Humankind has been obsessed with knots in religion, culture and daily life for millennia while physicists like Gauss, Kelvin and Maxwell involved them in models already centuries ago. Nowadays, colloidal particle can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be inter-linked (or not) with colloidal particle knots, revealing diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability, and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights on stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory.

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评论：液晶和胶体中的结和其他新拓扑效应

几千年来，人类一直痴迷于宗教、文化和日常生活中的结，而高斯、开尔文和麦克斯韦等物理学家早在几个世纪前就将它们纳入模型。如今，胶体颗粒可以被制造成具有任意复杂度的结和链接形状。在液晶中，奇异涡旋线的闭环可以通过使用胶体粒子和激光镊子以及通过将向列流体限制在具有复杂拓扑结构的微米级液滴中来打结。打结和连接的胶体颗粒会引起奇异缺陷的结和链接，它们可以与胶体颗粒结相互链接（或不链接），揭示了打结场的拓扑与胶体物体的拓扑非平凡表面之间相互作用的多样性。在液晶和胶体铁磁体的非奇异分子排列和磁化场中出现了更多不同的打结结构。拓扑孤子包括hopfions、skyrmions、heliknotons、torons等空间局域连续结构，基于同伦理论分类，以整数值拓扑不变量为特征，通常包含打结或链接的原像，对应于单点的非奇异空间区域顺序参数空间。液晶、胶体和铁磁体中的拓扑孤子动物园预示着新品种的信息显示和大量的数据存储、电光和光子应用。它们类似粒子的集体动力学与活性物质中的连贯运动相呼应，从人群到鱼群。这篇评论讨论了该领域的最新进展，并强调了打结软物质物理学的最新发展和悬而未决的问题。我们系统地概述了打结场配置、它们之间允许的转换、它们的物理稳定性，以及如何使用一种形式的打结场来建模、创建和印记其他形式。液晶和胶体可获得的大量对称性提供了对具有基本和应用重要性的完全非奇异和奇异结域的稳定性、转变和涌现动力学的见解。这篇评论的共同点是能够在真实空间中通过实验将这些结可视化。