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Creators/Authors contains: "Lipton, Max"

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  1. (, Partial Differential Equations and Applications)
    Abstract Consider a knotKin$$S^3$$ 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 get a lower bound on the size of the critical set based on the projection’s crossings, improving a 2021 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. 
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  2. ; ; (, Physical Review Research)
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  3. ; ; (, Chaos: An Interdisciplinary Journal of Nonlinear Science)
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  4. (, Journal of Knot Theory and Its Ramifications)
    null (Ed.)
    Consider a knot [Formula: see text] in [Formula: see text] with charge uniformly distributed on it. 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. We show the number of critical points of the potential is at least [Formula: see text], where [Formula: see text] is the tunnel number, defined as the smallest number of arcs one must add to [Formula: see text] such that its complement is a handlebody. The result is proven using Morse theory and stable manifold theory. 
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