Consider a knot $K$ in $S^3$ with uniformly distributed electric charge. From the standpoint of both physics and knot theory, it is natural to try to understand the critical points of the potential and their behavior. How many are there, what are their topological indices, and how does this relate to other properties of $K$?

In this talk, I will discuss my results from 2019 and 2020 which relate the critical set to knot invariants, and I will showcase the Python code I have written to help visualize and understand this problem.