22 Search Results for "Burton, Benjamin A."

Document
DOI: 10.4230/LIPIcs.SoCG.2026.42
Compressed Data Structures for Heegaard Splitting

Authors: Henrique Ennes and Clément Maria

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Heegaard splittings provide a natural representation of closed 3-manifolds by gluing handlebodies along a common surface. These splittings can be equivalently given by two finite sets of meridians lying on the surface, which define a Heegaard diagram. We present a data structure to effectively represent Heegaard diagrams as normal curves with respect to triangulations of a surface of complexity measured by the space required to express the normal coordinates' vectors in binary. This structure can be significantly more compressed than triangulations of 3-manifolds, giving exponential gains for some families. Even with this succinct definition of complexity, we establish polynomial-time algorithms for comparing and manipulating diagrams, performing stabilizations, detecting trivial stabilizations and reductions, and computing topological invariants of the underlying manifolds, such as their fundamental and homology groups. We also contrast early implementations of our techniques with standard software programs for 3-manifolds, achieving faster algorithms for the average cases and exponential gains in speed for some particular presentations of the inputs.
Cite as

Henrique Ennes and Clément Maria. Compressed Data Structures for Heegaard Splitting. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 42:1-42:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ennes_et_al:LIPIcs.SoCG.2026.42,
  author =	{Ennes, Henrique and Maria, Cl\'{e}ment},
  title =	{{Compressed Data Structures for Heegaard Splitting}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{42:1--42:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.42},
  URN =		{urn:nbn:de:0030-drops-258484},
  doi =		{10.4230/LIPIcs.SoCG.2026.42},
  annote =	{Keywords: 3-manifold, Heegaard splitting, curves on surfaces, surface theory, data structure, computational topology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.56
Optimal Randomized Clustering of Matrices

Authors: Mustafa Alper Gunes and Assaf Naor

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
If X = (𝖬_n(ℝ),‖⋅‖_X) is a unitarily invariant normed space, i.e., ‖𝖴𝖠𝖵‖_X = ‖𝖠‖_X for every matrix 𝖠 ∈ 𝖬_n(ℝ) and every two orthogonal matrices 𝖴,𝖵 ∈ 𝖬_n(ℝ), then we evaluate up to universal constant factors the smallest σ > 0 for which there is a probability distribution over partitions of X into clusters of diameter at most 1 yet for every two matrices 𝖠,𝖡 ∈ 𝖬_n(ℝ) the probability that they fall into distinct clusters is at most σ times the X-distance between 𝖠 and 𝖡. Specifically, we prove that this infimal σ, which is called the separation modulus of X and is denoted SEP(X), satisfies: (1) SEP(X) = Θ(√n⋅ ‖𝖨_n‖_X⋅ diam(B_X)), where 𝖨_n is the n-by-n identity matrix and diam(B_X) is the diameter with respect to the standard Euclidean metric on 𝖬_n(ℝ) of the unit ball B_ X of X. Our proof of (1) proceeds through an asymptotic evaluation of the spectral gap of the Laplacian with Dirichlet boundary conditions on B_ X, which we achieve by exact computations for a Jacobi orthogonal random matrix ensemble. Assuming oracle access to norm evaluations in X, by combining (1) with a new deterministic algorithm for a O(1)-approximation of the diameter of convex bodies in ℝⁿ that are given by a weak membership oracle and are symmetric with respect to coordinate permutations and reflections about the standard axes (this task is famously known to be impossible in the absence of such symmetries), we get an oracle polynomial time algorithm whose output is the separation modulus of X up to universal constant factors. Another example of a consequence of (1) is that for each m ∈ {1,…,n} the separation modulus of the m'th Ky Fan norm on 𝖬_n(ℝ) is bounded from above and from below by universal constant multiples of m√n if m ⩾ √n, and of n if m ⩽ √n. We also deduce from (1) an upper bound on the Lipschitz extension modulus of X that improves over the previously best-known bound even in the special case when X is 𝖬_n(ℝ) equipped with the 𝓁₂ⁿ → 𝓁₂ⁿ operator norm.
Cite as

Mustafa Alper Gunes and Assaf Naor. Optimal Randomized Clustering of Matrices. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{gunes_et_al:LIPIcs.SoCG.2026.56,
  author =	{Gunes, Mustafa Alper and Naor, Assaf},
  title =	{{Optimal Randomized Clustering of Matrices}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{56:1--56:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.56},
  URN =		{urn:nbn:de:0030-drops-258624},
  doi =		{10.4230/LIPIcs.SoCG.2026.56},
  annote =	{Keywords: Clustering, Unitarily Invariant Matrix Norms, Oracle Polynomial Time Approximation Algorithms for Radii of Convex Bodies, Extension of Lipschitz Functions, Random Matrices, Spectrum of the Laplacian with Dirichlet Boundary Conditions, Reverse Isoperimetry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.73
The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds

Authors: Shana Yunsheng Li

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We extend the complete census of orientable cusped hyperbolic 3-manifolds to 10 tetrahedra, giving the next 150,730 manifolds and their 496,638 minimal ideal triangulations. As applications, we find the precisely 439,898 exceptional Dehn fillings on them, revealing the next 1,849 simplest hyperbolic knot exteriors in S³. We also give the simplest example of an orientable cusped hyperbolic 3-manifold containing a closed totally geodesic surface.
Cite as

Shana Yunsheng Li. The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{li:LIPIcs.SoCG.2026.73,
  author =	{Li, Shana Yunsheng},
  title =	{{The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{73:1--73:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.73},
  URN =		{urn:nbn:de:0030-drops-258800},
  doi =		{10.4230/LIPIcs.SoCG.2026.73},
  annote =	{Keywords: hyperbolic manifolds, 3-manifolds, triangulation, census, tabulation, exact computation, computational topology, low-dimensional topology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.76
A Fast Algorithm for the Hecke Representation of the Braid Group, and Applications to the Computation of the HOMFLY-PT Polynomial and the Search for Interesting Braids

Authors: Clément Maria and Hoel Queffelec

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Knot theory is an active field of mathematics, in which combinatorial and computational methods play an important role. One side of computational knot theory, that has gained interest in recent years, both for complexity analysis and practical algorithms, is quantum topology and the computation of topological invariants issued from the theory. In this article, we leverage the rigidity brought by the representation-theoretic origins of the quantum invariants for algorithmic purposes. We do so by exploiting braids and the algebraic properties of the braid group to describe, analyze, and implement a fast algorithm to compute the Hecke representation of the braid group. We apply this construction to design a parameterized algorithm to compute the HOMFLY-PT polynomial of knots, and demonstrate its interest experimentally. Finally, we combine our fast Hecke representation algorithm with Garside theory, to implement a reservoir sampling search and find non-trivial braids with trivial Hecke representations with coefficients in ℤ/pℤ. We find explicitly several such braids, for the 4-strand and 5-strand braid groups.
Cite as

Clément Maria and Hoel Queffelec. A Fast Algorithm for the Hecke Representation of the Braid Group, and Applications to the Computation of the HOMFLY-PT Polynomial and the Search for Interesting Braids. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 76:1-76:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{maria_et_al:LIPIcs.SoCG.2026.76,
  author =	{Maria, Cl\'{e}ment and Queffelec, Hoel},
  title =	{{A Fast Algorithm for the Hecke Representation of the Braid Group, and Applications to the Computation of the HOMFLY-PT Polynomial and the Search for Interesting Braids}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{76:1--76:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.76},
  URN =		{urn:nbn:de:0030-drops-258838},
  doi =		{10.4230/LIPIcs.SoCG.2026.76},
  annote =	{Keywords: Hecke representation of the braid group, parameterized algorithm, HOMFLY-PT polynomial of knots, reservoir sampling, faithfulness of Hecke representation}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.85
ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes

Authors: Geevarghese Philip and Erlend Raa Vågset

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The Optimal Morse Matching (OMM) problem asks for a discrete gradient vector field on a simplicial complex that minimizes the number of critical simplices. It is NP-hard and has been studied extensively in heuristic, approximation, and parameterized complexity settings. Parameterized by treewidth k, OMM has long been known to be solvable on triangulations of 3-manifolds in 2^O(k²) n^O(1) time and in FPT time for triangulations of arbitrary manifolds, but the exact dependence on k has remained an open question. We resolve this by giving a new 2^O(k log k) n-time algorithm for any finite regular CW complex, and show that no 2^o(k log k) n^O(1)-time algorithm exists unless the Exponential Time Hypothesis (ETH) fails.
Cite as

Geevarghese Philip and Erlend Raa Vågset. ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{philip_et_al:LIPIcs.SoCG.2026.85,
  author =	{Philip, Geevarghese and V\r{a}gset, Erlend Raa},
  title =	{{ETH-Tight Complexity of Optimal Morse Matching on Bounded-Treewidth Complexes}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{85:1--85:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.85},
  URN =		{urn:nbn:de:0030-drops-258926},
  doi =		{10.4230/LIPIcs.SoCG.2026.85},
  annote =	{Keywords: Discrete Morse Theory, Simplicial Complexes, Optimal Morse Matching, Treewidth, Parameterized Algorithms, Computational Topology, Dynamic Programming, Exponential Time Hypothesis, Topological Data Analysis}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.93
Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations

Authors: Raphaël Tinarrage

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Simplicial approximation provides a framework for constructing simplicial complexes that are homotopy equivalent to a given manifold, provided a CW structure is explicitly known. However, its conventional implementation quickly becomes intractable on a computer: barycentric subdivision produces poorly shaped simplices, and the star condition introduces many vertices. To address these limitations, this article develops a subdivision scheme based on spherical Delaunay triangulations, which attains better refinement properties than barycentric subdivisions. Moreover, the star condition is reframed as two independent problems, one geometric and the other combinatorial, respectively tackled in the language of locally equiconnected spaces and the list homomorphism problem, allowing an exponential reduction in the number of vertices. Via a prototype implementation, we obtain simplicial complexes homotopy equivalent to Grassmannians and Stiefel manifolds up to dimension 5.
Cite as

Raphaël Tinarrage. Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 93:1-93:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{tinarrage:LIPIcs.SoCG.2026.93,
  author =	{Tinarrage, Rapha\"{e}l},
  title =	{{Simplicial Approximation to CW Complexes with Spherical Delaunay Triangulations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{93:1--93:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.93},
  URN =		{urn:nbn:de:0030-drops-258991},
  doi =		{10.4230/LIPIcs.SoCG.2026.93},
  annote =	{Keywords: Triangulation of manifolds, Simplicial approximation, CW complexes, Delaunay complexes, List homomorphism problem, Topological Data Analysis}
}
Document
Research
DOI: 10.4230/TGDK.3.3.4
Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web

Authors: Florian Ruosch, Cristina Sarasua, and Abraham Bernstein

Published in: TGDK, Volume 3, Issue 3 (2025). Transactions on Graph Data and Knowledge, Volume 3, Issue 3

Abstract
In Argument Mining, predicting argumentative relations between texts (or spans) remains one of the most challenging aspects, even more so in the cross-document setting. This paper makes three key contributions to advance research in this domain. We first extend an existing dataset, the Sci-Arg corpus, by annotating it with explicit inter-document argumentative relations, thereby allowing arguments to be distributed over several documents forming an Argument Web; these new annotations are published using Semantic Web technologies (RDF, OWL). Second, we explore and evaluate three automated approaches for predicting these inter-document argumentative relations, establishing critical baselines on the new dataset. We find that a simple classifier based on discourse indicators with access to context outperforms neural methods. Third, we conduct a comparative analysis of these approaches for both intra- and inter-document settings, identifying statistically significant differences in results that indicate the necessity of distinguishing between these two scenarios. Our findings highlight significant challenges in this complex domain and open crucial avenues for future research on the Argument Web of Science, particularly for those interested in leveraging Semantic Web technologies and knowledge graphs to understand scholarly discourse. With this, we provide the first stepping stones in the form of a benchmark dataset, three baseline methods, and an initial analysis for a systematic exploration of this field relevant to the Web of Data and Science.
Cite as

Florian Ruosch, Cristina Sarasua, and Abraham Bernstein. Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web. In Transactions on Graph Data and Knowledge (TGDK), Volume 3, Issue 3, pp. 4:1-4:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{ruosch_et_al:TGDK.3.3.4,
  author =	{Ruosch, Florian and Sarasua, Cristina and Bernstein, Abraham},
  title =	{{Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{4:1--4:33},
  ISSN =	{2942-7517},
  year =	{2025},
  volume =	{3},
  number =	{3},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.3.3.4},
  URN =		{urn:nbn:de:0030-drops-252159},
  doi =		{10.4230/TGDK.3.3.4},
  annote =	{Keywords: Argument Mining, Large Language Models, Knowledge Graphs, Link Prediction}
}
Document
DOI: 10.4230/LIPIcs.AFT.2025.1
Cache Timing Leakages in Zero-Knowledge Protocols

Authors: Shibam Mukherjee, Christian Rechberger, and Markus Schofnegger

Published in: LIPIcs, Volume 354, 7th Conference on Advances in Financial Technologies (AFT 2025)

Abstract
The area of modern zero-knowledge proof systems has seen a significant rise in popularity over the last couple of years, with new techniques and optimized constructions emerging on a regular basis. As the field matures, the aspect of implementation attacks becomes more relevant, however side-channel attacks on zero-knowledge proof systems have seen surprisingly little treatment so far. In this paper, we give an overview of potential attack vectors and show that some of the underlying finite field libraries, and implementations of heavily used components like hash functions using them, are vulnerable w.r.t. cache attacks on CPUs. On the positive side, we demonstrate that the computational overhead to protect against these attacks is relatively small.
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Shibam Mukherjee, Christian Rechberger, and Markus Schofnegger. Cache Timing Leakages in Zero-Knowledge Protocols. In 7th Conference on Advances in Financial Technologies (AFT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 354, pp. 1:1-1:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mukherjee_et_al:LIPIcs.AFT.2025.1,
  author =	{Mukherjee, Shibam and Rechberger, Christian and Schofnegger, Markus},
  title =	{{Cache Timing Leakages in Zero-Knowledge Protocols}},
  booktitle =	{7th Conference on Advances in Financial Technologies (AFT 2025)},
  pages =	{1:1--1:26},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-400-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{354},
  editor =	{Avarikioti, Zeta and Christin, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AFT.2025.1},
  URN =		{urn:nbn:de:0030-drops-247201},
  doi =		{10.4230/LIPIcs.AFT.2025.1},
  annote =	{Keywords: zero-knowledge, protocol, cache timing, side-channel, leakage}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.37
Hardness of Computation of Quantum Invariants on 3-Manifolds with Restricted Topology

Authors: Henrique Ennes and Clément Maria

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Quantum invariants in low-dimensional topology offer a wide variety of valuable invariants about knots and 3-manifolds, presented by explicit formulas that are readily computable. Their computational complexity has been actively studied and is tightly connected to topological quantum computing. In this article, we prove that for any 3-manifold quantum invariant in the Reshetikhin-Turaev model, there is a deterministic polynomial time algorithm that, given as input an arbitrary closed 3-manifold M, outputs a closed 3-manifold M' with the same quantum invariant, such that M' is hyperbolic, contains no low genus embedded incompressible surface, and is presented by a strongly irreducible Heegaard diagram. Our construction relies on properties of Heegaard splittings and the Hempel distance. At the level of computational complexity, this proves that the hardness of computing a given quantum invariant of 3-manifolds is preserved even when severely restricting the topology and the combinatorics of the input. This positively answers a question raised by Samperton [Samperton, 2023].
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Henrique Ennes and Clément Maria. Hardness of Computation of Quantum Invariants on 3-Manifolds with Restricted Topology. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ennes_et_al:LIPIcs.ESA.2025.37,
  author =	{Ennes, Henrique and Maria, Cl\'{e}ment},
  title =	{{Hardness of Computation of Quantum Invariants on 3-Manifolds with Restricted Topology}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{37:1--37:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.37},
  URN =		{urn:nbn:de:0030-drops-245057},
  doi =		{10.4230/LIPIcs.ESA.2025.37},
  annote =	{Keywords: 3-manifold, Heegaard splitting, Hempel distance, Quantum invariant, polynomial time reduction}
}
Artifact
Software
DOI: 10.4230/artifacts.23281
raburke/Dim4Census

Authors: Rhuaidi Antonio Burke, Benjamin A. Burton, and Jonathan Spreer

Abstract
Cite as

Rhuaidi Antonio Burke, Benjamin A. Burton, Jonathan Spreer. raburke/Dim4Census (Software). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@misc{dagstuhl-artifact-23281,
   title = {{raburke/Dim4Census}}, 
   author = {Burke, Rhuaidi Antonio and Burton, Benjamin A. and Spreer, Jonathan},
   note = {Software, swhId: \href{https://archive.softwareheritage.org/swh:1:dir:ee5ac0c76fdef9983c5de8a0be93f7684dd9a796;origin=https://github.com/raburke/Dim4Census;visit=swh:1:snp:a7fee9b4ed22b6bf281127e889c432095a216a58;anchor=swh:1:rev:54753c465209c14b34834a5f13cfe373b53ca4c6}{\texttt{swh:1:dir:ee5ac0c76fdef9983c5de8a0be93f7684dd9a796}} (visited on 2025-06-20)},
   url = {https://github.com/raburke/Dim4Census},
   doi = {10.4230/artifacts.23281},
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.67
Hard Diagrams of Split Links

Authors: Corentin Lunel, Arnaud de Mesmay, and Jonathan Spreer

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister moves, one sometimes needs to insert excess crossings, there are significant gaps between the best known lower and upper bounds on the required number of these added crossings. In this article, we study the problem of turning a diagram of a split link into a split diagram, and we show that there exist split links with diagrams requiring an arbitrarily large number of such additional crossings. More precisely, we provide a family of diagrams of split links, so that any sequence of Reidemeister moves transforming a diagram with c crossings into a split diagram requires going through a diagram with Ω(√c) extra crossings. Our proof relies on the framework of bubble tangles, as introduced by the first two authors, and a technique of Chambers and Liokumovitch to turn homotopies into isotopies in the context of Riemannian geometry.
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Corentin Lunel, Arnaud de Mesmay, and Jonathan Spreer. Hard Diagrams of Split Links. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lunel_et_al:LIPIcs.SoCG.2025.67,
  author =	{Lunel, Corentin and de Mesmay, Arnaud and Spreer, Jonathan},
  title =	{{Hard Diagrams of Split Links}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{67:1--67:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.67},
  URN =		{urn:nbn:de:0030-drops-232191},
  doi =		{10.4230/LIPIcs.SoCG.2025.67},
  annote =	{Keywords: Knot theory, hard knot and link diagrams, Reidemeister moves, extra crossings, split links, bubble tangles, compression representativity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.38
An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number

Authors: Colleen Delaney, Clément Maria, and Eric Samperton

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with ℤ/2ℤ coefficients. Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input.
Cite as

Colleen Delaney, Clément Maria, and Eric Samperton. An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{delaney_et_al:LIPIcs.SoCG.2025.38,
  author =	{Delaney, Colleen and Maria, Cl\'{e}ment and Samperton, Eric},
  title =	{{An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{38:1--38:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.38},
  URN =		{urn:nbn:de:0030-drops-231901},
  doi =		{10.4230/LIPIcs.SoCG.2025.38},
  annote =	{Keywords: 3-manifold, quantum invariant, fixed parameter tractable algorithm, topological parameter, Gauss sums, topological quantum field theory}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.28
Small Triangulations of 4-Manifolds and the 4-Manifold Census

Authors: Rhuaidi Antonio Burke, Benjamin A. Burton, and Jonathan Spreer

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated 4-manifolds with up to 6 pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, CP², and the rational homology sphere QS⁴(2), where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations - which we deem interesting in their own rights.
Cite as

Rhuaidi Antonio Burke, Benjamin A. Burton, and Jonathan Spreer. Small Triangulations of 4-Manifolds and the 4-Manifold Census. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{burke_et_al:LIPIcs.SoCG.2025.28,
  author =	{Burke, Rhuaidi Antonio and Burton, Benjamin A. and Spreer, Jonathan},
  title =	{{Small Triangulations of 4-Manifolds and the 4-Manifold Census}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{28:1--28:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.28},
  URN =		{urn:nbn:de:0030-drops-231805},
  doi =		{10.4230/LIPIcs.SoCG.2025.28},
  annote =	{Keywords: computational low-dimensional topology, triangulations, census of triangulations, 4-manifolds, PL standard 4-sphere, Pachner graph, mathematical software, experiments in low-dimensional topology}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.30
Effective Computation of the Heegaard Genus of 3-Manifolds

Authors: Benjamin A. Burton and Finn Thompson

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
The Heegaard genus is a fundamental invariant of 3-manifolds. However, computing the Heegaard genus of a triangulated 3-manifold is NP-hard, and while algorithms exist, little work has been done in making such an algorithm efficient and practical for implementation. Current algorithms use almost normal surfaces, which are an extension of the algorithm-friendly normal surface theory but which add considerable complexity for both running time and implementation. Here we take a different approach: instead of working with almost normal surfaces, we give a general method of modifying the input triangulation that allows us to avoid almost normal surfaces entirely. The cost is just four new tetrahedra, and the benefit is that important surfaces that were once almost normal can be moved to the simpler setting of normal surfaces in the new triangulation. We apply this technique to the computation of Heegaard genus, where we develop algorithms and heuristics that prove successful in practice when applied to a data set of 3,000 closed hyperbolic 3-manifolds; we precisely determine the genus for at least 2,705 of these.
Cite as

Benjamin A. Burton and Finn Thompson. Effective Computation of the Heegaard Genus of 3-Manifolds. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{burton_et_al:LIPIcs.SoCG.2024.30,
  author =	{Burton, Benjamin A. and Thompson, Finn},
  title =	{{Effective Computation of the Heegaard Genus of 3-Manifolds}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{30:1--30:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.30},
  URN =		{urn:nbn:de:0030-drops-199750},
  doi =		{10.4230/LIPIcs.SoCG.2024.30},
  annote =	{Keywords: 3-manifolds, triangulations, normal surfaces, computational topology, Heegaard genus}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.21
Finding Large Counterexamples by Selectively Exploring the Pachner Graph

Authors: Benjamin A. Burton and Alexander He

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
We often rely on censuses of triangulations to guide our intuition in 3-manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations - the current census only goes up to 10 tetrahedra. Here, we show that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations. We use this idea to find counterexamples to three conjectures which ask, for certain 3-manifolds, whether one-vertex triangulations always have a "distinctive" edge that would allow us to recognise the 3-manifold.
Cite as

Benjamin A. Burton and Alexander He. Finding Large Counterexamples by Selectively Exploring the Pachner Graph. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{burton_et_al:LIPIcs.SoCG.2023.21,
  author =	{Burton, Benjamin A. and He, Alexander},
  title =	{{Finding Large Counterexamples by Selectively Exploring the Pachner Graph}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.21},
  URN =		{urn:nbn:de:0030-drops-178712},
  doi =		{10.4230/LIPIcs.SoCG.2023.21},
  annote =	{Keywords: Computational topology, 3-manifolds, Triangulations, Counterexamples, Heuristics, Implementation, Pachner moves, Bistellar flips}
}
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