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Theoretical analysis reveals links with locally minimal bridge positions that resist monotonic simplification, demonstrating that bridge spheres must undergo perturbation to reach global minimality.
For any pair of integers m and n such that $3 < m < n$, we provide an infinite family of links where each link in the family has a locally minimal n-bridge position and a globally minimal m-bridge position. We accomplish this by applying the criterion of Takao et al. The n-bridge position is interesting because the corresponding bridge sphere is unperturbed, so it must be perturbed at least once before it can be de-perturbed to attain a globally minimal m-bridge sphere.
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Pongtanapaisan et al. (2026) studied this question.
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