For 1-manifolds in R^2, vineyard monodromy on small loops arises precisely when the loop intersects a singularity of the distance function on the symmetry set.
Persistent homology and euler integral transforms.Journal741 of Applied and Computational Topology, 2(1–2):55–60, August 2018
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The Singular Source of Vineyard Monodromy
For 1-manifolds in R^2, vineyard monodromy on small loops arises precisely when the loop intersects a singularity of the distance function on the symmetry set.