期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2015
卷号：112
期号：6
页码：1675-1680
DOI：10.1073/pnas.1417178112
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：SignificanceKnot theory is a branch of topology that deals with study and classification of closed loops in 3D Euclidean space. Creation and control of knots in physical systems is the pinnacle of technical expertise, pushing forward state-of-the-art experimental approaches as well as theoretical understanding of topology in selected medium. We show how several abstract concepts manifest elegantly as observable and measurable features in nematic colloids with knotted disclination lines. Construction of medial graphs, surfaces, and Jones polynomials is showcased directly on experimental images, and adapted for the specific system of colloidal crystals in a twisted nematic cell. Discussing the correspondence between topological concepts and experimental observation is essential for building the bridge between mathematical and physical communities. Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.
