A census of genus-two 3-manifolds up to 42 coloured tetrahedra

Bandieri, Paola; Cristofori, Paola; Gagliardi, Carlo

doi:10.1016/j.disc.2010.05.016

Cited by 7 publications

(13 citation statements)

References 21 publications

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“…Catalogues of 3-manifold rigid 5 crystallizations up to 32 vertices have already been generated and completely classified ( [15], [18], [19], [5]). Complete classification was also obtained for genus two 3-manifolds admitting a crystallization with at most 42 vertices ( [6]).…”

Section: -Dimensional Generation Algorithmmentioning

confidence: 99%

Cataloguing PL 4-Manifolds by Gem-Complexity

Casali¹,

Cristofori²

2015

Electron. J. Combin.

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We describe an algorithm to subdivide automatically a given set of PL n-manifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case n = 4, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices).Possible interactions with the (not completely known) relationship among the different classifications in the TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the K3-surface.

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Section: -Dimensional Generation Algorithmmentioning

confidence: 99%

Cataloguing PL 4-Manifolds by Gem-Complexity

Casali¹,

Cristofori²

2015

Electron. J. Combin.

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“…is isomorphic to the abelianized group of m, n, k . Therefore, (m, n, k) = (5, 3, 2) or (7,3,2) implies that abelianization of m, n, k is trivial. Thus, M 5, 3, 2 and M 7, 3, 2 are homology spheres, in fact, M 5, 3, 2 is the Poincaré homology sphere.…”

Section: Binary Polyhedral Groups and Generalized Quaternion Spacesmentioning

confidence: 99%

“…Since m, 2, 2 ( ∼ = Q 4m ), P 24 := 3, 3, 2 , P 48 := 4, 3, 2 and P 120 := 5, 3, 2 are finite groups, by the proof of elliptization conjecture of Perelman, M m, n, k is spherical, i.e., M m, n, k ∼ = S 3 / m, n, k for these groups m, n, k . It is not difficult to prove that, the abelianization of m, n, k ∼ = Z ⊕ H for some group H if and only if (m, n, k) = (6, 3, 2), (4,4,2) or (3,3,3). Therefore, in these three cases, the 3-manifold M m, n, k has a handle and in all the other cases, M m, n, k is handle-free.…”

Section: Binary Polyhedral Groups and Generalized Quaternion Spacesmentioning

confidence: 99%

“…Since m 3 is not such a vertex, i.e., m = 2. Therefore, x (1) 3 and x (2) 3 are joined to G 2 with edges of color 3. Let x…”

Section: Figure 4: Crystallization Of a Hyperbolic 3-manifoldmentioning

confidence: 99%

“…This contradicts that 9 vertices of G 2 have both the 2-adjacent vertices and the 3-adjacent vertices in G 1 . Therefore, m = 2 and n = 6 and hence x [6,Proposition 4] and from the catalogue in [2], these crystallizations are vertex-minimal regular genus two crystallizations when the number of vertices of the crystallizations are at most 42. The crystallizations constructed in Examples 4.3 and 4.5 are vertex-minimal for all known cases.…”

Section: Figure 4: Crystallization Of a Hyperbolic 3-manifoldmentioning

confidence: 99%

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An algorithmic approach to construct crystallizations of 3-manifolds from presentations of fundamental groups

Basak

2016

Proc Math Sci

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We have defined the weight of the pair ( S | R , R) for a given presentation S | R of a group, where the number of generators is equal to the number of relations. We present an algorithm to construct crystallizations of 3-manifolds whose fundamental group has a presentation with two generators and two relations. If the weight of ( S | R , R) is n then our algorithm constructs all the n-vertex crystallizations which yield ( S | R , R).As an application, we have constructed some new crystallizations of 3-manifolds.We have generalized our algorithm for presentations with three generators and a certain class of relations. For m ≥ 3 and m ≥ n ≥ k ≥ 2, our generalized algorithm gives a 2(2m + 2n + 2k − 6 + δ 2 n + δ 2 k )-vertex crystallization of the closed connected orientable 3-manifold M m, n, k having fundamental group x 1 , x 2 , x 3 | x m 1 = x n 2 = x k 3 = x 1 x 2 x 3 . These crystallizations are minimal and unique with respect to the given presentations. If 'n = 2' or 'k ≥ 3 and m ≥ 4' then our crystallization of M m, n, k is vertex-minimal for all the known cases.MSC 2010 : Primary 57Q15. Secondary 05C15, 57N10, 57Q05.

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An algorithmic method to compute plat slide moves in 3-manifolds of Heegaard genus two

Cavicchioli

2023

Discrete Mathematics

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A census of genus-two 3-manifolds up to 42 coloured tetrahedra

Abstract: We improve and extend to the non-orientable case a recent result of Karábaš, Maličký and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra

Cited by 7 publications

References 21 publications

Cataloguing PL 4-Manifolds by Gem-Complexity

Cataloguing PL 4-Manifolds by Gem-Complexity

An algorithmic approach to construct crystallizations of 3-manifolds from presentations of fundamental groups

An algorithmic method to compute plat slide moves in 3-manifolds of Heegaard genus two

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