Bode B, Dennis M R, Foster D, King R P
H H Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK.
Proc Math Phys Eng Sci. 2017 Jun;473(2202):20160829. doi: 10.1098/rspa.2016.0829. Epub 2017 Jun 7.
We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ℓ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme-Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.
我们给出了复映射的显式构造，其节点线具有双纽线纽结的形式。我们回顾了双纽线纽结的性质，双纽线纽结被定义为辫子的闭包，其中所有股线都遵循相同的横向(1, ℓ)李萨如图形，因此是推广环面纽结的螺旋纽结的一个子族。然后我们证明了这样的映射存在，并且实际上在适当选择参数的情况下是纤维化。我们描述了这在物理学中如何有助于创建纽结场，在量子力学、光学中，以及推广到有理映射并应用于斯格明子 - 法捷耶夫模型。我们还证明了这种构造如何扩展到具有弱孤立奇点的映射。
