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arXiv 2410.04670cond-mat.softmath-phmath.ATmath.GTmath.MP
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位错与纤维化：扭结近晶缺陷的拓扑结构

Dislocations and Fibrations: The Topological Structure of Knotted Smectic Defects

Paul G. Severino, Randall D. Kamien, Benjamin Bode

发表信息：
New J. Phys. 27 (2025) 064402
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AI总结：

本研究探究近晶液晶中扭结缺陷的拓扑性质，分析螺位错、刃型位错等缺陷与扭结纤维化的关联，给出点缺陷数量下界，深化了与焦锥畴的类比，为相关拓扑理论提供研究平台。

摘要：

在本研究中，我们探究了近晶液晶中扭结缺陷的拓扑性质。我们的研究从螺位错展开，对于纤维化扭结，可借助对应的扭结纤维化将其径向表面结构平滑地适配于$S^3$上。为理解近晶织构如何能以无纤维化的扭结形状呈现螺位错，我们首先研究了扭结刃型缺陷。与螺位错不同，对于任何非平凡扭结，扭结刃型位错都会在系统中引发奇点。我们给出了给定刃型位错扭结所需的这类点缺陷数量的下界，并将扭结刃型位错的点缺陷结构与焦锥畴的点缺陷结构进行了类比。通过证明刃型位错同样对扭结的纤维化性质敏感，我们将非纤维化螺位错扭结所需的所谓Morse-Novikov点重新解读为类似的近晶缺陷。随后我们将这些方法应用于近晶相中的$+1/2$和负电荷向错，进一步深化了扭结近晶缺陷与焦锥畴之间的类比，同时揭示了近晶液晶中点缺陷与线缺陷之间的复杂关联。近晶缺陷与扭结理论之间的联系不仅阐明了近晶缺陷独特的拓扑扭结特性，还为扭结理论与Morse-Novikov理论中的现代问题提供了数学与实验研究平台。

英文摘要：

In this work, we investigate the topological properties of knotted defects in smectic liquid crystals. Our story begins with screw dislocations, whose radial surface structure can be smoothly accommodated on $S^3$ for fibred knots by using the corresponding knot fibration. To understand how a smectic texture may take on a screw dislocation in the shape of a knot without a fibration, we study first knotted edge defects. Unlike screw defects, knotted edge dislocations force singular points in the system for any non-trivial knot. We provide a lower bound on the number of such point defects required for a given edge dislocation knot and draw an analogy between the point defect structure of knotted edge dislocations and that of focal conic domains. By showing that edge dislocations, too, are sensitive to knot fibredness, we reinterpret the so-called Morse-Novikov points required for non-fibred screw dislocation knots as analogous smectic defects. Our methods are then applied to $+1/2$ and negative-charge disclinations in the smectic phase, furthering the analogy between knotted smectic defects and focal conic domains and uncovering an intricate relationship between point and line defects in smectic liquid crystals. The connection between smectic defects and knot theory not only unravels the uniquely topological knotting of smectic defects but also provides a mathematical and experimental playground for modern questions in knot and Morse-Novikov theory.

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