Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Review

Gunnar Carlsson; Guo-Wei Wei; Jie Wu; Layal Bou Hamdan; Vasileios Maroulas; Xiang Liu; Zhe Su

REVIEW 4 major objections 8 minor 2 cited by

Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Review

T0 review · 4 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This review organizes topological data analysis beyond persistent homology into algebraic, differential, and geometric branches, and maps each family of methods to the data types it can handle.

desk verdict Useful survey of TDA beyond persistent homology, but the title overclaims TDL coverage and the selection leans heavily on the authors' own work. read the letter →

arxiv 2507.19504 v1 pith:RIVJRSDS submitted 2025-07-12 math.HO math.DGmath.GT

Zhe Su , Xiang Liu , Layal Bou Hamdan , Vasileios Maroulas , Jie Wu , Gunnar Carlsson , Guo-Wei Wei This is my paper · ORCID

classification math.HOmath.DGmath.GT MSC 62R4055-0857R1957K18

keywords TopologicaldataanalysisPersistenthomologyLaplacianDiracoperatordeRham-HodgetheoryKhovanovKnotdeeplearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that the field of topological data analysis and topological deep learning has outgrown persistent homology, and that the many generalizations now filling the field can be organized into three branches defined by the branch of topology they use: algebraic topology for point clouds, differential topology for data on manifolds, and geometric topology for curves embedded in 3-space. It argues that no previous review has covered these generalizations together with their machine-learning vectorizations, and that such a map is needed for practitioners to choose the right tool. The paper further claims that spectral methods, especially persistent Laplacians and persistent Dirac operators, extend persistent homology by encoding both topological invariants and the geometric evolution of shapes during filtration. If this organizational map is right, a reader can select a TDA method by asking what kind of data they have and which topological properties matter.

What carries the argument

The central organizing device is a two-axis classification: the branch of topology (algebraic, differential, geometric) paired with the input data type (point cloud, distance matrix, network, manifold, sequence, knot or link, and data with extra non-geometric information). The workhorse technical object is the persistent topological Laplacian, defined as a multiscale family of operators whose harmonic spectra reproduce persistent homology and whose non-harmonic spectra encode geometric or combinatorial evolution; the persistent Dirac operator, whose square yields the persistent Laplacians, is presented as a first-order refinement suited to local features and quantum computation.

What would settle it

Run the review's data-type-to-tool guidance on a standard benchmark for sequential data, such as neural spike decoding: if a simple topological neural network of the kind the paper excludes outperforms the review's recommended sequential-data methods (persistent path Laplacians, k-mer topology, or sliding-window persistence) on the same test set, the claim that the review serves as a complete guide to optimal tool selection is falsified.

Watch

Extended reading notes

Core claim

The paper's discovery, on its own terms, is that the post-persistent-homology landscape divides naturally into algebraic topology approaches (persistent combinatorial Laplacians, persistent Dirac operators, sheaf theory, Mayer topology, interaction topology), differential topology approaches (persistent de Rham cohomology, persistent Hodge Laplacians, Hodge decomposition), and geometric topology approaches for knots and links (multiscale Gauss linking integrals, persistent Jones polynomials, persistent Khovanov homology). For each family, the paper claims, the key advance is that topological information is no longer reduced to barcodes alone: persistent Laplacians and Dirac operators provide spectra whose zero-eigenvalue multiplicities recover persistent Betti numbers while nonzero eigenvalues capture homotopic geometric shape changes that persistent homology misses. The paper also claims that these methods, together with their vectorizations, form a complete practical pipeline, and that the choice of topological model should be driven by the intrinsic nature of the data: simplicial complexes and hypergraphs for point clouds, manifolds with boundary for images and volumetric data, and knot-theoretic objects for open or closed curves in space.

Load-bearing premise

The review's usefulness depends on the methods it selected being representative of the field, and it explicitly admits in Section 6 that many significant contributions are omitted, including topological neural networks; if those omissions change which tool is best for some data type, the organizational map fails even though each individual description may be correct.

Editorial extensions

If this is right

A practitioner with point-cloud data can choose between simplicial-complex methods and spectral Laplacian or Dirac methods, with the spectral methods adding geometric information that persistent homology alone cannot see.
For data that naturally lives on manifolds, such as MRI volumes or isosurfaces, the review points to persistent de Rham-Hodge theory and Hodge decomposition as the appropriate tools, with boundary conditions playing a role that combinatorial Laplacians cannot capture.
For curves embedded in 3-space, including proteins, DNA, and polymers, the review identifies multiscale Gauss linking integrals, persistent Jones polynomials, and persistent Khovanov homology as the applicable geometric-topology methods, and notes that these work for open curves where classical knot invariants fail.
Vectorization methods, ranging from persistence images and landscapes to eigenvalue and eigenvector summaries and Hodge-decomposition images, are what make each family usable inside machine-learning pipelines, and the review maps which representation fits which topological descriptor.
The paper identifies sequential data and local-topology methods as open directions, and flags the lack of robust computational algorithms for persistent Khovanov homology as an urgent gap.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the three-branch organization is correct, the field's main axis of innovation is not new topological invariants but new operators on existing complexes: Laplacians, Dirac operators, and their persistent versions are what turn topology into a quantitative, machine-learnable signal.
Because the paper deliberately excludes topological neural networks, its title's promise of 'topological deep learning' is narrower than the current field; readers should treat the data-type-to-tool map as a guide to feature-based TDA, not to end-to-end topological network architectures.
A testable consequence is that on a shared benchmark suite, methods from the differential branch should beat algebraic-topology methods on manifold-structured data, and geometric-topology methods should beat them on knot-like data; such a benchmark would directly probe the paper's organizational claim.
The persistent Khovanov homology framework, though promising for unknot detection and local entanglement, is likely to remain computationally prohibitive for large datasets until the efficient algorithms the paper calls for are developed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a genuinely useful survey, but it is a survey of TDA generalizations plus vectorization, not of TDL. The title and abstract promise more than the paper delivers. Section 6 explicitly says topological neural networks were excluded, so calling this a comprehensive TDL review is misleading. That is a framing problem, not a mathematical one, but it undercuts a central claim.

What the paper does well: the organizing frame—algebraic, differential, and geometric topology mapped to data types—is sensible and fills a real gap. No prior review ties persistent Laplacians/Dirac, de Rham–Hodge theory, and knot invariants to vectorization choices in one place. The mathematical summaries are mostly accurate at survey level, the software table is handy, and a new researcher could use this to orient themselves. The discussion of input data types (point clouds, manifolds, curves, sequential data) is a practical contribution.

The soft spots are real but not fatal. First, the TDL overclaim: the few architectural mentions in Section 5 do not amount to systematic coverage, and the paper's own statement in Section 6 confirms the restriction. Second, the selection leans heavily on the authors' own contributions. Claims like "one of the most impressive achievements" for SARS-CoV-2 variant forecasting and "consistently outperforms" for mGLI come from the authors' papers and are presented without independent validation. A survey should either soften these or flag them as self-reported. Third, there are many typos (Dinoysus, interSection, persitence, spacial) that look like a rushed final pass.

Who this is for: new researchers in TDA who want a map of methods beyond persistent homology, and practitioners selecting tools for manifold or curve data. It deserves serious peer review. With the framing fixed, the claims moderated, and the typos cleaned up, it would become a standard entry point. I would send it out.

Referee Report

4 major / 8 minor

Summary. This manuscript is a survey-style review of topological data analysis (TDA) and topological deep learning (TDL) methods that go beyond the standard persistent homology pipeline. It organizes the material into three broad families: algebraic topology approaches (persistent homology variants, persistent combinatorial/Laplacian and Dirac constructions, sheaf, Mayer, and interaction topology), differential topology approaches (persistent de Rham cohomology, persistent Hodge Laplacians, Hodge decomposition), and geometric topology approaches for curves in 3-space (multiscale Gauss linking integrals, localized/persistent Jones polynomials, persistent Khovanov homology). The final section discusses input data types, topological representations, software, and machine-learning vectorization. The authors claim that no prior review covers the generalizations of persistent homology together with their vectorization, and they present the paper as a comprehensive guide for selecting topological tools for different data types.

Significance. If the scope were made fully coherent, this review would be a useful entry point for researchers new to TDA beyond the standard Vietoris–Rips persistent homology pipeline. Its three-way organization by algebraic, differential, and geometric topology is pedagogically sensible, and it collects many recent developments from the spectral TDA literature in one place. The paper is also genuinely useful for its extensive catalog of input-data types, software packages, and vectorization strategies, including newer spectral representations. However, the significance of the central claim is undercut by a mismatch between the title/abstract and the actual coverage: topological deep learning in the sense of topological neural networks is explicitly excluded in Section 6, despite the title and abstract promising TDL beyond persistent homology. The review is therefore best assessed as a review of TDA generalizations and their vectorization rather than a comprehensive review of both TDA and TDL.

major comments (4)

[Abstract, Introduction, Section 6] The title and Abstract promise a comprehensive review of 'TDA and TDL beyond persistent homology,' but Section 6 states that the authors 'restricted ourselves from elaborating on' the development of topological neural networks, referring readers elsewhere. Topological neural networks, such as message-passing simplicial/cell complexes, persistent-homology layers, and topological losses, are exactly the TDL content that the title and Abstract lead readers to expect. The few mentions in Sections 5.1.4 and 5.1.5 (TCNN, simplicial message passing networks, cell complex encoders) are not systematic coverage. This is not a mathematical error, but it is a load-bearing inconsistency: under the stated scope, a reader selecting TDL methods cannot rely on this review as a comprehensive guide. The authors should either revise the title/Abstract to state that the survey covers TDA generalizations and their vectorization, with TDL treated only in passing, or substantially expand Section 5 to cover topological neural network architectures.
[Section 1, paragraph 2 and Section 6] The paper repeatedly highlights specific application-level successes as evidence for the reviewed methods, including the statement that 'one of the most impressive achievements of persistent topological Laplacians is the prediction of emerging dominant SARS-COV-2 variants BA4 and BA5 [65]' and similar claims about D3R Grand Challenges and SARS-CoV-2 forecasting. These achievements are cited to papers from the authors' own group, and no independent replication is discussed. For a review, this is not fatal, but the phrasing 'most impressive achievements' is stronger than the evidence supplied. It would be more appropriate to attribute these as reported results from the cited papers, or to add a caveat that independent validation of these application-level claims is still needed.
[Section 2.2.2, Eq. for persistent combinatorial Laplacian] The definition of the persistent boundary operator and the persistent combinatorial Laplacian is presented with a formula using the restriction of ∂^j_{k+1} to a subgroup C^{i,j}_{k+1}, but the precise domain and the role of the projection P^i_k are not fully formalized in the text. Since the paper is a review and explicitly avoids excessive technical detail, this level of presentation is defensible, but the notation in the displayed formula is confusing: the operator (∂^i_k)^* appears with a star that is not defined in the text, and the construction of C^{i,j}_{k+1} is only described informally. The authors should at least state that ∂^{i,j}_{k+1} is the composition of the boundary operator on K_j followed by projection onto chains of K_i, and define adjoints with respect to the chosen inner product.
[Section 5.3, overall featurization discussion] The paper claims to provide a guide for selecting topological tools for different input data, but the selection criteria in Section 5.1 are presented mostly as qualitative recommendations without comparative evidence. For example, Section 5.1.5 discusses several TSA methods (Takens embedding, sliding window, k-mer topology, Δ-complex approaches) without a clear statement of when one outperforms another. This is a limitation of the review's utility claim rather than a technical error. Adding a concise comparison table of methods versus data types would substantially strengthen the practical guide aspect.

minor comments (8)

[Section 2.1.2] There is a typo: 'interSection' should be 'intersection' in the description of the Čech complex.
[Section 2.1.2] The word 'persitence' appears instead of 'persistence' in the definition of birth and death.
[Section 2.1.3] In the paragraph on graph-complex-based persistent homology, 'maainly' should be 'mainly'.
[Section 2.3.2] The displayed formula for the persistent Dirac operator has a line break in the subscript '∂^{i,j}_{k+ 1}' that may cause confusion; it should be '∂^{i,j}_{k+1}'.
[Section 4.2] The text uses 'spacial scales' where 'spatial scales' is intended.
[Table 1] The table lists 'Dinoysus' but the correct software name is 'Dionysus' (ref. [229]). Also, the table caption would benefit from noting whether the listed software for persistent Laplacians includes the newer packages such as HERMES and PersistLap.
[Section 5.2.6] The description of Mapper is included under 'Software and code resources,' but Mapper is an algorithm/visualization tool rather than a package for computing persistent homology or Laplacians; this is a minor organizational issue.
[References] Reference [20] is cited as 'The Jones polynomial of collections of open curves in 3-space' but in Section 4.2 the author name is not given; ensure that all in-text citations use a consistent style.

Circularity Check

0 steps flagged · score 2.0 of 10

No derivation-chain circularity: the review contains no input-output derivation chain, its spectral-recovery claims are stated theorems or explicit constructions, and its self-cited flagship applications are externally published, time-ordered forecasts; the disclosed TDL-exclusion is a scope issue, not circularity.

full rationale

This is a review (math.HO) with no claimed derivation from first principles, so there is no input-output chain whose conclusion could equal its premise. The places where a reader might suspect 'recovery by construction' are all explicit theorems or constructions: the isomorphism ker L^{i,j}_k = H^{i,j}_k in Section 2.2.2 is attributed to prior proofs [190, 217]; the persistent Dirac operator is defined and its square is displayed in Section 2.3.2 with the persistent combinatorial Laplacian appearing as an explicit block, so the relationship is shown rather than disguised; and the persistent Hodge Laplacian kernel isomorphism in Section 3.3 is likewise stated as a theorem. None of these is a fitted parameter renamed as a prediction. The paper's evaluative claims (e.g., 'one of the most impressive achievements of persistent topological Laplacians is the prediction of emerging dominant SARS-COV-2 variants BA4 and BA5 [65]', and the BA.2 forecast [62]) rest on the authors' own prior papers, and the method selection is concentrated on the authors' line of work; this is a real coverage/objectivity concern and is the reason the score is not 0. However, those cited results are externally published, time-ordered predictions about variants that subsequently emerged, so under the reviewing rules they count as independent, falsifiable evidence rather than self-citation circularity. The review also explicitly admits in Section 6 that it 'inevitably omit[s] many significant contributions' and that 'to focus on TDA, we have restricted ourselves from elaborating on' topological neural networks, which makes the title's 'TDL ... beyond persistent homology' a scope overclaim; that internal inconsistency is flagged here, but it is a scope/coverage defect, not a circular reduction of a result to its own input. No circular step can therefore be exhibited with a specific reduction, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities appear because this is a literature review. The review's claims rest on standard background results in algebraic, differential, and geometric topology, all of which are cited.

assumptions (4)

standard math Persistent homology and its stability theory are standard background (Sec. 2.1).
The review summarizes persistent homology without proving it.
standard math The kernel of a combinatorial Laplacian is isomorphic to simplicial homology (Sec. 2.2.1).
Used to justify that harmonic spectra recover persistent homology.
standard math Hodge decomposition holds on manifolds with boundary under absolute and relative boundary conditions (Sec. 3.1.3).
The review relies on this for the 5-component Hodge decomposition discussions.
standard math Khovanov homology is a link invariant that categorifies the Jones polynomial (Sec. 4.3).
Background for persistent Khovanov homology and Khovanov Laplacians.

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Cite this review

Pith. "Pith review of Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Review." pith.science (2026). https://pith.science/paper/RIVJRSDS

@misc{pith2026250719504,
  author       = {Pith},
  title        = {Pith review of: Topological Data Analysis and Topological Deep Learning Beyond Persistent Homology -- A Review},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RIVJRSDS}},
  note         = {Machine review of arXiv:2507.19504}
}

read the original abstract

Topological data analysis (TDA) is a rapidly evolving field in applied mathematics and data science that leverages tools from topology to uncover robust, shape-driven insights in complex datasets. The main workhorse is persistent homology, a technique rooted in algebraic topology. Paired with topological deep learning (TDL) or topological machine learning, persistent homology has achieved tremendous success in a wide variety of applications in science, engineering, medicine, and industry. However, persistent homology has many limitations due to its high-level abstraction, insensitivity to non-topological changes, and reliance on point cloud data. This paper presents a comprehensive review of TDA and TDL beyond persistent homology. It analyzes how persistent topological Laplacians and Dirac operators provide spectral representations to capture both topological invariants and homotopic evolution. Other formulations are presented in terms of sheaf theory, Mayer topology, and interaction topology. For data on differentiable manifolds, techniques rooted in differential topology, such as persistent de Rham cohomology, persistent Hodge Laplacian, and Hodge decomposition, are reviewed. For one-dimensional (1D) curves embedded in 3-space, approaches from geometric topology are discussed, including multiscale Gauss-link integrals, persistent Jones polynomials, and persistent Khovanov homology. This paper further discusses the appropriate selection of topological tools for different input data, such as point clouds, sequential data, data on manifolds, curves embedded in 3-space, and data with additional non-geometric information. A review is also given of various topological representations, software packages, and machine learning vectorizations. Finally, this review ends with concluding remarks.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Persistent Laplacian features predict VLP stoichiometry with reported AUC up to 0.98, but the stability conclusion rests on classifier accuracy rather than physical stability, and the input features may leak the label.
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CAKL, a persistent Stanley-Reisner k-mer representation, outperforms five baseline methods on 11 genomic benchmarks, especially viral classification.

Reference graph

Works this paper leans on

300 extracted references · 67 canonical work pages · cited by 2 Pith papers

[65]

4 and BA

Chen, J., Qiu, Y., Wang, R., Wei, G.-W.: Persistent Laplacian pro- jected Omicron BA. 4 and BA. 5 to become new dominating variants. Computers in Biology and Medicine 151, 106262 (2022)

2022
[1]

American Mathematical Society, (2004)

Adams, C.C.: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society, (2004)

2004
[2]

SIAM Journal on Imaging Sciences 2(1), 110–117 (2009)

Adams, H., Carlsson, G.: On the nonlinear statistics of range image patches. SIAM Journal on Imaging Sciences 2(1), 110–117 (2009)

2009
[3]

In: Mathematical Software– ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9,

Adams, H., Tausz, A., Vejdemo-Johansson, M.: Javaplex: a research soft- ware package for persistent (co)homology. In: Mathematical Software– ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9,

2014
[4]

Journal of Machine Learning Research 18(8), 1–35 (2017)

Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Ship- man, P., Chepushtanova, S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research 18(8), 1–35 (2017)

2017
[5]

Persistent equivariant cohomology

Adams, H., Lagoda, E., Moy, M., Sadovek, N., De Saha, A.: Persistent equivariant cohomology. arXiv preprint arXiv:2408.17331 (2024)

work page Pith review arXiv 2024
[6]

Homology, Homotopy and Applications 18(1), 381–402 (2016)

Adcock, A., Carlsson, E., Carlsson, G.: The ring of algebraic functions on persistence bar codes. Homology, Homotopy and Applications 18(1), 381–402 (2016)

2016
[7]

Transactions of the American Mathematical Society 30(2), 275–306 (1928) CONTENTS 61

Alexander, J.W.: Topological invariants of knots and links. Transactions of the American Mathematical Society 30(2), 275–306 (1928) CONTENTS 61

1928

Show all 300 references

[8]

IEEE Transactions on Pattern Analysis and Machine Intelligence 45(12), 14069–14080 (2023)

Ali, D., Asaad, A., Jimenez, M.-J., Nanda, V., Paluzo-Hidalgo, E., Soriano-Trigueros, M.: A survey of vectorization methods in topologi- cal data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence 45(12), 14069–14080 (2023)

2023
[9]

Journal of Applied and Computational Topology8(7), 1961–1980 (2024)

Ameneyro, B., Maroulas, V., Siopsis, G.: Quantum persistent homology. Journal of Applied and Computational Topology8(7), 1961–1980 (2024)

2024
[10]

In: APS March Meeting Abstracts, vol

Ameneyro, B., Siopsis, G., Maroulas, V.: Quantum persistent homology for time series. In: APS March Meeting Abstracts, vol. 2023, pp. 73–003 (2023)

2023
[11]

SIAM, (2018)

Arnold, D.N.: Finite Element Exterior Calculus. SIAM, (2018)

2018
[12]

Acta numerica 15, 1–155 (2006)

Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calcu- lus, homological techniques, and applications. Acta numerica 15, 1–155 (2006)

2006
[13]

Proceedings of the National Academy of Sciences 102(26), 9165–9169 (2005)

Arsuaga, J., Vazquez, M., McGuirk, P., Trigueros, S., Sumners, D.W., Roca, J.: DNA knots reveal a chiral organization of DNA in phage capsids. Proceedings of the National Academy of Sciences 102(26), 9165–9169 (2005)

2005
[14]

Mathematics 10(21), 4039 (2022)

Asaad, A., Ali, D., Majeed, T., Rashid, R.: Persistent homology for breast tumor classification using mammogram scans. Mathematics 10(21), 4039 (2022)

2022
[15]

Journal of Intelligent Information Systems 52, 637–655 (2019)

Atienza, N., Gonzalez-Diaz, R., Rucco, M.: Persistent entropy for sep- arating topological features from noise in Vietoris-Rips complexes. Journal of Intelligent Information Systems 52, 637–655 (2019)

2019
[16]

Physical Review E 106(3), 034319 (2022)

Baccini, F., Geraci, F., Bianconi, G.: Weighted simplicial complexes and their representation power of higher-order network data and topology. Physical Review E 106(3), 034319 (2022)

2022
[17]

In: Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp

Bae, W., Yoo, J., Chul Ye, J.: Beyond deep residual learning for image restoration: Persistent homology-guided manifold simplification. In: Pro- ceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp. 145–153 (2017)

2017
[18]

Journal of Theoretical Biology 529, 110854 (2021)

Baldwin, Q., Panagiotou, E.: The local topological free energy of proteins. Journal of Theoretical Biology 529, 110854 (2021)

2021
[19]

Polymers 14(15), 3014 (2022)

Baldwin, Q., Sumpter, B., Panagiotou, E.: The local topological free energy of the SARS-CoV-2 spike protein. Polymers 14(15), 3014 (2022)

2022
[20]

Proceedings of the Royal Society A 478(2267), 20220302 (2022)

Barkataki, K., Panagiotou, E.: The Jones polynomial of collections of 62 CONTENTS open curves in 3-space. Proceedings of the Royal Society A 478(2267), 20220302 (2022)

2022
[21]

Frontiers in Artificial Intel- ligence 4, 681174 (2021)

Barnes, D., Polanco, L., Perea, J.A.: A comparative study of machine learning methods for persistence diagrams. Frontiers in Artificial Intel- ligence 4, 681174 (2021)

2021
[22]

Journal of Applied and Computational Topology 5(3), 391–423 (2021)

Bauer, U.: Ripser: efficient computation of Vietoris–Rips persistence bar- codes. Journal of Applied and Computational Topology 5(3), 391–423 (2021)

2021
[23]

Software available at https://github

Bauer, U., Kerber, M., Reininghaus, J.: Dipha (a distributed per- sistent homology algorithm). Software available at https://github. com/DIPHA/dipha (2014)

2014
[24]

Journal of symbolic computation 78, 76– 90 (2017)

Bauer, U., Kerber, M., Reininghaus, J., Wagner, H.: Phat–persistent homology algorithms toolbox. Journal of symbolic computation 78, 76– 90 (2017)

2017
[25]

Physical review letters 98(14), 146401 (2007)

Behler, J., Parrinello, M.: Generalized neural-network representation of high-dimensional potential-energy surfaces. Physical review letters 98(14), 146401 (2007)

2007
[26]

IEEE Transactions on visualization and computer graphics 19(8), 1386–1404 (2012)

Bhatia, H., Norgard, G., Pascucci, V., Bremer, P.-T.: The Helmholtz- Hodge decomposition - a survey. IEEE Transactions on visualization and computer graphics 19(8), 1386–1404 (2012)

2012
[27]

arXiv preprint arXiv:2205.10796 (2022)

Bi, W., Li, J., Liu, J., Wu, J.: On the Cayley-persistence algebra. arXiv preprint arXiv:2205.10796 (2022)

2022 arXiv
[28]

Journal of Physics: Complexity 2(3), 035022 (2021)

Bianconi, G.: The topological Dirac equation of networks and simplicial complexes. Journal of Physics: Complexity 2(3), 035022 (2021)

2021
[29]

Theoretical computer science 392(1-3), 5–22 (2008)

Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theoretical computer science 392(1-3), 5–22 (2008)

2008
[30]

In: Proceedings of the 38th International Conference on Machine Learning, pp

Bodnar, C., Frasca, F., Wang, Y., Otter, N., Mont´ ufar, G.F., Li` o, P., Bronstein, M.: Weisfeiler and lehman go topological: Message passing simplicial networks. In: Proceedings of the 38th International Conference on Machine Learning, pp. 1026–1037 (2021)

2021
[31]

In: Representations of Algebras and Related Structures, pp

Botnan, M., Lesnick, M.: An introduction to multiparameter persistence. In: Representations of Algebras and Related Structures, pp. 77–150 (2023) CONTENTS 63

2023
[32]

In: Leibniz International Proceedings in Informatics, vol

Botnan, M.B., Lebovici, V., Oudot, S.: On rectangle-decomposable 2- parameter persistence modules. In: Leibniz International Proceedings in Informatics, vol. 164, pp. 22–1 (2020). Leibniz-Zentrum f¨ ur Informatik

2020
[33]

Asian Journal of Mathematics 23(3), 479–500 (2019)

Bressan, S., Li, J., Ren, S., Wu, J.: The embedded homology of hyper- graphs and applications. Asian Journal of Mathematics 23(3), 479–500 (2019)

2019
[34]

Journal of Machine Learning Research 16(1), 77–102 (2015)

Bubenik, P.: Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research 16(1), 77–102 (2015)

2015
[35]

Journal of Symbolic Computation 78, 91–114 (2017)

Bubenik, P., D lotko, P.: A persistence landscapes toolbox for topological statistics. Journal of Symbolic Computation 78, 91–114 (2017)

2017
[36]

New Journal of Physics 25(9), 093013 (2023)

Calmon, L., Schaub, M.T., Bianconi, G.: Dirac signal processing of higher-order topological signals. New Journal of Physics 25(9), 093013 (2023)

2023
[37]

PLoS computational biology 13(7), 1005690 (2017)

Cang, Z., Wei, G.-W.: Topologynet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions. PLoS computational biology 13(7), 1005690 (2017)

2017
[38]

International journal for numerical methods in biomedical engineering 34(2), 2914 (2018)

Cang, Z., Wei, G.-W.: Integration of element specific persistent homol- ogy and machine learning for protein-ligand binding affinity prediction. International journal for numerical methods in biomedical engineering 34(2), 2914 (2018)

2018
[39]

SIAM journal on mathematics of data science 2(2), 396–418 (2020)

Cang, Z., Wei, G.-W.: Persistent cohomology for data with multicompo- nent heterogeneous information. SIAM journal on mathematics of data science 2(2), 396–418 (2020)

2020
[40]

PLoS computational biology 14(1), 1005929 (2018)

Cang, Z., Mu, L., Wei, G.-W.: Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening. PLoS computational biology 14(1), 1005929 (2018)

2018
[41]

Journal of applied and computational topology 4(4), 481–507 (2020)

Cang, Z., Munch, E., Wei, G.-W.: Evolutionary homology on cou- pled dynamical systems with applications to protein flexibility analysis. Journal of applied and computational topology 4(4), 481–507 (2020)

2020
[42]

Computational and Mathematical Biophysics 3(1) (2015)

Cang, Z., Mu, L., Wu, K., Opron, K., Xia, K., Wei, G.-W.: A topologi- cal approach for protein classification. Computational and Mathematical Biophysics 3(1) (2015)

2015
[43]

The American Mathematical Monthly 109(5), 409–442 (2002) 64 CONTENTS

Cantarella, J., DeTurck, D., Gluck, H.: Vector calculus and the topology of domains in 3-space. The American Mathematical Monthly 109(5), 409–442 (2002) 64 CONTENTS

2002
[44]

Bulletin of the American Mathematical Society 46(2), 255–308 (2009)

Carlsson, G.: Topology and data. Bulletin of the American Mathematical Society 46(2), 255–308 (2009)

2009
[45]

Foundations of computa- tional mathematics 10, 367–405 (2010)

Carlsson, G., De Silva, V.: Zigzag persistence. Foundations of computa- tional mathematics 10, 367–405 (2010)

2010
[46]

In: Proceedings of the Twenty-third Annual Symposium on Computational Geometry, pp

Carlsson, G., Zomorodian, A.: The theory of multidimensional per- sistence. In: Proceedings of the Twenty-third Annual Symposium on Computational Geometry, pp. 184–193 (2007)

2007
[47]

In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp

Carlsson, G., De Silva, V., Morozov, D.: Zigzag persistent homology and real-valued functions. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp. 247–256 (2009)

2009
[48]

In: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp

Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.: Persistence barcodes for shapes. In: Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, pp. 124–135 (2004)

2004
[49]

International journal of computer vision 76, 1–12 (2008)

Carlsson, G., Ishkhanov, T., De Silva, V., Zomorodian, A.: On the local behavior of spaces of natural images. International journal of computer vision 76, 1–12 (2008)

2008
[50]

Algebraic & Geometric Topology19(2), 657–700 (2019)

Carlsson, G., De Silva, V., Kaliˇ snik, S., Morozov, D.: Parametrized homology via zigzag persistence. Algebraic & Geometric Topology19(2), 657–700 (2019)

2019
[51]

In: GUDHI User and Reference Manual, (2025)

Carri` ere, M.: Cover complex. In: GUDHI User and Reference Manual, (2025)

2025
[52]

Advances in Neural Information Processing Systems 33, 22432–22444 (2020)

Carriere, M., Blumberg, A.: Multiparameter persistence image for topo- logical machine learning. Advances in Neural Information Processing Systems 33, 22432–22444 (2020)

2020
[53]

In: International Conference on Machine Learning, pp

Carriere, M., Cuturi, M., Oudot, S.: Sliced Wasserstein kernel for persis- tence diagrams. In: International Conference on Machine Learning, pp. 664–673 (2017). PMLR

2017
[54]

Proceedings of the National Academy of Sciences 110(46), 18566–18571 (2013)

Chan, J.M., Carlsson, G., Rabadan, R.: Topology of viral evolution. Proceedings of the National Academy of Sciences 110(46), 18566–18571 (2013)

2013
[55]

In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp

Chazal, F., Fasy, B.T., Lecci, F., Rinaldo, A., Wasserman, L.: Stochastic convergence of persistence landscapes and silhouettes. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 474–483 (2014)

2014
[56]

Chazal, F., De Silva, V., Glisse, M., Oudot, S.: The Structure and CONTENTS 65 Stability of Persistence Modules vol. 10. Springer, (2016)

2016
[57]

Journal of Machine Learning Research 18(159), 1–40 (2018)

Chazal, F., Fasy, B.T., Lecci, F., Michel, B., Rinaldo, A., Wasserman, L.: Robust topological inference: Distance to a measure and kernel distance. Journal of Machine Learning Research 18(159), 1–40 (2018)

2018
[58]

Nature Machine Intelligence 6(7), 799–810 (2024)

Chen, D., Liu, J., Wei, G.-W.: Multiscale topology-enabled structure-to- sequence transformer for protein–ligand interaction predictions. Nature Machine Intelligence 6(7), 799–810 (2024)

2024
[59]

The journal of physical chemistry letters 14(4), 954–964 (2023)

Chen, D., Liu, J., Wu, J., Wei, G.-W., Pan, F., Yau, S.-T.: Path topology in molecular and materials sciences. The journal of physical chemistry letters 14(4), 954–964 (2023)

2023
[60]

Foundations of Data Science 5(4), 558–588 (2023)

Chen, D., Liu, J., Wu, J., Wei, G.-W.: Persistent hyperdigraph homology and persistent hyperdigraph Laplacians. Foundations of Data Science 5(4), 558–588 (2023)

2023
[61]

IEEE Transactions on Visualization and Computer Graphics 14(4), 848–862 (2008)

Chen, G., Mischaikow, K., Laramee, R.S., Zhang, E.: Efficient Morse decompositions of vector fields. IEEE Transactions on Visualization and Computer Graphics 14(4), 848–862 (2008)

2008
[62]

Chen, J., Wei, G.-W.: Omicron BA. 2 (B. 1.1. 529.2): high potential for becoming the next dominant variant. The journal of physical chemistry letters 13(17), 3840–3849 (2022)

2022
[63]

Journal of molecular biology 432(19), 5212– 5226 (2020)

Chen, J., Wang, R., Wang, M., Wei, G.-W.: Mutations strengthened SARS-CoV-2 infectivity. Journal of molecular biology 432(19), 5212– 5226 (2020)

2020
[64]

Discrete and continuous dynamical systems

Chen, J., Zhao, R., Tong, Y., Wei, G.-W.: Evolutionary de Rham-Hodge method. Discrete and continuous dynamical systems. Series B 26(7), 3785 (2021)

2021
[66]

Computers in biology and medicine 164, 107258 (2023)

Chen, J., Woldring, D.R., Huang, F., Huang, X., Wei, G.-W.: Topological deep learning based deep mutational scanning. Computers in biology and medicine 164, 107258 (2023)

2023
[67]

In: International Conference on Machine Learning, pp

Chen, Y., Segovia, I., Gel, Y.R.: Z-GCNETs: time zigzags at graph convolutional networks for time series forecasting. In: International Conference on Machine Learning, pp. 1684–1694 (2021). PMLR 66 CONTENTS

2021
[68]

IEEE transactions on pattern analysis and machine intelligence 42(1), 192–202 (2018)

Chevyrev, I., Nanda, V., Oberhauser, H.: Persistence paths and signa- ture features in topological data analysis. IEEE transactions on pattern analysis and machine intelligence 42(1), 192–202 (2018)

2018
[69]

In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp

Chowdhury, S., M´ emoli, F.: Persistent path homology of directed networks. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1152–1169 (2018). SIAM

2018
[70]

Advances in Computational Mathematics 48(1), 6 (2022)

Chung, Y.-M., Lawson, A.: Persistence curves: A canonical frame- work for summarizing persistence diagrams. Advances in Computational Mathematics 48(1), 6 (2022)

2022
[71]

SIAM, (2002)

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, (2002)

2002
[72]

IEEE transactions on pattern analysis and machine intelligence 44(12), 8766–8778 (2020)

Clough, J.R., Byrne, N., Oksuz, I., Zimmer, V.A., Schnabel, J.A., King, A.P.: A topological loss function for deep-learning based image segmen- tation using persistent homology. IEEE transactions on pattern analysis and machine intelligence 44(12), 8766–8778 (2020)

2020
[73]

In: Proceedings of the Twenty-first Annual Symposium on Computational Geometry, pp

Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. In: Proceedings of the Twenty-first Annual Symposium on Computational Geometry, pp. 263–271 (2005)

2005
[74]

Foundations of Computational Mathematics 9(1), 79–103 (2009)

Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Extending persistence using Poincar´ e and Lefschetz duality. Foundations of Computational Mathematics 9(1), 79–103 (2009)

2009
[75]

Computers & Graphics 28(6), 881–894 (2004)

Collins, A., Zomorodian, A., Carlsson, G., Guibas, L.J.: A barcode shape descriptor for curve point cloud data. Computers & Graphics 28(6), 881–894 (2004)

2004
[76]

Conley, C.C.: Isolated Invariant Sets and the Morse Index vol. 38. American Mathematical Soc., (1978)

1978
[77]

Mathematics 10(17), 3086 (2022)

Conti, F., Moroni, D., Pascali, M.A.: A topological machine learning pipeline for classification. Mathematics 10(17), 3086 (2022)

2022
[78]

arXiv preprint arXiv:2501.02581 (2025)

Cooperband, Z., Ghrist, R.: Unified origami kinematics via cosheaf homology. arXiv preprint arXiv:2501.02581 (2025)

2025 arXiv
[79]

arXiv preprint arXiv:2311.12946 (2023)

Cooperband, Z., Ghrist, R., Hansen, J.: A cosheaf theory of recipro- cal figures: Planar and higher genus graphic statics. arXiv preprint arXiv:2311.12946 (2023)

2023 arXiv
[80]

Computers in CONTENTS 67 biology and medicine 175, 108497 (2024)

Cottrell, S., Hozumi, Y., Wei, G.-W.: K-nearest-neighbors induced topo- logical PCA for single cell RNA-sequence data analysis. Computers in CONTENTS 67 biology and medicine 175, 108497 (2024)

2024
[81]

Journal of chemical information and modeling 64(7), 2405–2420 (2023)

Cottrell, S., Wang, R., Wei, G.-W.: PLPCA: persistent Laplacian- enhanced PCA for microarray data analysis. Journal of chemical information and modeling 64(7), 2405–2420 (2023)

2023
[82]

Crowell, R.H., Fox, R.H.: Introduction to Knot Theory vol. 57. Springer, (2012)

2012
[83]

University of Penn- sylvania, (2014)

Curry, J.M.: Sheaves, Cosheaves and Applications. University of Penn- sylvania, (2014)

2014
[84]

Dabaghian, Y., M´ emoli, F., Frank, L., Carlsson, G.: A topological paradigm for hippocampal spatial map formation using persistent homol- ogy (2012)

2012
[85]

Nucleic acids research, 976 (2016)

Dabrowski-Tumanski, P., Jarmolinska, A.I., Niemyska, W., Rawdon, E.J., Millett, K.C., Sulkowska, J.I.: Linkprot: a database collecting information about biological links. Nucleic acids research, 976 (2016)

2016
[86]

Briefings in Bioinformatics 22(3), 196 (2021)

Dabrowski-Tumanski, P., Rubach, P., Niemyska, W., Gren, B.A., Sulkowska, J.I.: Topoly: Python package to analyze topology of polymers. Briefings in Bioinformatics 22(3), 196 (2021)

2021
[87]

Cahiers de Topologie et G´ eom´ etrie Diff´ erentielle Cat´ egoriques31(3), 229–243 (1990)

Dawson, R.J.M.: Homology of weighted simplicial complexes. Cahiers de Topologie et G´ eom´ etrie Diff´ erentielle Cat´ egoriques31(3), 229–243 (1990)

1990
[88]

In: PBG, pp

De Silva, V., Carlsson, G.E.: Topological estimation using witness complexes. In: PBG, pp. 157–166 (2004)

2004
[89]

In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp

De Silva, V., Vejdemo-Johansson, M.: Persistent cohomology and circular coordinates. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, pp. 227–236 (2009)

2009
[90]

Inverse Problems 27(12), 124003 (2011)

De Silva, V., Morozov, D., Vejdemo-Johansson, M.: Dualities in persis- tent (co)homology. Inverse Problems 27(12), 124003 (2011)

2011
[91]

Izvestia Akademii Nauk SSSR 7, 793– 800 (1934)

Delaunay, B.: Sur la sphere vide. Izvestia Akademii Nauk SSSR 7, 793– 800 (1934)

1934
[92]

Deligne, P.: La conjecture de weil. i. Publications Math´ ematiques de l’Institut des Hautes ´Etudes Scientifiques 43, 273–307 (1974)

1974
[93]

Publications Math´ ematiques de l’IH´ES 52, 137–252 (1980) 68 CONTENTS

Deligne, P.: La conjecture de weil: Ii. Publications Math´ ematiques de l’IH´ES 52, 137–252 (1980) 68 CONTENTS

1980
[94]

In: ACM SIGGRAPH 2006 Courses, pp

Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for com- putational modeling. In: ACM SIGGRAPH 2006 Courses, pp. 39–54 (2006)

2006
[95]

In: 37th International Symposium on Computational Geom- etry (2021)

Dey, T.K., Hou, T.: Computing zigzag persistence on graphs in near- linear time. In: 37th International Symposium on Computational Geom- etry (2021)

2021
[96]

arXiv preprint arXiv:2112.02352 (2021)

Dey, T.K., Hou, T.: Updating zigzag persistence and maintaining rep- resentatives over changing filtrations. arXiv preprint arXiv:2112.02352 (2021)

2021 arXiv
[97]

In: 30th Annual European Symposium on Algorithms (ESA 2022)

Dey, T.K., Hou, T.: Fast computation of zigzag persistence. In: 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), vol. 244 (2022). pp. 43:1–43:15

2022
[98]

Discrete & Computational Geometry 71(1), 67–94 (2024)

Dey, T.K., Kim, W., M´ emoli, F.: Computing generalized rank invari- ant for 2-parameter persistence modules via zigzag persistence and its applications. Discrete & Computational Geometry 71(1), 67–94 (2024)

2024
[99]

Discrete & Computational Geometry 68(4), 1102–1132 (2022)

Dey, T.K., Li, T., Wang, Y.: An efficient algorithm for 1-dimensional (persistent) path homology. Discrete & Computational Geometry 68(4), 1102–1132 (2022)

2022
[100]

In: 36th International Sympo- sium on Computational Geometry (SoCG 2020)

Dey, T.K., Mrozek, M., Slechta, R.: Persistence of the Conley index in combinatorial dynamical systems. In: 36th International Sympo- sium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), vol. 164 (2020). pp. 37:1–37:17

2020
[101]

SIAM Journal on Applied Dynam- ical Systems 21(2), 817–839 (2022)

Dey, T.K., Mrozek, M., Slechta, R.: Persistence of Conley-Morse graphs in combinatorial dynamical systems. SIAM Journal on Applied Dynam- ical Systems 21(2), 817–839 (2022)

2022
[102]

SIAM Journal on Applied Dynamical Systems 18(1), 510–530 (2019)

Dey, T.K., Juda, M., Kapela, T., Kubica, J., Lipi´ nski, M., Mrozek, M.: Persistent homology of Morse decompositions in combinatorial dynamics. SIAM Journal on Applied Dynamical Systems 18(1), 510–530 (2019)

2019
[103]

Cambridge University Press, (2022)

Dey, T.K., Wang, Y.: Computational Topology for Data Analysis. Cambridge University Press, (2022)

2022
[104]

Journal of Algebra 653, 156–199 (2024) CONTENTS 69

Di, S., Ivanov, S.O., Mukoseev, L., Zhang, M.: On the path homology of cayley digraphs and covering digraphs. Journal of Algebra 653, 156–199 (2024) CONTENTS 69

2024
[105]

In: Image Analysis and Processing-ICIAP 2015: 18th Inter- national Conference, Genoa, Italy, September 7-11, 2015, Proceedings, Part I 18, pp

Di Fabio, B., Ferri, M.: Comparing persistence diagrams through com- plex vectors. In: Image Analysis and Processing-ICIAP 2015: 18th Inter- national Conference, Genoa, Italy, September 7-11, 2015, Proceedings, Part I 18, pp. 294–305 (2015). Springer

2015
[106]

Proceedings of the Royal Society of London

Dirac, P.A.M.: The quantum theory of the electron. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117(778), 610–624 (1928)

1928
[107]

Journal of Computational Geometry 10(2), 1–31 (2019)

Divol, V., Chazal, F.: The density of expected persistence diagrams and its kernel based estimation. Journal of Computational Geometry 10(2), 1–31 (2019). https://doi.org/10.20382/jocg.v10i2a7

2019 doi
[108]

Journal of Applied and Computational Topology 5(1), 1–53 (2021)

Divol, V., Lacombe, T.: Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport. Journal of Applied and Computational Topology 5(1), 1–53 (2021). https://doi. org/10.1007/s41468-020-00061-z

2021 doi
[109]

Physica D: Nonlinear Phenomena 334, 60–81 (2016)

D lotko, P., Wanner, T.: Topological microstructure analysis using persis- tence landscapes. Physica D: Nonlinear Phenomena 334, 60–81 (2016)

2016
[110]

Briefings in Bioinformatics 25(2), 054 (2024)

Du, H., Wei, G.-W., Hou, T.: Multiscale topology in interactomic net- work: from transcriptome to antiaddiction drug repurposing. Briefings in Bioinformatics 25(2), 054 (2024)

2024
[111]

Commentarii Mathematici Helvetici 17(1), 240–255 (1944)

Eckmann, B.: Harmonische funktionen und randwertaufgaben in einem komplex. Commentarii Mathematici Helvetici 17(1), 240–255 (1944)

1944
[112]

Discrete & computational geometry 28, 511–533 (2002)

Edelsbrunner, Letscher, Zomorodian: Topological persistence and sim- plification. Discrete & computational geometry 28, 511–533 (2002)

2002
[113]

In: Tessellations in the Sciences: Virtues, Techniques and Applications of Geometric Tilings, (2011)

Edelsbrunner, H.: Alpha shapes - a survey. In: Tessellations in the Sciences: Virtues, Techniques and Applications of Geometric Tilings, (2011)

2011
[114]

American Mathematical Soc., (2010)

Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Soc., (2010)

2010
[115]

eScholarship, University of California, (2013)

Edelsbrunner, H., Morozov, D.: Persistent Homology: Theory and Prac- tice. eScholarship, University of California, (2013)

2013
[116]

: Persistent homology - a survey

Edelsbrunner, H., Harer, J., et al. : Persistent homology - a survey. Contemporary mathematics 453(26), 257–282 (2008)

2008
[117]

arXiv preprint arXiv:2503.12919 (2025) 70 CONTENTS

Einzade, A., Thanou, D., Malliaros, F.D., Giraldo, J.H.: Cosmos: Con- tinuous simplicial neural networks. arXiv preprint arXiv:2503.12919 (2025) 70 CONTENTS

2025
[118]

Linear algebra and its applications 436(9), 3373–3391 (2012)

Estrada, E.: Path Laplacian matrices: introduction and application to the analysis of consensus in networks. Linear algebra and its applications 436(9), 3373–3391 (2012)

2012
[119]

In: Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp

Fabri, A., Pion, S.: CGAL: the computational geometry algorithms library. In: Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 538– 539 (2009)

2009
[120]

The Annals of Statis- tics 42(6), 2301–2339 (2014)

Fasy, B.T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., Singh, A.: Confidence sets for persistence diagrams. The Annals of Statis- tics 42(6), 2301–2339 (2014). https://doi.org/10.1214/14-AOS1252

2014 doi
[121]

arXiv preprint arXiv:1411.1830 (2014)

Fasy, B.T., Kim, J., Lecci, F., Maria, C.: Introduction to the R package TDA. arXiv preprint arXiv:1411.1830 (2014)

2014 arXiv
[122]

Journal of Computational Biophysics and Chemistry, 1–14 (2024)

Feng, H., Shen, L., Liu, J., Wei, G.-W.: Mayer-homology learning pre- diction of protein-ligand binding affinities. Journal of Computational Biophysics and Chemistry, 1–14 (2024)

2024
[123]

Proceedings of the National Academy of Sciences 121(40), 2412220121 (2024)

Feng, L., Gong, H., Zhang, S., Liu, X., Wang, Y., Che, J., Dong, A., Griffin, C.H., Gragnoli, C., Wu, J., Yau, S.-T., Wu, R.: Hypernetwork modeling and topology of high-order interactions for complex systems. Proceedings of the National Academy of Sciences 121(40), 2412220121 (2024)

2024
[124]

Physics reports 1101, 1–65 (2025)

Feng, L., Yang, D., Wu, S., Xue, C., Sang, M., Liu, X., Che, J., Wu, J., Gragnoli, C., Griffin, C., Wang, C., Yau, S.-T., Wu, R.: Network modeling and topology of aging. Physics reports 1101, 1–65 (2025)

2025
[125]

Ferri, M., Landi, C.: Representing size functions by complex polynomials. Proc. Math. Met. in Pattern Recognition 9, 16–19 (1999)

1999
[126]

Commu- nications on Pure and Applied Mathematics 8(4), 551–590 (1955)

Friedrichs, K.O.: Differential forms on Riemannian manifolds. Commu- nications on Pure and Applied Mathematics 8(4), 551–590 (1955)

1955
[127]

Bulletin of the Belgian Mathematical Society- Simon Stevin 6(3), 455–464 (1999)

Frosini, P., Mulazzani, M.: Size homotopy groups for computation of natural size distances. Bulletin of the Belgian Mathematical Society- Simon Stevin 6(3), 455–464 (1999)

1999
[128]

Japan Journal of Industrial and Applied Mathematics 32, 1–17 (2015)

Gameiro, M., Hiraoka, Y., Izumi, S., Kramar, M., Mischaikow, K., Nanda, V.: A topological measurement of protein compressibility. Japan Journal of Industrial and Applied Mathematics 32, 1–17 (2015)

2015
[129]

Gauss, C.F.: Zur Mathematischen Theorie der Electrodynamischen Wirkungen, pp. 601–630. Springer, Berlin, Heidelberg (1877) CONTENTS 71
[130]

Advances in neural information processing systems 24 (2011)

Ge, X., Safa, I., Belkin, M., Wang, Y.: Data skeletonization via Reeb graphs. Advances in neural information processing systems 24 (2011)

2011
[131]

International journal for numerical methods in engineering 79(11), 1309–1331 (2009)

Geuzaine, C., Remacle, J.-F.: Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. International journal for numerical methods in engineering 79(11), 1309–1331 (2009)

2009
[132]

Bulletin of the American Mathematical Society 45(1), 61–75 (2008)

Ghrist, R.: Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society 45(1), 61–75 (2008)

2008
[133]

Ghrist, R.W.: Elementary Applied Topology vol. 1. Createspace Seattle, (2014)

2014
[134]

PhD thesis, Citeseer (2002)

Goldberg, T.E.: Combinatorial Laplacians of simplicial complexes. PhD thesis, Citeseer (2002)

2002
[135]

Physics of Life Reviews (2024)

Gong, H., Wang, H., Wang, Y., Zhang, S., Liu, X., Che, J., Wu, S., Wu, J., Sun, X., Zhang, S., Yau, S.-T., Wu, R.: Topological change of soil microbiota networks for forest resilience under global warming. Physics of Life Reviews (2024)

2024
[136]

Foundations of data science (Springfield, Mo.) 4(2), 165 (2022)

Grbi´ c, J., Wu, J., Xia, K., Wei, G.-W.: Aspects of topological approaches for data science. Foundations of data science (Springfield, Mo.) 4(2), 165 (2022)

2022
[137]

arXiv preprint arXiv:1207.2834 (2012)

Grigor’yan, A., Lin, Y., Muranov, Y., Yau, S.-T.: Homologies of path complexes and digraphs. arXiv preprint arXiv:1207.2834 (2012)

2012 arXiv
[138]

Notices of the International Consortium of Chinese Mathematicians 10(2), 61–124 (2022)

Grigor’yan, A.: Advances in path homology theory of digraphs. Notices of the International Consortium of Chinese Mathematicians 10(2), 61–124 (2022)

2022
[139]

Journal of Homotopy and Related Struc- tures 11(2), 209–230 (2016)

Grigor’yan, A., Muranov, Y., Yau, S.-T.: On a cohomology of digraphs and hochschild cohomology. Journal of Homotopy and Related Struc- tures 11(2), 209–230 (2016)

2016
[140]

Pure and Applied Mathematics Quarterly 10(4), 619–674 (2023)

Grigor’yan, A., Lin, Y., Muranov, Y., Yau, S.-T.: Homotopy theory for digraphs. Pure and Applied Mathematics Quarterly 10(4), 619–674 (2023)

2023
[141]

arXiv preprint arXiv:2311.06870 (2023)

G¨ ulen, A.B., M´ emoli, F., Wan, Z.: Orthogonal m¨ obius inversion and Grassmannian persistence diagrams. arXiv preprint arXiv:2311.06870 (2023)

2023 arXiv
[142]

arXiv preprint arXiv:2504.06077 (2025) 72 CONTENTS

G¨ ulen, A.B., M´ emoli, F., Wan, Z.: Grassmannian persistence dia- grams: Special properties in the 1-parameter setting. arXiv preprint arXiv:2504.06077 (2025) 72 CONTENTS

2025 arXiv
[143]

In: 39th International Sympo- sium on Computational Geometry (SoCG 2023)

G¨ ulen, A.B., M´ emoli, F., Wan, Z., Wang, Y.: A Generalization of the Persistent Laplacian to Simplicial Maps. In: 39th International Sympo- sium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), vol. 258 (2023). pp. 37:1–37:17

2023
[144]

In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp

Gundert, A., Szedl´ ak, M.: Higher dimensional Cheeger inequalities. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 181–188 (2014)

2014
[145]

The Visual Computer 28, 959–969 (2012)

G¨ unther, D., Reininghaus, J., Wagner, H., Hotz, I.: Efficient computation of 3D Morse–Smale complexes and persistent homology using discrete Morse theory. The Visual Computer 28, 959–969 (2012)

2012
[146]

Japan Journal of Industrial and Applied Mathematics 40(1), 41–93 (2023)

Hang, H., Mio, W.: Correspondence modules and persistence sheaves: a unifying perspective on one-parameter persistent homology. Japan Journal of Industrial and Applied Mathematics 40(1), 41–93 (2023)

2023
[147]

PhD thesis, University of Pennsylvania (2020)

Hansen, J.: Laplacians of cellular sheaves: Theory and applications. PhD thesis, University of Pennsylvania (2020)

2020
[148]

Journal of Applied and Computational Topology 3(4), 315–358 (2019)

Hansen, J., Ghrist, R.: Toward a spectral theory of cellular sheaves. Journal of Applied and Computational Topology 3(4), 315–358 (2019)

2019
[149]

Quantum 6, 873 (2022)

Hayakawa, R.: Quantum algorithm for persistent Betti numbers and topological data analysis. Quantum 6, 873 (2022)

2022
[150]

AIMS Mathematics 10(1), 1384–1406 (2025)

He, Y., Liu, J.: Multi-scale hochschild spectral analysis on graph data. AIMS Mathematics 10(1), 1384–1406 (2025)

2025
[151]

Frontiers in Artificial Intelligence 4, 681108 (2021)

Hensel, F., Moor, M., Rieck, B.: A survey of topological machine learning methods. Frontiers in Artificial Intelligence 4, 681108 (2021)

2021
[152]

arXiv preprint arXiv:1606.00199 (2016)

Henselman, G., Ghrist, R.: Matroid filtrations and computational per- sistent homology. arXiv preprint arXiv:1606.00199 (2016)

2016 arXiv
[153]

arXiv preprint arXiv:2505.10467 (2025)

Hern´ andez-Garc´ ıa, P., Serrano, D.H., G´ omez, D.S.: From persistence to resilience: New betti numbers for analyzing robustness in simplicial complex networks. arXiv preprint arXiv:2505.10467 (2025)

2025 arXiv
[154]

Physics reports 519(3), 97–125 (2012)

Holme, P., Saram¨ aki, J.: Temporal networks. Physics reports 519(3), 97–125 (2012)

2012
[155]

Advances in Mathematics 244, 303–336 (2013)

Horak, D., Jost, J.: Spectra of combinatorial Laplace operators on simplicial complexes. Advances in Mathematics 244, 303–336 (2013)

2013
[156]

Journal of Statistical Mechanics: Theory and Experiment 2009(03), 03034 (2009) CONTENTS 73

Horak, D., Maleti´ c, S., Rajkovi´ c, M.: Persistent homology of complex networks. Journal of Statistical Mechanics: Theory and Experiment 2009(03), 03034 (2009) CONTENTS 73

2009
[157]

arXiv preprint arXiv:2412.20202 (2024)

Hozumi, Y., Wei, G.-W.: Revealing the shape of genome space via k-mer topology. arXiv preprint arXiv:2412.20202 (2024)

2024 arXiv
[158]

Journal of the London Mathematical Society 109(1), 12812 (2024)

Ivanov, S.O., Pavutnitskiy, F.: Simplicial approach to path homology of quivers, marked categories, groups and algebras. Journal of the London Mathematical Society 109(1), 12812 (2024)

2024
[159]

In: GUDHI User and Reference Manual, (2025)

Jamin, C.: Tangential complex. In: GUDHI User and Reference Manual, (2025)

2025
[160]

Nucleic acids research 43(D1), 306–314 (2015)

Jamroz, M., Niemyska, W., Rawdon, E.J., Stasiak, A., Millett, K.C., Su lkowski, P., Sulkowska, J.I.: Knotprot: a database of proteins with knots and slipknots. Nucleic acids research 43(D1), 306–314 (2015)

2015
[161]

npj computational materials 7(1), 28 (2021)

Jiang, Y., Chen, D., Chen, X., Li, T., Wei, G.-W., Pan, F.: Topolog- ical representations of crystalline compounds for the machine-learning prediction of materials properties. npj computational materials 7(1), 28 (2021)

2021
[162]

Journal of Physics: Complexity 6(2), 025014 (2025)

Jones, B., Wei, G.-W.: Khovanov Laplacian and Khovanov Dirac for knots and links. Journal of Physics: Complexity 6(2), 025014 (2025)

2025
[163]

Foundations of Data Science 7(3), 737–758 (2025)

Jones, B., Wei, G.-W.: Persistent directed flag Laplacian. Foundations of Data Science 7(3), 737–758 (2025)

2025
[164]

In: Fields Medallists’ Lectures, pp

Jones, V.F.: A polynomial invariant for knots via von Neumann algebras. In: Fields Medallists’ Lectures, pp. 448–458. World Scientific, (1997)

1997
[165]

Jonsson, J.: Simplicial Complexes of Graphs vol. 1928. Springer, (2008)

2008
[166]

arXiv preprint arXiv:2005.12692 (2020)

Kaji, S., Sudo, T., Ahara, K.: Cubical Ripser: Software for comput- ing persistent homology of image and volume data. arXiv preprint arXiv:2005.12692 (2020)

2020 arXiv
[167]

Foundations of Computational Mathematics 19(1), 101–129 (2019)

Kaliˇ snik, S.: Tropical coordinates on the space of persistence barcodes. Foundations of Computational Mathematics 19(1), 101–129 (2019)

2019
[168]

Scientific reports 9(1), 13817 (2019)

Kannan, H., Saucan, E., Roy, I., Samal, A.: Persistent homology of unweighted complex networks via discrete Morse theory. Scientific reports 9(1), 13817 (2019)

2019
[169]

Master’s thesis, Middle East Technical University (Turkey) (2021)

Karag¨ uler, D.: A survey on multidimensional persistence theory. Master’s thesis, Middle East Technical University (Turkey) (2021)

2021
[170]

Expert Systems with Applications 183, 115326 (2021)

Karan, A., Kaygun, A.: Time series classification via topological data analysis. Expert Systems with Applications 183, 115326 (2021). https: //doi.org/10.1016/j.eswa.2021.115326 74 CONTENTS

2021
[171]

In: ACM SIGGRAPH 2023 Conference Proceedings, pp

Keros, A., Subr, K.: Spectral coarsening with Hodge Laplacians. In: ACM SIGGRAPH 2023 Conference Proceedings, pp. 1–11 (2023)

2023
[172]

Khovanov, M.: A categorification of the Jones polynomial (2000)

2000
[173]

Advances in Neural Information Processing Systems 33, 15965–15977 (2020)

Kim, K., Kim, J., Zaheer, M., Kim, J., Chazal, F., Wasserman, L.: Pllay: Efficient topological layer based on persistent landscapes. Advances in Neural Information Processing Systems 33, 15965–15977 (2020)

2020
[174]

Journal of Applied and Computational Topology 5(4), 533–581 (2021)

Kim, W., M´ emoli, F.: Generalized persistence diagrams for persistence modules over posets. Journal of Applied and Computational Topology 5(4), 533–581 (2021)

2021
[175]

Notices of the American Mathematical Society 70(08) (2023)

Kim, W., M´ emoli, F.: Persistence over posets. Notices of the American Mathematical Society 70(08) (2023)

2023
[176]

Annalen der Physik 148(12), 497–508 (1847)

Kirchhoff, G.: Ueber die aufl¨ osung der gleichungen, auf welche man bei der untersuchung der linearen vertheilung galvanischer str¨ ome gef¨ uhrt wird. Annalen der Physik 148(12), 497–508 (1847)
[177]

arXiv preprint arXiv:1803.06788 (2018)

Knill, O.: The cohomology for Wu characteristics. arXiv preprint arXiv:1803.06788 (2018)

2018 arXiv
[178]

Statistical applications in genetics and molecular biology 15(1), 19–38 (2016)

Kovacev-Nikolic, V., Bubenik, P., Nikoli´ c, D., Heo, G.: Using persistent homology and dynamical distances to analyze protein binding. Statistical applications in genetics and molecular biology 15(1), 19–38 (2016)

2016
[179]

Chaos, Solitons & Fractals 177, 114296 (2023)

Krishnagopal, S., Bianconi, G.: Topology and dynamics of higher-order multiplex networks. Chaos, Solitons & Fractals 177, 114296 (2023)

2023
[180]

In: International Conference on Machine Learning, pp

Kusano, G., Hiraoka, Y., Fukumizu, K.: Persistence weighted Gaus- sian kernel for topological data analysis. In: International Conference on Machine Learning, pp. 2004–2013 (2016). PMLR

2016
[181]

Gordon & Breach, (1969)

Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompress- ible Flow. Gordon & Breach, (1969)

1969
[182]

Foundations of Data Science 7(2), 536–567 (2025)

Le, M.Q., Taylor, D.: Persistent homology with k-nearest-neighbor filtra- tions reveals topological convergence of pagerank. Foundations of Data Science 7(2), 536–567 (2025)

2025
[183]

Advances in neural information processing systems 31 (2018)

Le, T., Yamada, M.: Persistence Fisher kernel: A Riemannian mani- fold kernel for persistence diagrams. Advances in neural information processing systems 31 (2018)

2018
[184]

Nature communications 8(1), 1–8 (2017) CONTENTS 75

Lee, Y., Barthel, S.D., D lotko, P., Moosavi, S.M., Hess, K., Smit, B.: Quantifying similarity of pore-geometry in nanoporous materials. Nature communications 8(1), 1–8 (2017) CONTENTS 75

2017
[185]

Leray, J.: L’anneau d’homologie d’une repr´ esentation. C.R. Acad. Sci. Paris (222), 1366–1368 (1946)

1946
[186]

Li, J., Muranov, Y., Wu, J., Yau, S.-T.: On singular homology theories of digraphs and quivers (2024)

2024
[187]

arXiv preprint arXiv:2411.18955 (2024)

Li, J., Muranov, Y., Wu, J., Yau, S.-T.: Primitive path homology. arXiv preprint arXiv:2411.18955 (2024)

2024 arXiv
[188]

Homology, Homotopy & Applications 19(2) (2017)

LI, J., VERSHININ, V., WU, J.: Twisted simplicial groups and twisted homology of categories. Homology, Homotopy & Applications 19(2) (2017)

2017
[189]

Journal of the American Chemical Society 116(24), 11189–11190 (1994)

Liang, C., Mislow, K.: Knots in proteins. Journal of the American Chemical Society 116(24), 11189–11190 (1994)

1994
[190]

Lieutier, A.: Talk: Persistent harmonic forms (2014)

2014
[191]

Siam Review 62(3), 685–715 (2020)

Lim, L.-H.: Hodge Laplacians on graphs. Siam Review 62(3), 685–715 (2020)

2020
[192]

arXiv preprint arXiv:1910.09891 (2019)

Lin, Y., Ren, S., Wang, C., Wu, J.: Weighted path homology of weighted digraphs and persistence. arXiv preprint arXiv:1910.09891 (2019)

2019 arXiv
[193]

arXiv preprint arXiv:2311.16322 (2023)

Liu, J., Chen, D., Wei, G.-W.: Interaction homotopy and interaction homology. arXiv preprint arXiv:2311.16322 (2023)

2023 arXiv
[194]

arXiv preprint arXiv:2404.11799 (2024)

Liu, J., Chen, D., Wei, G.-W.: Persistent interaction topology in data analysis. arXiv preprint arXiv:2404.11799 (2024)

2024 arXiv
[195]

Homology, Homotopy and Applications 26(2), 297–323 (2024)

Liu, J., Li, J., Wu, J.: The algebraic stability for persistent Laplacians. Homology, Homotopy and Applications 26(2), 297–323 (2024)

2024
[196]

Foundations of data science 6(2), 221–250 (2024)

Liu, J., Shen, L., Wei, G.-W.: ChatGPT for computational topology. Foundations of data science 6(2), 221–250 (2024)

2024
[197]

arXiv preprint arXiv:2409.18312 (2024)

Liu, J., Shen, L., Wei, G.-W.: Persistent Khovanov homology of tangles. arXiv preprint arXiv:2409.18312 (2024)

2024
[198]

arXiv preprint arXiv:2507.05452 (2025)

Liu, J., Shen, L., Chen, D., Wei, G.-W.: Topological sequence analysis of genomes: Delta complex approaches. arXiv preprint arXiv:2507.05452 (2025)

2025 arXiv
[199]

Journal of chem- ical information and modeling 63(3), 1066–1075 (2023)

Liu, R., Liu, X., Wu, J.: Persistent path-spectral (pps) based machine learning for protein–ligand binding affinity prediction. Journal of chem- ical information and modeling 63(3), 1066–1075 (2023)

2023
[200]

In: International Workshop on Interpretability of Machine Intelligence in Medical Image Computing, and Topological Data Analysis and Its Applications for Medical Data, pp

Liu, X., Xia, K.: Neighborhood complex based machine learning (NCML) 76 CONTENTS models for drug design. In: International Workshop on Interpretability of Machine Intelligence in Medical Image Computing, and Topological Data Analysis and Its Applications for Medical Data, pp. ...

2021
[201]

Briefings in Bioinformatics 22(5), 411 (2021)

Liu, X., Wang, X., Wu, J., Xia, K.: Hypergraph-based persistent coho- mology (HPC) for molecular representations in drug design. Briefings in Bioinformatics 22(5), 411 (2021)

2021
[202]

Briefings in Bioinformatics 22(5), 127 (2021)

Liu, X., Feng, H., Wu, J., Xia, K.: Persistent spectral hypergraph based machine learning (PSH-ML) for protein-ligand binding affinity prediction. Briefings in Bioinformatics 22(5), 127 (2021)

2021
[203]

PLoS computational biology 18(4), 1009943 (2022)

Liu, X., Feng, H., Wu, J., Xia, K.: Dowker complex based machine learn- ing (DCML) models for protein-ligand binding affinity prediction. PLoS computational biology 18(4), 1009943 (2022)

2022
[204]

Journal of chemical information and modeling 62(17), 3961–3969 (2022)

Liu, X., Feng, H., Wu, J., Xia, K.: Hom-complex-based machine learning (HCML) for the prediction of protein-protein binding affinity changes upon mutation. Journal of chemical information and modeling 62(17), 3961–3969 (2022)

2022
[205]

Foundations of Data Science 6(2), 172–194 (2024)

Liu, X., Feng, H., Wu, J., Xia, K.: Computing hypergraph homology. Foundations of Data Science 6(2), 172–194 (2024)

2024
[206]

arXiv preprint arXiv:2412.02806 (2024)

Liu, X., Liu, R., Li, J., Wu, R., Wu, J.: Intcomplex for high-order interactions. arXiv preprint arXiv:2412.02806 (2024)

2024 arXiv
[207]

arXiv preprint arXiv:2503.00175 (2025)

Liu, X., Su, Z., Shi, Y., Tong, Y., Wang, G., Wei, G.-W.: Man- ifold topological deep learning for biomedical data. arXiv preprint arXiv:2503.00175 (2025)

2025 arXiv
[208]

Nature communications 7(1), 10138 (2016)

Lloyd, S., Garnerone, S., Zanardi, P.: Quantum algorithms for topologi- cal and geometric analysis of data. Nature communications 7(1), 10138 (2016)

2016
[209]

Journal of Machine Learning Research 24(59), 1–35 (2023)

Love, E.R., Filippenko, B., Maroulas, V., Carlsson, G.: Topological con- volutional layers for deep learning. Journal of Machine Learning Research 24(59), 1–35 (2023)

2023
[210]

Algorithms 13(1), 19 (2020)

L¨ utgehetmann, D., Govc, D., Smith, J.P., Levi, R.: Computing persistent homology of directed flag complexes. Algorithms 13(1), 19 (2020)

2020
[211]

In: Mathematical Software–ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014

Maria, C., Boissonnat, J.-D., Glisse, M., Yvinec, M.: The GUDHI library: simplicial complexes and persistent homology. In: Mathematical Software–ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9, 2014. Proceedings 4, pp. 167–174 (2014). Springer CONTENTS 77

2014
[212]

Bayesian Analysis 17(3), 711–736 (2022)

Maroulas, V., Micucci, C.P., Nasrin, F.: Bayesian topological learning for classifying the structure of biological networks. Bayesian Analysis 17(3), 711–736 (2022). https://doi.org/10.1214/21-BA1270

2022 doi
[213]

Journal of Machine Learning Research 20(196), 1–49 (2019)

Maroulas, V., Mike, J.L., Oballe, C.: Nonparametric estimation of prob- ability density functions of random persistence diagrams. Journal of Machine Learning Research 20(196), 1–49 (2019)

2019
[214]

SIAM Journal on Mathematics of Data Science 2(1), 48–74 (2020)

Maroulas, V., Nasrin, F., Oballe, C.: A Bayesian framework for persistent homology. SIAM Journal on Mathematics of Data Science 2(1), 48–74 (2020). https://doi.org/10.1137/19M1268719

2020 doi
[215]

Annals of Mathematics 43(2), 370– 380 (1942)

Mayer, W.: A new homology theory. Annals of Mathematics 43(2), 370– 380 (1942)

1942
[216]

McCleary, J.: A User’s Guide to Spectral Sequences vol. 58. Cambridge University Press, (2001)

2001
[217]

SIAM Journal on Mathematics of Data Science 4(2), 858–884 (2022)

M´ emoli, F., Wan, Z., Wang, Y.: Persistent Laplacians: Properties, algo- rithms and implications. SIAM Journal on Mathematics of Data Science 4(2), 858–884 (2022)

2022
[218]

arXiv preprint arXiv:1708.04710 (2017)

Mendoza-Smith, R., Tanner, J.: Parallel multi-scale reduction of persis- tent homology filtrations. arXiv preprint arXiv:1708.04710 (2017)

2017 arXiv
[219]

Science advances 7(19), 5329 (2021)

Meng, Z., Xia, K.: Persistent spectral-based machine learning (PerSpect ML) for protein-ligand binding affinity prediction. Science advances 7(19), 5329 (2021)

2021
[220]

Scientific reports 10(1), 2079 (2020)

Meng, Z., Anand, D.V., Lu, Y., Wu, J., Xia, K.: Weighted persistent homology for biomolecular data analysis. Scientific reports 10(1), 2079 (2020)

2020
[221]

Inverse Problems 27(12), 124007 (2011)

Mileyko, Y., Mukherjee, S., Harer, J.: Probability measures on the space of persistence diagrams. Inverse Problems 27(12), 124007 (2011)

2011
[222]

American Journal of Physics 52(10), 948–950 (1984)

Miller, B.P.: Interpretations from Helmholtz’ theorem in classical elec- tromagnetism. American Journal of Physics 52(10), 948–950 (1984)

1984
[223]

Biochemical Society Transactions 41(2), 533–537 (2013)

Millett, K.C., Rawdon, E.J., Stasiak, A., Su lkowska, J.I.: Identifying knots in proteins. Biochemical Society Transactions 41(2), 533–537 (2013)

2013
[224]

Discrete & Computational Geometry 50, 330–353 (2013)

Mischaikow, K., Nanda, V.: Morse theory for filtrations and efficient com- putation of persistent homology. Discrete & Computational Geometry 50, 330–353 (2013)

2013
[225]

78 CONTENTS

Mischaikow, K., Mrozek, M., Zgliczy´ nski, P.: Conley Index Theory vol. 78 CONTENTS
[226]

Bio- physical Journal 123(17), 2781–2789 (2024)

Mitchell, E.C., Story, B., Boothe, D., Franaszczuk, P.J., Maroulas, V.: A topological deep learning framework for neural spike decoding. Bio- physical Journal 123(17), 2781–2789 (2024). https://doi.org/10.1016/j. bpj.2024.01.025

2024 doi
[227]

SIAM Journal on Applied Algebra and Geometry 3(2), 337–371 (2019)

Monod, A., Kaliˇ snik, S., Pati˜ no-Galindo, J.´A., Crawford, L.: Tropical sufficient statistics for persistent homology. SIAM Journal on Applied Algebra and Geometry 3(2), 337–371 (2019). https://doi.org/10.1137/ 17M1148037

2019
[228]

arXiv preprint arXiv:2409.12033 (2024)

Montagna, M., Scardapane, S., Telyatnikov, L.: Topological deep learn- ing with state-space models: A mamba approach for simplicial com- plexes. arXiv preprint arXiv:2409.12033 (2024)

2024 arXiv
[229]

Morozov, D.: Dionysus, a C++ library for computing persistent homol- ogy (2007)

2007
[230]

American Journal of Mathematics 78(1), 137–170 (1956)

Morrey, C.B.: A variational method in the theory of harmonic integrals, ii. American Journal of Mathematics 78(1), 137–170 (1956)

1956
[231]

Transactions of the American Mathematical Society 27(3), 345–396 (1925)

Morse, M.: Relations between the critical points of a real function of n independent variables. Transactions of the American Mathematical Society 27(3), 345–396 (1925)

1925
[232]

Electronic Journal of Statistics 9, 1173–1204 (2015)

Munch, E., Turner, K., Bendich, P., Mukherjee, S., Mattingly, J., Harer, J.: Probabilistic Fr´ echet means for time varying persistence diagrams. Electronic Journal of Statistics 9, 1173–1204 (2015)

2015
[233]

Journal of computer-aided molecular design 33, 71–82 (2019)

Nguyen, D.D., Cang, Z., Wu, K., Wang, M., Cao, Y., Wei, G.-W.: Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges. Journal of computer-aided molecular design 33, 71–82 (2019)

2019
[234]

Journal of computer-aided molecular design 34, 131–147 (2020)

Nguyen, D.D., Gao, K., Wang, M., Wei, G.-W.: Mathdl: mathematical deep learning for D3R Grand Challenge 4. Journal of computer-aided molecular design 34, 131–147 (2020)

2020
[235]

Cambridge university press, (2010)

Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge university press, (2010)

2010
[236]

Discrete and Con- tinuous Dynamical Systems - Series S 15(4), 797–817 (2022)

Oballe, C., Cherne, A., Boothe, D., Kerick, S., Franaszczuk, P.J., Maroulas, V.: Bayesian topological signal processing. Discrete and Con- tinuous Dynamical Systems - Series S 15(4), 797–817 (2022). https: //doi.org/10.3934/dcdss.2021084 CONTENTS 79

2022 doi
[237]

Ohtsuki, T.: Quantum Invariants: a Study of Knots, 3-manifolds, and Their Sets vol. 29. World Scientific, (2001)

2001
[238]

Advances in Mathematics 186(1), 58–116 (2004)

Ozsv´ ath, P., Szab´ o, Z.: Holomorphic disks and knot invariants. Advances in Mathematics 186(1), 58–116 (2004)

2004
[239]

Proceedings of the Royal Society A 476(2240), 20200124 (2020)

Panagiotou, E., Kauffman, L.H.: Knot polynomials of open and closed curves. Proceedings of the Royal Society A 476(2240), 20200124 (2020)

2020
[240]

Panagiotou, E., Plaxco, K.W.: A topological study of protein folding kinetics. Topol. Geom. Biopolym. AMS Contemp. Math. Ser 746, 223– 233 (2020)

2020
[241]

Polymers 11(3), 437 (2019)

Panagiotou, E., Millett, K.C., Atzberger, P.J.: Topological methods for polymeric materials: characterizing the relationship between polymer entanglement and viscoelasticity. Polymers 11(3), 437 (2019)

2019
[242]

Statistics and Computing 32(88) (2022)

Papamarkou, T., Nasrin, F., Lawson, A., Gong, N., Rios, O., Maroulas, V.: A random persistence diagram generator. Statistics and Computing 32(88) (2022). https://doi.org/10.1007/s11222-022-10141-y

2022 doi
[243]

Proceedings of Machine Learning Research 235, 39529–39555 (2024)

Papamarkou, T., Birdal, T., Bronstein, M.M., Carlsson, G., Curry, J., Gao, Y., Hajij, M., Kwitt, R., Li` o, P., Di Lorenzo, P., Maroulas, V., Miolane, N., Nasrin, F., Natesan Ramamurthy, K., Rieck, B., Scarda- pane, S., Schaub, M.T., Veliˇ ckovi´ c, P., Wang, B., Wang, Y., Wei...

2024
[244]

Journal of Applied and Computational Topology 1(3), 397–419 (2018)

Patel, A.: Generalized persistence diagrams. Journal of Applied and Computational Topology 1(3), 397–419 (2018)

2018
[245]

In: 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (icassp), pp

Perea, J.A.: Persistent homology of toroidal sliding window embeddings. In: 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (icassp), pp. 6435–6439 (2016). IEEE

2016
[246]

Foundations of computational mathematics 15, 799–838 (2015)

Perea, J.A., Harer, J.: Sliding windows and persistence: an application of topological methods to signal analysis. Foundations of computational mathematics 15, 799–838 (2015)

2015
[247]

PloS one 8(6), 66506 (2013)

Petri, G., Scolamiero, M., Donato, I., Vaccarino, F.: Topological strata of weighted complex networks. PloS one 8(6), 66506 (2013)

2013
[248]

PhD thesis (2017) 80 CONTENTS

Poelke, K.: Hodge-type decompositions for piecewise constant vector fields on simplicial surfaces and solids with boundary. PhD thesis (2017) 80 CONTENTS

2017
[249]

Computer-Aided Design 78, 126–136 (2016)

Poelke, K., Polthier, K.: Boundary-aware Hodge decompositions for piecewise constant vector fields. Computer-Aided Design 78, 126–136 (2016)

2016
[250]

Artificial Intelligence Review 55(7), 5169–5213 (2022)

Pun, C.S., Lee, S.X., Xia, K.: Persistent-homology-based machine learn- ing: a survey and a comparative study. Artificial Intelligence Review 55(7), 5169–5213 (2022)

2022
[251]

PloS one 15(8), 0237747 (2020)

Pun, C.S., Yong, B.Y.S., Xia, K.: Weighted-persistent-homology-based machine learning for RNA flexibility analysis. PloS one 15(8), 0237747 (2020)

2020
[252]

Puzyn, T., Leszczynski, J., Cronin, M.T.: Recent advances in QSAR studies: methods and applications (2010)

2010
[253]

Nature computational science 3(2), 149–163 (2023)

Qiu, Y., Wei, G.-W.: Persistent spectral theory-guided protein engineer- ing. Nature computational science 3(2), 149–163 (2023)

2023
[254]

Comptes Rendus Acad

Reeb, G.: Sur les points singuliers d’une forme de pfaff completement integrable ou d’une fonction numerique [on the singular points of a com- pletely integrable pfaff form or of a numerical function]. Comptes Rendus Acad. Sciences Paris 222, 847–849 (1946)

1946
[255]

In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp

Reininghaus, J., Huber, S., Bauer, U., Kwitt, R.: A stable multi-scale kernel for topological machine learning. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4741–4748 (2015)

2015
[256]

arXiv preprint arXiv:2002.02237 (2020)

Ren, S., Wu, J.: The stability of persistent homology of hypergraphs. arXiv preprint arXiv:2002.02237 (2020)

2020 arXiv
[257]

The Rocky Mountain Journal of Mathematics 48(8), 2661–2687 (2018)

Ren, S., Wu, C., Wu, J.: Weighted persistent homology. The Rocky Mountain Journal of Mathematics 48(8), 2661–2687 (2018)

2018
[258]

arXiv preprint arXiv:2108.02384 (2021)

Ren, S., Wang, C., Wu, C., Wu, J.: On the discrete Morse functions for hypergraphs. arXiv preprint arXiv:2108.02384 (2021)

2021 arXiv
[259]

Computer-Aided Design 38(4), 342– 366 (2006)

Reuter, M., Wolter, F.-E., Peinecke, N.: Laplace-Beltrami spectra as ‘Shape-DNA’ of surfaces and solids. Computer-Aided Design 38(4), 342– 366 (2006)

2006
[260]

SIAM Review 66(3), 575–601 (2024)

Ribando-Gros, E., Wang, R., Chen, J., Tong, Y., Wei, G.-W.: Combi- natorial and Hodge Laplacians: Similarity and difference. SIAM Review 66(3), 575–601 (2024)

2024
[261]

arXiv preprint arXiv:2204.13446 (2022) CONTENTS 81

Russold, F.: Persistent sheaf cohomology. arXiv preprint arXiv:2204.13446 (2022) CONTENTS 81

2022 arXiv
[262]

In: Symposium on Geometry Processing, vol

Rustamov, R.M., et al.: Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In: Symposium on Geometry Processing, vol. 257, pp. 225–233 (2007)

2007
[263]

Springer, (2022)

Schenck, H.: Algebraic Foundations for Applied Topology and Data Analysis. Springer, (2022)

2022
[264]

Journal of the American Chemical Society 143(30), 11404–11422 (2021)

Schlick, T., Zhu, Q., Dey, A., Jain, S., Yan, S., Laederach, A.: To knot or not to knot: multiple conformations of the SARS-CoV-2 frameshift- ing RNA element. Journal of the American Chemical Society 143(30), 11404–11422 (2021)

2021
[265]

Mathematische Zeitschrift 61(1), 245–288 (1954)

Schubert, H.: ¨Uber eine numerische knoteninvariante. Mathematische Zeitschrift 61(1), 245–288 (1954)

1954
[266]

Springer, (2006)

Schwarz, G.: Hodge Decomposition - A Method for Solving Boundary Value Problems. Springer, (2006)

2006
[267]

Publications Math´ ematiques de l’IH´ES 34, 105–112 (1968)

Segal, G.: Classifying spaces and spectral sequences. Publications Math´ ematiques de l’IH´ES 34, 105–112 (1968)

1968
[268]

AIMS Mathematics 9(9), 26139–26165 (2024)

Shen, L., Liu, J., Wei, G.-W.: Evolutionary Khovanov homology. AIMS Mathematics 9(9), 26139–26165 (2024)

2024
[269]

Foundations of Data Science 6(4), 584–612 (2024)

Shen, L., Liu, J., Wei, G.-W.: Persistent Mayer homology and persistent Mayer Laplacian. Foundations of Data Science 6(4), 584–612 (2024)

2024
[270]

Proceedings of the National Academy of Sciences 121(42), 2408431121 (2024)

Shen, L., Feng, H., Li, F., Lei, F., Wu, J., Wei, G.-W.: Knot data anal- ysis using multiscale Gauss link integral. Proceedings of the National Academy of Sciences 121(42), 2408431121 (2024)

2024
[271]

Brown University, (1985)

Shepard, A.D.: A Cellular Description of the Derived Category of a Stratified Space. Brown University, (1985)

1985
[272]

PhD thesis, University of Pennsylvania (2009)

Shonkwiler, C.: Poincar´ e duality angles on Riemannian manifolds with boundary. PhD thesis, University of Pennsylvania (2009)

2009
[273]

Journal of Knot Theory and Its Ramifications 28(09), 1950058 (2019)

Silver, D.S., Williams, S.G.: Knot invariants from Laplacian matrices. Journal of Knot Theory and Its Ramifications 28(09), 1950058 (2019)

2019
[274]

: Topological methods for the analysis of high dimensional data sets and 3D object recognition

Singh, G., M´ emoli, F., Carlsson, G.E., et al. : Topological methods for the analysis of high dimensional data sets and 3D object recognition. PBG@ Eurographics 2, 091–100 (2007)

2007
[275]

Insights into Imaging 14(1), 58 (2023) 82 CONTENTS

Singh, Y., Farrelly, C.M., Hathaway, Q.A., Leiner, T., Jagtap, J., Carls- son, G.E., Erickson, B.J.: Topological data analysis in medical imaging: current state of the art. Insights into Imaging 14(1), 58 (2023) 82 CONTENTS

2023
[276]

In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp

Som, A., Choi, H., Ramamurthy, K.N., Buman, M.P., Turaga, P.: Pi- net: A deep learning approach to extract topological persistence images. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp. 834–835 (2020)

2020
[277]

AIMS Mathematics 10(1), 1463–1487 (2025)

Song, R., Li, F., Wu, J., Lei, F., Wei, G.-W.: Multi-scale jones poly- nomial and persistent jones polynomial for knot data analysis. AIMS Mathematics 10(1), 1463–1487 (2025)

2025
[278]

Spanier, E.H.: The Mayer homology theory. Bull. Amer. Math. Soc. 55(12), 102–112 (1949)

1949
[279]

Chaos: An Interdisciplinary Journal of Nonlinear Science 27(4) (2017)

Stolz, B.J., Harrington, H.A., Porter, M.A.: Persistent homology of time- dependent functional networks constructed from coupled time series. Chaos: An Interdisciplinary Journal of Nonlinear Science 27(4) (2017)

2017
[280]

Journal of chemical information and modeling64(8), 3558–3568 (2024)

Su, Z., Tong, Y., Wei, G.-W.: Hodge decomposition of single-cell RNA velocity. Journal of chemical information and modeling64(8), 3558–3568 (2024)

2024
[281]

In: SIGGRAPH Asia 2024 Conference Papers, pp

Su, Z., Tong, Y., Wei, G.-W.: Hodge decomposition of vector fields in Cartesian grids. In: SIGGRAPH Asia 2024 Conference Papers, pp. 1–10 (2024)

2024
[282]

AIMS Mathematics 9(10), 27438–27470 (2024)

Su, Z., Tong, Y., Wei, G.-W.: Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning. AIMS Mathematics 9(10), 27438–27470 (2024)

2024
[283]

arXiv preprint arXiv:2408.14356 (2024)

Su, Z., Tong, Y., Wei, G.-W.: Topology-preserving Hodge decomposition in the Eulerian representation. arXiv preprint arXiv:2408.14356 (2024)

2024 arXiv
[284]

Biophysical Journal 102(3), 253 (2012)

Sulkowska, J.I., Rawdon, E.J., Millet, K.C., Onuchic, J.N., Stasiak, A.: Conservation of complex knotting and slipknotting patterns in proteins. Biophysical Journal 102(3), 253 (2012)

2012
[285]

In: Geometry and Topology, pp

Sumners, D.: The role of knot theory in DNA research. In: Geometry and Topology, pp. 297–318. CRC Press, (2020)

2020
[286]

Foundations of data science (Springfield, Mo.) 6(2), 124 (2024)

Suwayyid, F., Wei, G.-W.: Persistent Dirac of paths on digraphs and hypergraphs. Foundations of data science (Springfield, Mo.) 6(2), 124 (2024)

2024
[287]

Journal of Physics: Complexity 5(4), 045005 (2024)

Suwayyid, F., Wei, G.-W.: Persistent Mayer Dirac. Journal of Physics: Complexity 5(4), 045005 (2024)

2024
[288]

In: Dynamical Systems and Turbulence, Warwick 1980: Proceedings of a Symposium CONTENTS 83 Held at the University of Warwick 1979/80, pp

Takens, F.: Detecting strange attractors in turbulence. In: Dynamical Systems and Turbulence, Warwick 1980: Proceedings of a Symposium CONTENTS 83 Held at the University of Warwick 1979/80, pp. 366–381 (2006). Springer

2006
[289]

Journal of Machine Learning Research 22(39), 1–6 (2021)

Tauzin, G., Lupo, U., Tunstall, L., P´ erez, J.B., Caorsi, M., Medina- Mardones, A.M., Dassatti, A., Hess, K.: giotto-tda:: A topological data analysis toolkit for machine learning and data exploration. Journal of Machine Learning Research 22(39), 1–6 (2021)

2021
[290]

IEEE transactions on visualization and computer graphics 24(1), 832–842 (2017)

Tierny, J., Favelier, G., Levine, J.A., Gueunet, C., Michaux, M.: The topology toolkit. IEEE transactions on visualization and computer graphics 24(1), 832–842 (2017)

2017
[291]

Nature communications 11(1), 3230 (2020)

Townsend, J., Micucci, C.P., Hymel, J.H., Maroulas, V., Vogiatzis, K.D.: Representation of molecular structures with persistent homology for machine learning applications in chemistry. Nature communications 11(1), 3230 (2020)

2020
[292]

Journal of Applied and Computational Topology 8(7), 2111–2154 (2024)

Turner, K., Robins, V., Morgan, J.: The extended persistent homol- ogy transform of manifolds with boundary. Journal of Applied and Computational Topology 8(7), 2111–2154 (2024)

2024
[293]

Discrete & Computational Geometry 52(1), 44–70 (2014)

Turner, K., Mileyko, Y., Mukherjee, S., Harer, J.: Fr´ echet means for dis- tributions of persistence diagrams. Discrete & Computational Geometry 52(1), 44–70 (2014)

2014
[294]

Journal of Open Source Software 4(42), 1315 (2019)

Van Veen, H.J., Saul, N., Eargle, D., Mangham, S.W.: Kepler Mapper: a flexible Python implementation of the Mapper algorithm. Journal of Open Source Software 4(42), 1315 (2019)

2019
[295]

Mathematische Annalen 97(1), 454–472 (1927)

Vietoris, L.: ¨Uber den h¨ oheren zusammenhang kompakter r¨ aume und eine klasse von zusammenhangstreuen abbildungen. Mathematische Annalen 97(1), 454–472 (1927)

1927
[296]

Journal of Machine Learning Research 21(61), 1–38 (2020)

Vipond, O.: Multiparameter persistence landscapes. Journal of Machine Learning Research 21(61), 1–38 (2020)

2020
[297]

In: 39th International Symposium on Computational Geometry (SoCG 2023) (2023)

Wagner, H.: Slice, simplify and stitch: Topology-preserving simplifica- tion scheme for massive voxel data. In: 39th International Symposium on Computational Geometry (SoCG 2023) (2023). Schloss Dagstuhl- Leibniz-Zentrum f¨ ur Informatik

2023
[298]

In: Topological Methods in Data Analysis and Visualization II: Theory, Algorithms, and Applications, pp

Wagner, H., Chen, C., Vu¸ cini, E.: Efficient computation of persistent homology for cubical data. In: Topological Methods in Data Analysis and Visualization II: Theory, Algorithms, and Applications, pp. 91–106. Springer, (2011)

2011
[299]

Journal of computational physics 305, 276–299 (2016) 84 CONTENTS

Wang, B., Wei, G.-W.: Object-oriented persistent homology. Journal of computational physics 305, 276–299 (2016) 84 CONTENTS

2016
[2014]

129–136 (2014)

Proceedings 4, pp. 129–136 (2014). Springer

2014

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [65]

4 and BA

Chen, J., Qiu, Y., Wang, R., Wei, G.-W.: Persistent Laplacian pro- jected Omicron BA. 4 and BA. 5 to become new dominating variants. Computers in Biology and Medicine 151, 106262 (2022)

2022

[2] [1]

American Mathematical Society, (2004)

Adams, C.C.: The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society, (2004)

2004

[3] [2]

SIAM Journal on Imaging Sciences 2(1), 110–117 (2009)

Adams, H., Carlsson, G.: On the nonlinear statistics of range image patches. SIAM Journal on Imaging Sciences 2(1), 110–117 (2009)

2009

[4] [3]

In: Mathematical Software– ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9,

Adams, H., Tausz, A., Vejdemo-Johansson, M.: Javaplex: a research soft- ware package for persistent (co)homology. In: Mathematical Software– ICMS 2014: 4th International Congress, Seoul, South Korea, August 5-9,

2014

[5] [4]

Journal of Machine Learning Research 18(8), 1–35 (2017)

Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Ship- man, P., Chepushtanova, S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research 18(8), 1–35 (2017)

2017

[6] [5]

Persistent equivariant cohomology

Adams, H., Lagoda, E., Moy, M., Sadovek, N., De Saha, A.: Persistent equivariant cohomology. arXiv preprint arXiv:2408.17331 (2024)

work page Pith review arXiv 2024

[7] [6]

Homology, Homotopy and Applications 18(1), 381–402 (2016)

Adcock, A., Carlsson, E., Carlsson, G.: The ring of algebraic functions on persistence bar codes. Homology, Homotopy and Applications 18(1), 381–402 (2016)

2016

[8] [7]

Transactions of the American Mathematical Society 30(2), 275–306 (1928) CONTENTS 61

Alexander, J.W.: Topological invariants of knots and links. Transactions of the American Mathematical Society 30(2), 275–306 (1928) CONTENTS 61

1928