Totally tangential $\mathbb{C}$-links and electromagnetic knots

Abstract: The set of real-analytic Legendrian links with respect to the standard contact structure on the 3-sphere $S^3$ corresponds both to the set of totally tangential $\mathbb{C}$-links as defined by Rudolph and to the set of stable knotted field lines in Bateman electromagnetic fields of Hopf type. It is known that every isotopy class has a real-analytic Legendrian representative, so that every link type $L$ admits a holomorphic function $G:\mathbb{C}^2\to\mathbb{C}$ whose zeros intersect $S^3$ tangentially in $L$ and there is a Bateman electromagnetic field $\mathbf{F}$ with closed field lines in the shape of $L$. However, so far the family of torus links are the only examples where explicit expressions of $G$ and $\mathbf{F}$ have been found. In this paper, we present an algorithm that finds for every given link type $L$ a real-analytic Legendrian representative, parametrised in terms of trigonometric polynomials. We then prove that (good candidates for) examples of $G$ and $\mathbf{F}$ can be obtained by solving a system of linear equations, which is homogeneous in the case of $G$ and inhomogeneous in the case of $\mathbf{F}$. We also use the real-analytic Legendrian parametrisations to study the dynamics of knots in Bateman electromagnetic fields of Hopf type. In particular, we show that no compact subset of $\mathbb{R}^3$ can contain an electromagnetic knot indefinitely.