Source-free so-called ModMax theories of nonLinear electrodynamics (NLE) in the four dimensional Minkowski spacetime vacuum are the only possible continuous deformations—and as a function of a single real and positive parameter—of source-free Maxwell linear electrodynamics (MLE) in the same vacuum, which preserve all the same Poincar and conformal spacetime symmetries as well as the continuous duality invariance of Maxwell’s theory. Null field configurations of the latter however, including null electromagnetic knots, are singular for the Lagrangian formulation of any spacetime Poincar and conformal invariant theory of NLE. In particular null hopfion-Raada (HR) knots are a distinguished and fascinating class on their own of topologically nontrivial solutions to Maxwell’s equations. This work addresses the fate of these configurations within ModMax theories. A doubled class of ModMax deformed HR knots is thereby identified, each of which coalescing back in a continuous fashion to the original HR knot when the nonlinear deformation parameter is turned off.

四维 Minkowski 时空真空中无源的所谓非线性电动力学 (NLE) 的 ModMax 理论是无源麦克斯韦线性电动力学 (MLE) 唯一可能的连续变形 - 并且作为单个实参数和正参数的函数在同一个真空中，保留了所有相同的庞加莱和共形时空对称性以及麦克斯韦理论的连续对偶不变性。然而，后者的零场配置，包括零电磁结，对于任何时空庞加莱的拉格朗日公式和 NLE 的共形不变理论来说都是奇异的。尤其是 null hopfion-Raada (HR) 结是麦克斯韦方程的拓扑非平凡解的杰出而迷人的一类。这项工作解决了 ModMax 理论中这些配置的命运。从而识别出双重类别的 ModMax 变形 HR 结，当关闭非线性变形参数时，每个结都以连续方式合并回原始 HR 结。