编织螺旋度可调的纽结向量场
Weaving knotted vector fields with tunable helicity
摘要：
本文从复标量场出发，给出了无发散纽结向量场的一般构造，其闭场线编码了多种纽结和链环，包括环面纽结、环索、8字形纽结及其推广.作为有限能量的物理场，它们代表了诸如等离子体中的磁场或流体中的涡量场之类的场的初始状态。我们给出了一个系统的程序计算矢量势，从复杂的标量函数与打结零丝，从而使这些打结字段的螺旋度的明确计算。该构造可用于生成孤立的打结通量管，由编码在向量场的线中的结填充。最后，我们给出了螺旋度为零的明显纽结向量场的例子。我们的研究结果为分析模型和模拟提供了基础。
英文摘要：
We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot and its generalizations. As finite-energy physical fields they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike.
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