A model for random three--manifolds

Petri, Bram; Raimbault, Jean

doi:10.48550/arxiv.2009.11923

Cited by 3 publications

(3 citation statements)

References 30 publications

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“…On the probabilistic side, it would be interesting to generalize this work to other models of random 3-manifolds (e.g. see [AFW15, Section 7.4], [PR20]). Do they produce the same probability measure?…”

Section: Existential Theorymentioning

confidence: 99%

Finite quotients of 3-manifold groups

Sawin¹,

Wood²

2022

Preprint

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For G and H 1 , . . . , H n finite groups, does there exist a 3-manifold group with G as a quotient but no H i as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments.2. Properties of the fundamental group of closed 3-manifolds

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Section: Existential Theorymentioning

confidence: 99%

Finite quotients of 3-manifold groups

Sawin¹,

Wood²

2022

Preprint

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“…In dimension three and higher, random gluing of simplices along their boundaries leads to singularities with high probability, making these constructions less useful. In three dimensions, one may glue together 3-simplices with truncated corners, yielding 3-manifolds with a boundary surface of a linearly high genus [PR20].…”

Section: Introductionmentioning

confidence: 99%

Random Colorings in Manifolds

Even-Zohar¹,

Hass²

2022

Preprint

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We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional space by Sheffield and Yadin (2014). We determine conditions on which submanifolds can arise, in terms of Stiefel-Whitney classes and other properties. We then consider the random submanifolds that arise from randomly coloring the vertices. Since this model generates submanifolds, it allows for studying properties and using tools that are not available in processes that produce general random subcomplexes. The case of 3 colors in a triangulated 3-ball gives rise to random knots and links. In this setting, we answer a question raised by de Crouy-Chanel and Simon (2019), showing that the probability of generating an unknot decays exponentially. In the general case of k colors in d-dimensional manifolds, we investigate the random submanifolds of different codimensions, as the number of vertices in the triangulation grows. We compute the expected Euler characteristic, and discuss relations to homological percolation and other topological properties. Finally, we explore a method to search for solutions to topological problems by generating random submanifolds. We describe computer experiments that search for a low-genus surface in the 4-dimensional ball whose boundary is a given knot in the 3-dimensional sphere.

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“…Dunfield and Thurston [DT06] list several models for random three-manifolds that could be developed further. Work by Petri with Baik, Bauer, Gekhtman, Hamenstädt, Hensel, Kastenholz, and Valenzuela [BBG + 18], Thaele [PT18], Mirzakhani [MP19], and Raimbault [PR20] showcases the benefits of exploring various models.…”

Section: Introductionmentioning

confidence: 99%

A lower bound on the average genus of a 2-bridge knot

Cohen¹

2021

Preprint

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Experimental data from Dunfield et al using random grid diagrams suggests that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number.

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A model for random three--manifolds

Cited by 3 publications

References 30 publications

Finite quotients of 3-manifold groups

Finite quotients of 3-manifold groups

Random Colorings in Manifolds

A lower bound on the average genus of a 2-bridge knot

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