strength of perpendicular surface boundary conditions can be used to control the structure and topology of solitonic and other localized field configurations. By combining numerical modeling and threedimensional imaging of the director field, we reveal structural stability diagrams and intertransformation of twisted walls and fingers, torons and skyrmions and their crystalline organizations upon changing boundary conditions. Our findings provide a recipe for controllably realizing skyrmions, torons and hybrid solitonic structures possessing features of both of them, which will aid in fundamental explorations and technological uses of such topological solitons.Moreover, with limited examples, we discuss how similar principles can be systematically used to tune stability of twisted walls versus cholesteric fingers and hopfions versus skyrmions, torons and twistions.