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Documents authored by Maria, Clément

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.2026.58
On Sparse Representations of 3‑Manifolds

Authors: Kristóf Huszár and Clément Maria

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

Abstract
3-manifolds are commonly represented as triangulations, consisting of abstract tetrahedra whose triangular faces are identified in pairs. The combinatorial sparsity of a triangulation, as measured by the treewidth of its dual graph, plays a fundamental role in the design of parameterized algorithms. In this work, we investigate algorithmic procedures that transform or modify a given triangulation while controlling specific sparsity parameters. First, we revisit a standard, linear-time algorithm that converts a given triangulation into a Heegaard diagram of the underlying 3-manifold, showing that the construction preserves treewidth. We apply this construction to exhibit a fixed-parameter tractable framework for computing Kuperberg’s quantum invariants of 3-manifolds. Second, we present a quasi-linear-time algorithm that retriangulates a given triangulation into one with maximum edge valence of at most nine, while only moderately increasing the treewidth of the dual graph. Combining these two algorithms yields a quasi-linear-time algorithm that produces, from a given triangulation, a Heegaard diagram in which every attaching curve intersects at most nine others.
Cite as

Kristóf Huszár and Clément Maria. On Sparse Representations of 3‑Manifolds. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{huszar_et_al:LIPIcs.SoCG.2026.58,
  author =	{Husz\'{a}r, Krist\'{o}f and Maria, Cl\'{e}ment},
  title =	{{On Sparse Representations of 3‑Manifolds}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{58:1--58:18},
  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.2026.58},
  URN =		{urn:nbn:de:0030-drops-258659},
  doi =		{10.4230/LIPIcs.SoCG.2026.58},
  annote =	{Keywords: computational 3-manifold topology, fixed-parameter tractability, Heegaard splittings and diagrams, triangulations, edge valence, treewidth, quantum invariants, tensor networks}
}
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.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].
Cite as

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}
}
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.ESA.2022.78
Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds

Authors: Clément Maria and Owen Rouillé

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic 3-manifold with torus boundaries. This family of 3-manifolds includes the knot complements. This computation of a hyperbolic structure requires the resolution of gluing equations on a triangulation of the space, but not all triangulations admit a solution to the equations. In this paper, we propose a new method to find a triangulation that admits a solution to the gluing equations, using convex optimization and localized combinatorial modifications. It is based on Casson and Rivin’s reformulation of the equations. We provide a novel approach to modify a triangulation and update its geometry, along with experimental results to support the new method.
Cite as

Clément Maria and Owen Rouillé. Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 78:1-78:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{maria_et_al:LIPIcs.ESA.2022.78,
  author =	{Maria, Cl\'{e}ment and Rouill\'{e}, Owen},
  title =	{{Localized Geometric Moves to Compute Hyperbolic Structures on Triangulated 3-Manifolds}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{78:1--78:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.78},
  URN =		{urn:nbn:de:0030-drops-170168},
  doi =		{10.4230/LIPIcs.ESA.2022.78},
  annote =	{Keywords: knots and 3-manifolds, triangulation, hyperbolic structure, Thurston equations}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.53
Parameterized Complexity of Quantum Knot Invariants

Authors: Clément Maria

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by planar diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a O(N^{3/2 cw} poly(n)) ∈ N^O(√n) time algorithm to compute any Reshetikhin-Turaev invariant - derived from a simple Lie algebra 𝔤 - of a link presented by a planar diagram with n crossings and carving-width cw, and whose components are coloured with 𝔤-modules of dimension at most N. For example, this includes the N^{th}-coloured Jones polynomial.
Cite as

Clément Maria. Parameterized Complexity of Quantum Knot Invariants. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 53:1-53:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{maria:LIPIcs.SoCG.2021.53,
  author =	{Maria, Cl\'{e}ment},
  title =	{{Parameterized Complexity of Quantum Knot Invariants}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{53:1--53:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.53},
  URN =		{urn:nbn:de:0030-drops-138527},
  doi =		{10.4230/LIPIcs.SoCG.2021.53},
  annote =	{Keywords: computational knot theory, parameterized complexity, quantum invariants}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2020.56
Intrinsic Topological Transforms via the Distance Kernel Embedding

Authors: Clément Maria, Steve Oudot, and Elchanan Solomon

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT), both of which apply to shapes embedded in Euclidean space. The contribution of this paper is to define topological transforms for abstract metric measure spaces. Our proposed pipeline is to pre-compose the PHT or ECT with a Euclidean embedding derived from the eigenfunctions and eigenvalues of an integral operator. To that end, we define and study an integral operator called the distance kernel operator, and demonstrate that it gives rise to stable and quasi-injective topological transforms. We conclude with some numerical experiments, wherein we compute and compare the eigenfunctions and eigenvalues of our operator across a range of standard 2- and 3-manifolds.
Cite as

Clément Maria, Steve Oudot, and Elchanan Solomon. Intrinsic Topological Transforms via the Distance Kernel Embedding. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{maria_et_al:LIPIcs.SoCG.2020.56,
  author =	{Maria, Cl\'{e}ment and Oudot, Steve and Solomon, Elchanan},
  title =	{{Intrinsic Topological Transforms via the Distance Kernel Embedding}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{56:1--56:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.56},
  URN =		{urn:nbn:de:0030-drops-122145},
  doi =		{10.4230/LIPIcs.SoCG.2020.56},
  annote =	{Keywords: Topological Transforms, Persistent Homology, Inverse Problems, Spectral Geometry, Algebraic Topology, Topological Data Analysis}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.64
Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants

Authors: Clément Maria and Jonathan Spreer

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
Turaev-Viro invariants are amongst the most powerful tools to distinguish 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them rely on the enumeration of an extremely large set of combinatorial data defined on the triangulation, regardless of the underlying topology of the manifold. In the article, we propose a finer study of these combinatorial data, called admissible colourings, in relation with the cohomology of the manifold. We prove that the set of admissible colourings to be considered is substantially smaller than previously known, by furnishing new upper bounds on its size that are aware of the topology of the manifold. Moreover, we deduce new topology-sensitive enumeration algorithms based on these bounds. The paper provides a theoretical analysis, as well as a detailed experimental study of the approach. We give strong experimental evidence on large manifold censuses that our upper bounds are tighter than the previously known ones, and that our algorithms outperform significantly state of the art implementations to compute Turaev-Viro invariants.
Cite as

Clément Maria and Jonathan Spreer. Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{maria_et_al:LIPIcs.ESA.2016.64,
  author =	{Maria, Cl\'{e}ment and Spreer, Jonathan},
  title =	{{Admissible Colourings of 3-Manifold Triangulations for Turaev-Viro Type Invariants}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{64:1--64:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.64},
  URN =		{urn:nbn:de:0030-drops-64050},
  doi =		{10.4230/LIPIcs.ESA.2016.64},
  annote =	{Keywords: low-dimensional topology, triangulations of 3-manifolds, cohomology theory, Turaev-Viro invariants, combinatorial algorithms}
}
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