论文标题
打结电荷分布的拓扑方法
Topological approaches to knotted electric charge distributions
论文作者
论文摘要
考虑一个$ s^3 $的结$ k $，并带有均匀分布的电荷。尽管很容易就Dirichlet积分来提供拉普拉斯方程的解决方案，但了解潜力的定性行为，尤其是在关键点和等电位表面方面，仍然具有理论和物理的利益。在本文中，我们演示了几何拓扑的技术如何从静电的角度产生新的见解。具体来说，我们表明，当结足够接近平面投影时，我们证明了基于投影的交叉点对关键集的大小的下限，从而改善了作者的2019年结果。然后，我们通过跟踪结的拓扑结构如何限制与电势临界点相关的摩尔斯手术，通过跟踪带电的结分布的等电位表面进行分类。关键词：物理结理论，静电学，摩尔斯理论，动力学系统，几何拓扑，CERF理论
Consider a knot $K$ in $S^3$ with uniformly distributed electric charge. Whilst solutions to the Laplace equation in terms of Dirichlet integrals are readily available, it is still of theoretical and physical interest to understand the qualitative behavior of the potential, particularly with respect to critical points and equipotential surfaces. In this paper, we demonstrate how techniques from geometric topology can yield novel insights from the perspective of electrostatics. Specifically, we show that when the knot is sufficiently close to a planar projection, we prove a lower bound on the size of the critical set based on the projection's crossings, improving a 2019 result of the author. We then classify the equipotential surfaces of a charged knot distribution by tracking how the topology of the knot complement restricts the Morse surgeries associated to the critical points of the potential. keywords: Physical knot theory, electrostatics, Morse theory, dynamical systems, geometric topology, Cerf theory
