  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
