In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in \({\mathbb {R}}^3\), there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue \(+1\) of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

在本文中，我们证明了实解析Beltrami场的零集的分类定理。也就是说，我们表明，实解析交3流形上无边界的实解析Beltrami场的零集在除去其孤立点后为空，或者可以写为可微分嵌入的，连接1的可数局部有限并集具有（可能是空的）边界和驯服结的三维子流形。此外，我们考虑这些温驯结可能有多复杂的问题。为此，我们证明在\（{\ mathbb {R}} ^ 3 \）中的标准（开放）实心环形环上，存在任何对（p，  q）可以无限地计数许多不同的实际分析指标的共质数整数，因此对于每个此类指标，都有一个真实的分析Beltrami字段，该字段对应于curl运算符的特征值\（+ 1 \），其零集精确地由一个标准的（p，  q）-torus结。度量标准和相应的Beltrami字段是显式构造的，并且可以仅通过基本函数将其记录在笛卡尔坐标中。