Morse theory without (PS) condition at isolated values and strong resonance problems
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We introduce an algorithm for construction of the Morse hierarchy, i.e., a hierarchy of Morse decompositions of a piecewise constant vector field on a surface driven by stability of the Morse sets with respect to perturbation of the vector field. Our approach builds upon earlier work on stable Morse decompositions, which can be used to obtain Morse sets of user-prescribed stability. More stable Morse decompositions are coarser, i.e., they consist of larger Morse sets. In this work, we develop an algorithm for tracking the growth of Morse sets and topological events (mergers) that they undergo as their stability is gradually increased. The resulting Morse hierarchy can be explored interactively. We provide examples demonstrating that it can provide a useful coarse overview of the vector field topology.
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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.
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The questions when two Morse function on closed manifolds are conjugated is investigated. Using the handle decompositions of manifolds the condition of conjugation is formulated. For each Morse function on 3-manifold the ordered generalized Heegaard diagram is built. The criteria of Morse function conjugation are given in the terms of equivalence of such diagrams.
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