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arxiv: 2607.01046 · v1 · pith:J23TEXJPnew · submitted 2026-07-01 · 💻 cs.CG · math.DG· math.GT

The Singular Source of Vineyard Monodromy

Erin W. Chambers , Christopher Fillmore , Shankha Shubhra Mukherjee , Rohit Roy , Elizabeth Stephenson , Mathijs Wintraecken This is my paper

Pith reviewed 2026-07-02 01:31 UTC · model grok-4.3

classification 💻 cs.CG math.DGmath.GT
keywords vineyardmonodromysymmetry setdistance functionpersistence diagrams1-manifoldstopological data analysis
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The pith

Vineyard monodromy for small loops on 1-manifolds arises only from specific distance singularities on the symmetry set

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for a 1-manifold in the plane, the vineyard associated to a sufficiently small closed loop cannot exhibit monodromy unless the loop contains a singularity of the distance function. These singularities are classified by the symmetry set, the locus of points that are centers of spheres tangent to the manifold at more than one location. This result classifies the geometric causes of diagram point permutation in time-varying persistence, which can obstruct feature tracking in data analysis. A reader would care because it reduces an open problem about when monodromy occurs to a checkable condition on the symmetry set.

Core claim

We prove that the vineyard of a sufficiently small loop γ cannot exhibit monodromy unless it contains a specific singularity of the distance function. The central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely when monodromy occurs.

What carries the argument

The symmetry set, the locus of centers of spheres tangent in more than one point to the manifold, which classifies singularities of the distance function and dictates when monodromy occurs in the vineyard.

Load-bearing premise

The symmetry set is the complete classifier of all distance-function singularities that can produce monodromy for small loops.

What would settle it

A counterexample would be a sufficiently small loop around a 1-manifold in R^2 that avoids all points of the symmetry set yet produces a vineyard with monodromy.

Figures

Figures reproduced from arXiv: 2607.01046 by Christopher Fillmore, Elizabeth Stephenson, Erin W. Chambers, Mathijs Wintraecken, Rohit Roy, Shankha Shubhra Mukherjee.

Figure 1.1
Figure 1.1. Figure 1.1: The singularities of the symmetry and focal set in the plane. We adopt Arnold’s notation for these [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Visualizing the focal set (evolute) and symmetry set of a spline-defined curve [PITH_FULL_IMAGE:figures/full_fig_p005_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Radial persistence of an ellipse (shown in black) with respect to a base point (shown in yellow). Note [PITH_FULL_IMAGE:figures/full_fig_p006_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: A canonical vineyard with trivial topology in the sense of Definition 2.6. [PITH_FULL_IMAGE:figures/full_fig_p007_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: The singularities of the symmetry and focal set in the plane. We adopt Arnold’s notation for these [PITH_FULL_IMAGE:figures/full_fig_p009_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: The singularities of the symmetry and focal set in the plane of a curve. Adjusted from Figure 3 of [PITH_FULL_IMAGE:figures/full_fig_p010_2_5.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A loop γ enclosing the symmetry set or an A2 1 singularity (left) and the associated vineyard (right). 3.2 Type A2 1/A2 387 1 As mentioned in Section 2.4, this is a transversal intersection point p of two manifold 388 pieces of the symmetry set. We will refer to the 4 manifold pieces of the symmetry set that arise by removing the intersection point p as the 4 segments. As these 4 pieces are each of type … view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Some necessary conditions for the configuration of [PITH_FULL_IMAGE:figures/full_fig_p014_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: The part of the symmetry set depicted in the figure is not to scale. [PITH_FULL_IMAGE:figures/full_fig_p015_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Left: The complete symmetry (blue) and focal (purple) sets for the example where a single singularity [PITH_FULL_IMAGE:figures/full_fig_p016_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Left: An example of a deformed (ellipsoidal) spiral inspired from the example by [9]. The dark [PITH_FULL_IMAGE:figures/full_fig_p018_3_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: ). As [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Each point in the persistence diagram has a zone around it (either in the birth direction or the death [PITH_FULL_IMAGE:figures/full_fig_p019_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: A loop γ enclosing an A3 1 singularity (left) and the associated vineyard (right). 534 these crossings do not force reordering or repairing among distinct, pre–existing off–diagonal points. Because the 535 newly created pair interacts only with itself, the elder rule does not come into effect for any pre–existing classes 536 at these events [PITH_FULL_IMAGE:figures/full_fig_p019_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: A loop enclosing the focal set or a singularity of type [PITH_FULL_IMAGE:figures/full_fig_p020_3_8.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: A loop enclosing the focal set or a singularity of type [PITH_FULL_IMAGE:figures/full_fig_p021_3_9.png] view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: A loop enclosing the focal set or a singularity of type [PITH_FULL_IMAGE:figures/full_fig_p021_3_10.png] view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: A loop enclosing the focal set or a singularity of type [PITH_FULL_IMAGE:figures/full_fig_p022_3_11.png] view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Schematic of an A2 1/A2 singularity 599 3.9 Main result From the analysis above for all the singularities we can immediately conclude our main 600 result: Theorem 1.1 (A2 1/A2 1 is the unique local planar monodromy generator). Let M ⊂ R 2 601 be a generic smooth closed curve and let γ : S 1 → R 2 602 be a generic sufficiently small loop. If the interior of γ contains no singularity of type A2 1/A2 603 1… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: When γ is a big enough loop enclosing the curve which acted as an example for a singularity of type A2 1/A2 1 which contains monodromy, then the vineyard of the larger loop no longer contains monodromy, even though it contains the A2 1/A2 1 singularity which locally generates monodromy. 619 In this paper we have restricted ourselves to two dimensions, a setting which as one has seen already exhibits 620 … view at source ↗
read the original abstract

Vineyards, or time-varying families of persistence diagrams, are widely used in topological data analysis (TDA) pipelines to track how topological features change and evolve as a parameter varies. When the parameter traces a closed loop, a vineyard can exhibit monodromy: diagram points permute over the course of a full traversal, which obstructs feature tracking and can complicate downstream analysis of such data. Chambers et al. considered the periodic vineyards that arise from the radial persistence transform, which maps the manifold to a family of persistence diagrams, where each diagram fixes a base point and considers the filtration that is based on Euclidean distance to that point, and showed that monodromy and knotting can occur. Other recent work by Arya et al. considers geometric conditions that exclude monodromy in two dimensions, in an effort to better understand when this effect happens. That said, understanding when and why monodromy occurs is a fundamental open problem with direct practical consequences for many data analysis pipelines. In this work, we study this question for 1-manifolds in $\mathbb{R}^2$, using a surprising connection with tools from singularity theory, and provide a classification for the causes of monodromy in vineyards. More precisely, we prove that the vineyard of a sufficiently small loop $\gamma$ cannot exhibit monodromy unless it contains a specific singularity of the distance function. The central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely when monodromy occurs. This characterization opens the door to the development of algorithmic criteria for detecting and utilizing (or avoiding) monodromy in TDA pipelines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims to classify the causes of monodromy in vineyards arising from the radial persistence transform on 1-manifolds in R^2. It proves that for a sufficiently small loop γ, the vineyard exhibits monodromy if and only if the loop encounters a specific singularity of the distance function; the symmetry set (locus of centers of spheres tangent to the manifold at more than one point) is the central object that classifies all such distance-function singularities capable of producing monodromy.

Significance. If the classification holds, the result supplies a geometric criterion linking monodromy directly to singularities of the distance function via established tools from singularity theory. This could support algorithmic detection or avoidance of monodromy in TDA pipelines and clarifies when feature tracking is obstructed for closed parameter loops.

major comments (1)
  1. [Abstract] Abstract (central claim paragraph): the manuscript asserts a complete classification proof but supplies no derivation steps, lemmas, or verification that the symmetry set is exhaustive for all distance-function singularities producing monodromy; without these the load-bearing claim that monodromy occurs precisely from symmetry-set singularities cannot be evaluated.
minor comments (1)
  1. [Abstract] The abstract references prior work by Chambers et al. and Arya et al. but does not indicate how the new classification improves upon or differs from the geometric conditions already studied in two dimensions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments. We address the major comment on the abstract below, noting that the full proof appears in the body of the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (central claim paragraph): the manuscript asserts a complete classification proof but supplies no derivation steps, lemmas, or verification that the symmetry set is exhaustive for all distance-function singularities producing monodromy; without these the load-bearing claim that monodromy occurs precisely from symmetry-set singularities cannot be evaluated.

    Authors: The abstract is a concise summary of the main result: for 1-manifolds in R^2, a sufficiently small loop γ produces vineyard monodromy if and only if it intersects a singularity of the distance function, with all such singularities classified by the symmetry set. The derivation steps, lemmas establishing exhaustiveness via singularity theory, and the if-and-only-if argument are contained in the body of the paper (using the geometry of the symmetry set as the locus of centers of spheres tangent at multiple points). We are happy to revise the abstract to include a brief pointer to the relevant sections or a one-sentence outline of the key steps if that would facilitate evaluation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; classification rests on external singularity theory

full rationale

The paper's central claim is a mathematical proof that monodromy in vineyards of small loops on 1-manifolds in R^2 occurs precisely when the symmetry set of the distance function exhibits specific singularities. This classification invokes established results from singularity theory to position the symmetry set as the complete classifier of relevant distance-function singularities, without reducing any derived statement to a fitted parameter, self-defined quantity, or load-bearing self-citation. Prior references to Chambers et al. establish the existence of monodromy phenomena but do not substitute for the new geometric argument. The derivation chain is therefore self-contained against external mathematical benchmarks rather than internally circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard background from differential geometry and singularity theory plus prior TDA definitions; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (2)
  • standard math The symmetry set classifies singularities of the distance function to a 1-manifold
    Invoked as the central geometric object that dictates when monodromy occurs.
  • domain assumption Radial persistence transform produces vineyards whose monodromy is governed by distance-function singularities
    Builds directly on the definition used in Chambers et al.

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Works this paper leans on

68 extracted references · 46 canonical work pages · 2 internal anchors

  1. [1]

    doubling

    If there are contacts between circles with223 different radii, but sharing the same centre, and the manifold (curve in this case) then we indicate this by a slash,224 for exampleA 2 1/A2 1, indicates that there are two concentric circles, each of which with an ordinary contact with225 the curveM.226 Having now described the notation, we can go over the ty...

  2. [2]

    Agarwal, Herbert Edelsbrunner, John Harer, and Yusu Wang

    Pankaj K. Agarwal, Herbert Edelsbrunner, John Harer, and Yusu Wang. Extreme elevation on a 2-638 manifold.Discrete & Computational Geometry, 36(4):553–572, Sept 2006. URL: http://dx.doi.org/10.1007/639 s00454-006-1265-8,doi:10.1007/s00454-006-1265-8.640

    work page doi:10.1007/639 2006

  3. [3]

    Amenta and M

    N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering.Discrete & Computational Geometry,641 22(4):481–504, December 1999. URL: http://dx.doi.org/10.1007/PL00009475,doi:10.1007/pl00009475.642

    work page doi:10.1007/pl00009475 1999

  4. [4]

    V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko.The bifurcation sets and the monodromy group of a643 singularity, page 67–113. Birkhäuser Boston, 2012. URL: http://dx.doi.org/10.1007/978-0-8176-8343-6_3,644 doi:10.1007/978-0-8176-8343-6_3.645

    work page doi:10.1007/978-0-8176-8343-6_3 2012

  5. [5]

    Arnold, S.M

    V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko.Singularities of Differentiable Maps, Volume 1:646 Classification of Critical Points, Caustics and Wave Fronts. Birkhäuser Boston, 2012. URL: http:647 //dx.doi.org/10.1007/978-0-8176-8340-5,doi:10.1007/978-0-8176-8340-5.648

    work page doi:10.1007/978-0-8176-8340-5 2012

  6. [6]

    Springer Science & Business Media, 2003.649

    Vladimir I Arnold.Catastrophe theory. Springer Science & Business Media, 2003.649

    2003

  7. [7]

    Springer Science & Business Media, 1998.651

    Vladimir Igorevič Arnold, Victor V Goryunov, OV Lyashko, and Valerij Aleksandrovič Vasil’ev.Singularity650 theory I, volume 1. Springer Science & Business Media, 1998.651

    1998

  8. [8]

    Singularity theory II classification652 and applications

    Vladimir Igorevich Arnold, VV Goryunov, OV Lyashko, and VA Vasil’ev. Singularity theory II classification652 and applications. InDynamical Systems VIII: Singularity Theory II. Applications, pages 1–235. Springer,653 1993.654

    1993

  9. [9]

    V. I. Arnol’d. Normal forms for functions near degenerate critical points, the Weyl groups ofa k,d k,655 ek and lagrangian singularities.Functional Analysis and Its Applications, 6(4):254–272, Oct 1972. URL:656 http://dx.doi.org/10.1007/BF01077644,doi:10.1007/bf01077644.657

    work page doi:10.1007/bf01077644 1972

  10. [10]

    URL: https://arxiv.org/abs/659 2408.14995,doi:10.48550/ARXIV.2408.14995.660

    Shreya Arya, Barbara Giunti, Abigail Hickok, Lida Kanari, Sarah McGuire, and Katharine Turner.658 Decomposing the persistent homology transform of star-shaped objects, 2024. URL: https://arxiv.org/abs/659 2408.14995,doi:10.48550/ARXIV.2408.14995.660

    work page doi:10.48550/arxiv.2408.14995.660 2024

  11. [11]

    Millman, Daniel Salinas,661 Anna Schenfisch, Jordan Schupbach, and Lucia Williams

    Robin Lynne Belton, Brittany Terese Fasy, Rostik Mertz, Samuel Micka, David L. Millman, Daniel Salinas,661 Anna Schenfisch, Jordan Schupbach, and Lucia Williams. Reconstructing embedded graphs from persistence662 diagrams.Computational Geometry, 90:101658, October 2020. URL: http://dx.doi.org/10.1016/j.comgeo.663 2020.101658,doi:10.1016/j.comgeo.2020.101658.664

    work page doi:10.1016/j.comgeo.663 2020

  12. [12]

    Bergomi and Adriano Baratè

    Mattia G. Bergomi and Adriano Baratè. Homological persistence in time series: an application to music665 classification.Journal of Mathematics and Music, 14(2):204–221, May 2020. URL: http://dx.doi.org/10.666 1080/17459737.2020.1786745,doi:10.1080/17459737.2020.1786745.667

    work page doi:10.1080/17459737.2020.1786745.667 2020

  13. [13]

    URL: http://dx.doi.org/10.1017/9781108297806,doi:10.1017/669 9781108297806.670

    Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec.Geometric and Topological Inference.668 Cambridge University Press, Sept 2018. URL: http://dx.doi.org/10.1017/9781108297806,doi:10.1017/669 9781108297806.670

    work page doi:10.1017/9781108297806 2018

  14. [14]

    Cambridge University Press, July 1975

    Theodor Bröcker.Differentiable Germs and Catastrophes. Cambridge University Press, July 1975. Translated671 by: L. Lander. URL: http://dx.doi.org/10.1017/CBO9781107325418,doi:10.1017/cbo9781107325418.672

    work page doi:10.1017/cbo9781107325418 1975

  15. [15]

    Cambridge university press, 1992.674

    James William Bruce and Peter J Giblin.Curves and Singularities: a geometrical introduction to singularity673 theory. Cambridge university press, 1992.674

    1992

  16. [16]

    Bruce, P.J

    J.W. Bruce, P.J. Giblin, and C.G. Gibson. Symmetry sets.Proceedings of the Royal Society of Edinburgh:675 Section A Mathematics, 104A:179–204, 1985.676

    1985

  17. [17]

    Dan Burghelea and Tamal K. Dey. Topological persistence for circle-valued maps.Discrete & Computational677 Geometry, 50(1):69–98, April 2013. URL: http://dx.doi.org/10.1007/s00454-013-9497-x,doi:10.1007/678 s00454-013-9497-x.679

    work page doi:10.1007/s00454-013-9497-x 2013

  18. [18]

    A study of monodromy in the computation of680 multidimensional persistence

    Andrea Cerri, Marc Ethier, and Patrizio Frosini. A study of monodromy in the computation of680 multidimensional persistence. InDiscrete Geometry for Computer Imagery, pages 192–202, Berlin,681 Heidelberg, 2013. Springer Berlin Heidelberg.682

    2013

  19. [19]

    Counting triangulations of fixed683 cardinal degrees.Journal of Computational Geometry, 17, 2026

    Erin Chambers, Tim Ophelders, Anna Schenfisch, and Julia Sollberger. Counting triangulations of fixed683 cardinal degrees.Journal of Computational Geometry, 17, 2026. URL: https://jocg.org/index.php/jocg/684 article/view/5627,doi:10.20382/JOCG.V17I1A6.685

    work page doi:10.20382/jocg.v17i1a6.685 2026

  20. [20]

    Chambers, Christopher Fillmore, Elizabeth Stephenson, and Mathijs Wintraecken.Braiding686 Vineyards, page 6240–6263

    Erin W. Chambers, Christopher Fillmore, Elizabeth Stephenson, and Mathijs Wintraecken.Braiding686 Vineyards, page 6240–6263. Society for Industrial and Applied Mathematics, January 2026. URL:687 http://dx.doi.org/10.1137/1.9781611978971.225,doi:10.1137/1.9781611978971.225.688

    work page doi:10.1137/1.9781611978971.225 2026

  21. [21]

    Bouquet: A Visualization Tool for Symmetry Sets and Vineyards

    Erin Wolf Chambers, Christopher Fillmore, Shankha Shubhra Mukherjee, Rohit Roy, Elizabeth Stephenson,689 and Mathijs Wintraecken. Bouquet: A Visualization Tool for Symmetry Sets and Vineyards. InEuroCG690 2026 - 42nd European Workshop on Computational Geometry, Hagen, Germany, March 2026. URL:691 https://hal.science/hal-05546179.692

    2026

  22. [22]

    Vines and vineyards by updating703 persistence in linear time

    Frédéric Chazal and André Lieutier. Topology guaranteeing manifold reconstruction using distance function693 to noisy data. InProceedings of the twenty-second annual symposium on Computational geometry, SoCG06,694 page 112–118. ACM, 2006. URL: http://dx.doi.org/10.1145/1137856.1137876,doi:10.1145/1137856.695 1137876.696

    work page doi:10.1145/1137856.1137876 2006

  23. [23]

    Stability of persistence diagrams.Dis-697 crete & Computational Geometry, 37(1):103–120, December 2006

    David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams.Dis-697 crete & Computational Geometry, 37(1):103–120, December 2006. URL: http://dx.doi.org/10.1007/698 s00454-006-1276-5,doi:10.1007/s00454-006-1276-5.699

    work page doi:10.1007/698 2006

  24. [24]

    Extending persistence using Poincaré and700 Lefschetz duality.Foundations of Computational Mathematics, 9(1):79–103, April 2008

    David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Extending persistence using Poincaré and700 Lefschetz duality.Foundations of Computational Mathematics, 9(1):79–103, April 2008. URL: http:701 //dx.doi.org/10.1007/s10208-008-9027-z,doi:10.1007/s10208-008-9027-z.702

    work page doi:10.1007/s10208-008-9027-z 2008

  25. [25]

    Vines and vineyards by updating703 persistence in linear time

    David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov. Vines and vineyards by updating703 persistence in linear time. InProceedings of the twenty-second annual symposium on Computational704 geometry, SoCG06, page 119–126. ACM, June 2006. URL: http://dx.doi.org/10.1145/1137856.1137877,705 doi:10.1145/1137856.1137877.706

    work page doi:10.1145/1137856.1137877 2006

  26. [26]

    Justin Curry, Sayan Mukherjee, and Katharine Turner. How many directions determine a shape and other707 sufficiency results for two topological transforms.Transactions of the American Mathematical Society, Series708 B, 9(32):1006–1043, October 2022. URL: http://dx.doi.org/10.1090/btran/122,doi:10.1090/btran/122.709

    work page doi:10.1090/btran/122 2022

  27. [27]

    The global medial structure of regions inR3.Geometry & Topology, 10(4):2385–2429, 2006.710 doi:10.2140/gt.2006.10.2385.711

    James Damon. The global medial structure of regions inR3.Geometry & Topology, 10(4):2385–2429, 2006.710 doi:10.2140/gt.2006.10.2385.711

    work page doi:10.2140/gt.2006.10.2385.711 2006

  28. [28]

    Medial/skeletal linking structures for multi-region configurations.712 Memoirs of the American Mathematical Society, 250, 02 2014.doi:10.1090/memo/1193.713

    James Damon and Ellen Gasparovic. Medial/skeletal linking structures for multi-region configurations.712 Memoirs of the American Mathematical Society, 250, 02 2014.doi:10.1090/memo/1193.713

    work page doi:10.1090/memo/1193.713 2014

  29. [29]

    Corrêa, and David M

    Shannon Dee Algar, Débora C. Corrêa, and David M. Walker. On detecting dynamical regime change714 using a transformation cost metric between persistent homology diagrams.Chaos: An Interdisciplinary715 Journal of Nonlinear Science, 31(12), December 2021. URL: http://dx.doi.org/10.1063/5.0073247,doi:716 10.1063/5.0073247.717

    work page doi:10.1063/5.0073247 2021

  30. [30]

    Cambridge University718 Press, February 2022

    Tamal Krishna Dey and Yusu Wang.Computational Topology for Data Analysis. Cambridge University718 Press, February 2022. URL: http://dx.doi.org/10.1017/9781009099950,doi:10.1017/9781009099950.719

    work page doi:10.1017/9781009099950 2022

  31. [31]

    Springer Berlin Heidelberg,720

    André Diatta and Peter Giblin.Pre-symmetry Sets of 3D Shapes, page 36–48. Springer Berlin Heidelberg,720

  32. [32]

    URL: http://dx.doi.org/10.1007/11577812_4,doi:10.1007/11577812_4.721

    work page doi:10.1007/11577812_4

  33. [33]

    Springer, 1992.722

    Manfredo Perdigao Do Carmo.Riemannian geometry, volume 2. Springer, 1992.722

    1992

  34. [34]

    Topological persistence and simplification.Discrete & Com-723 putational Geometry, 28(4):511–533, November 2002

    Edelsbrunner, Letscher, and Zomorodian. Topological persistence and simplification.Discrete & Com-723 putational Geometry, 28(4):511–533, November 2002. URL: http://dx.doi.org/10.1007/s00454-002-2885-2,724 doi:10.1007/s00454-002-2885-2.725

    work page doi:10.1007/s00454-002-2885-2 2002

  35. [35]

    The726 Depth Poset Under Transpositions in the Filter

    Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin. The726 Depth Poset Under Transpositions in the Filter. In Hee-Kap Ahn, Michael Hoffmann, and Amir Nayyeri,727 editors,42nd International Symposium on Computational Geometry (SoCG 2026), volume 367 ofLeibniz728 International Proceedings in Informatics (LIPIcs),...

    work page doi:10.4230/730 2026

  36. [36]

    The poset of cancellations induced by gradient dynamics in a filtered Lefschetz complex

    Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, and Manuel Soriano-Trigueros. The poset of732 cancellations induced by gradient dynamics in a filtered Lefschetz complex, 2023. URL: https://arxiv.org/733 abs/2311.14364,doi:10.48550/ARXIV.2311.14364.734

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2311.14364.734 2023

  37. [37]

    Herbert Edelsbrunner, Elizabeth Stephenson, and Martin Hafskjold Thoresen. The mid-sphere cousin of735 the medial axis transform.4th International Joint Conference on Discrete Geometry and Mathematical736 Morphology, pages 133–147, 2025.doi:10.1007/978-3-032-09544-2_10.737

    work page doi:10.1007/978-3-032-09544-2_10.737 2025

  38. [38]

    Brittany Terese Fasy, Maksym Makarchuk, Samuel Micka, and David L. Millman. Too many or too738 few? sampling bounds for topological descriptors, 2025. URL: https://arxiv.org/abs/2511.12059,doi:739 10.48550/ARXIV.2511.12059.740

    work page doi:10.48550/arxiv.2511.12059.740 2025

  39. [39]

    Persistent homology and euler integral transforms.Journal741 of Applied and Computational Topology, 2(1–2):55–60, August 2018

    Robert Ghrist, Rachel Levanger, and Huy Mai. Persistent homology and euler integral transforms.Journal741 of Applied and Computational Topology, 2(1–2):55–60, August 2018. URL: http://dx.doi.org/10.1007/742 s41468-018-0017-1,doi:10.1007/s41468-018-0017-1.743

    work page doi:10.1007/742 2018

  40. [40]

    Symmetry sets and medial axes in two and three dimensions

    Peter Giblin. Symmetry sets and medial axes in two and three dimensions. In Roberto Cipolla and Ralph744 Martin, editors,The Mathematics of Surfaces IX, pages 306–321, London, 2000. Springer London.745

    2000

  41. [41]

    Peter Giblin and André Diatta. Symmetry sets and medial axes in two dimensions, 2003.746 URL: https://www.researchgate.net/profile/Peter-Giblin-2/publication/221403718_Symmetry_747 Sets_and_Medial_Axes_in_Two_and_Three_Dimensions/links/5cc0825b299bf120977d5eb5/748 Symmetry-Sets-and-Medial-Axes-in-Two-and-Three-Dimensions.pdf.749

    work page arXiv 2003

  42. [42]

    Kimia.Transitions of the 3D Medial Axis under a One-Parameter Family750 of Deformations, page 718–734

    Peter Giblin and Benjamin B. Kimia.Transitions of the 3D Medial Axis under a One-Parameter Family750 of Deformations, page 718–734. Springer Berlin Heidelberg, 2002. URL: http://dx.doi.org/10.1007/751 3-540-47967-8_48,doi:10.1007/3-540-47967-8_48.752

    work page doi:10.1007/751 2002

  43. [43]

    Giblin and Benjamin B

    Peter J. Giblin and Benjamin B. Kimia. On the local form and transitions of symmetry sets, medial753 axes, and shocks.International Journal of Computer Vision, 54(1-3):143–157, August 2003. URL:754 http://dx.doi.org/10.1023/A:1023761518825,doi:10.1023/a:1023761518825.755

    work page doi:10.1023/a:1023761518825 2003

  44. [44]

    DONUT: Database of Original & Non-Theoretical Uses756 of Topology, 2022

    Barbara Giunti, J¯ anis Lazovskis, and Bastian Rieck. DONUT: Database of Original & Non-Theoretical Uses756 of Topology, 2022. https://donut.topology.rocks.757

    2022

  45. [45]

    Algorithms for recognizing knots and 3-manifolds

    Joel Hass. Algorithms for recognizing knots and 3-manifolds.arXiv preprint math/9712269, 1997.758

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  46. [46]

    Hofer, Roland Kwitt, and Marc Niethammer

    Christoph D. Hofer, Roland Kwitt, and Marc Niethammer. Learning representations of persistence barcodes.759 Journal of Machine Learning Research, 20(126):1–45, 2019. URL: http://jmlr.org/papers/v20/18-358.html.760

    2019

  47. [47]

    The computational complexity of classical knot761 recognition.Journal of Knot Theory and Its Ramifications, 32(11), October 2023

    Kazuhiro Ichihara, Yuya Nishimura, and Seiichi Tani. The computational complexity of classical knot761 recognition.Journal of Knot Theory and Its Ramifications, 32(11), October 2023. URL: http://dx.doi.762 org/10.1142/S0218216523500694,doi:10.1142/s0218216523500694.763

    work page doi:10.1142/s0218216523500694 2023

  48. [48]

    NP–hard problems naturally arising in knot theory.Transactions of764 the American Mathematical Society, Series B, 8(15):420–441, May 2021

    Dale Koenig and Anastasiia Tsvietkova. NP–hard problems naturally arising in knot theory.Transactions of764 the American Mathematical Society, Series B, 8(15):420–441, May 2021. URL: http://dx.doi.org/10.1090/765 btran/71,doi:10.1090/btran/71.766

    work page doi:10.1090/765 2021

  49. [49]

    Kuijper and O.F

    A. Kuijper and O.F. Olsen. Transitions of the pre-symmetry set. InProceedings of the 17th International767 Conference on Pattern Recognition, 2004. ICPR 2004., page 190–193 Vol.3. IEEE, 2004. URL: http:768 //dx.doi.org/10.1109/ICPR.2004.1334500,doi:10.1109/icpr.2004.1334500.769

    work page doi:10.1109/icpr.2004.1334500 2004

  50. [50]

    Springer Berlin Heidelberg, 2004

    Arjan Kuijper, Ole Fogh Olsen, Peter Giblin, Philip Bille, and Mads Nielsen.From a 2D Shape to770 a String Structure Using the Symmetry Set, page 313–325. Springer Berlin Heidelberg, 2004. URL:771 http://dx.doi.org/10.1007/978-3-540-24671-8_25,doi:10.1007/978-3-540-24671-8_25.772

    work page doi:10.1007/978-3-540-24671-8_25 2004

  51. [51]

    The efficient certification of knottedness and thurston norm.Advances in Mathematics,773 387:107796, August 2021

    Marc Lackenby. The efficient certification of knottedness and thurston norm.Advances in Mathematics,773 387:107796, August 2021. URL: http://dx.doi.org/10.1016/j.aim.2021.107796,doi:10.1016/j.aim.2021.774 107796.775

    work page doi:10.1016/j.aim.2021.107796 2021

  52. [52]

    PhD thesis, Universiteit van776 Amsterdam, 1974.777

    Eduard Looijenga.Structural Stability of smooth families ofC ∞-functions. PhD thesis, Universiteit van776 Amsterdam, 1974.777

    1974

  53. [53]

    Distance from a submanifold in Euclidean-space

    John N Mather. Distance from a submanifold in Euclidean-space. InProceedings of symposia in pure778 mathematics, volume 40, pages 199–216. American Mathematical Society, 1983.779

    1983

  54. [54]

    Sara Scaramuccia (04/23/25): Monodromy in bi-parameter persistence780 modules, 2025

    Applied Algebraic Topology Network. Sara Scaramuccia (04/23/25): Monodromy in bi-parameter persistence780 modules, 2025. URL: https://youtu.be/14dSHjeFUoI?si=XUdRBLtG1ImwrR88.781

    2025

  55. [55]

    Finding the homology of submanifolds with high782 confidence from random samples.Discrete & Computational Geometry, 39(1-3):419–441, March 2008

    Partha Niyogi, Stephen Smale, and Shmuel Weinberger. Finding the homology of submanifolds with high782 confidence from random samples.Discrete & Computational Geometry, 39(1-3):419–441, March 2008. URL:783 http://dx.doi.org/10.1007/s00454-008-9053-2,doi:10.1007/s00454-008-9053-2.784

    work page doi:10.1007/s00454-008-9053-2 2008

  56. [56]

    Shoving tubes through shapes gives a sufficient and efficient785 shape statistic, 2024

    Adam Onus, Nina Otter, and Renata Turkes. Shoving tubes through shapes gives a sufficient and efficient785 shape statistic, 2024. URL: https://arxiv.org/abs/2412.18452,doi:10.48550/ARXIV.2412.18452.786

    work page doi:10.48550/arxiv.2412.18452.786 2024

  57. [57]

    Cambridge University787 Press, 2001.788

    Ian R Porteous.Geometric differentiation: for the intelligence of curves and surfaces. Cambridge University787 Press, 2001.788

    2001

  58. [58]

    Towards computing homology from approximations.Topology Proceedings, 24, 01 1999.789

    Vanessa Robins. Towards computing homology from approximations.Topology Proceedings, 24, 01 1999.789

    1999

  59. [59]

    McGraw-Hill inc., 3rd edition, 1976.790

    Walter Rudin.Principles of mathematical analysis. McGraw-Hill inc., 3rd edition, 1976.790

    1976

  60. [60]

    Stratified braid groups: monodromy, 2023

    Nick Salter. Stratified braid groups: monodromy, 2023. URL: https://arxiv.org/abs/2304.04627,arXiv:791 2304.04627.792

    work page arXiv 2023

  61. [61]

    Monodromy of stratified braid groups, ii.Research in the Mathematical Sciences, 11(4), October793

    Nick Salter. Monodromy of stratified braid groups, ii.Research in the Mathematical Sciences, 11(4), October793

  62. [62]

    URL: http://dx.doi.org/10.1007/s40687-024-00477-4,doi:10.1007/s40687-024-00477-4.794

    work page doi:10.1007/s40687-024-00477-4

  63. [63]

    Stephenson

    Elizabeth R. Stephenson. Generalizing medial axes with homology switches. Master’s thesis, IST Austria,795

  64. [64]

    URL: https://research-explorer.ista.ac.at/record/14226,doi:10.15479/AT:ISTA:14226.796

    work page doi:10.15479/at:ista:14226.796

  65. [65]

    Representing vineyard modules.arXiv preprint arXiv:2307.06020, 2023.797

    Katharine Turner. Representing vineyard modules.arXiv preprint arXiv:2307.06020, 2023.797

    work page arXiv 2023

  66. [66]

    The extended persistent homology transform of798 manifolds with boundary.Journal of Applied and Computational Topology, 8(7):2111–2154, May 2024

    Katharine Turner, Vanessa Robins, and James Morgan. The extended persistent homology transform of798 manifolds with boundary.Journal of Applied and Computational Topology, 8(7):2111–2154, May 2024. URL:799 http://dx.doi.org/10.1007/s41468-024-00175-8,doi:10.1007/s41468-024-00175-8.800

    work page doi:10.1007/s41468-024-00175-8 2024

  67. [67]

    On the local structure of a generic central set.Compositio Mathematica, 43(2):225–238, 1981.801 URL: http://www.numdam.org/item/CM_1981__43_2_225_0/.802

    Yosef Yomdin. On the local structure of a generic central set.Compositio Mathematica, 43(2):225–238, 1981.801 URL: http://www.numdam.org/item/CM_1981__43_2_225_0/.802

    1981

  68. [68]

    Topological persistence vineyard for803 dynamic functional brain connectivity during resting and gaming stages.Journal of Neuroscience Methods,804 267:1–13, July 2016

    Jaejun Yoo, Eun Young Kim, Yong Min Ahn, and Jong Chul Ye. Topological persistence vineyard for803 dynamic functional brain connectivity during resting and gaming stages.Journal of Neuroscience Methods,804 267:1–13, July 2016. URL: http://dx.doi.org/10.1016/j.jneumeth.2016.04.001,doi:10.1016/j.jneumeth.805 2016.04.001.806

    work page doi:10.1016/j.jneumeth.2016.04.001 2016