research-article

Embeddability in the 3-Sphere Is Decidable

Authors:

Jiří Matoušek,

Uli WagnerAuthors Info & Claims

Journal of the ACM (JACM), Volume 65, Issue 1

Article No.: 5, Pages 1 - 49

https://doi.org/10.1145/3078632

Published: 23 January 2018 Publication History

Abstract

We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in R³? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S³. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S³, then there is also an embedding in which X has a short meridian, that is, an essential curve in the boundary of X bounding a disk in S³ \ X with length bounded by a computable function of the number of tetrahedra of X.

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Cited By

Arseneva EKleist LKlemz BLöffler MSchulz AVogtenhuber BWolff A(2023)Adjacency Graphs of Polyhedral SurfacesDiscrete & Computational Geometry10.1007/s00454-023-00537-671:4(1429-1455)Online publication date: 18-Oct-2023
https://doi.org/10.1007/s00454-023-00537-6
Fulek RTóth C(2022)Atomic Embeddability, Clustered Planarity, and ThickenabilityJournal of the ACM10.1145/350226469:2(1-34)Online publication date: 31-Jan-2022
https://dl.acm.org/doi/10.1145/3502264
Fulek RTóth CChawla S(2020)Atomic embeddability, clustered planarity, and thickenabilityProceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3381089.3381264(2876-2895)Online publication date: 5-Jan-2020
https://dl.acm.org/doi/10.5555/3381089.3381264
Show More Cited By

Index Terms

Embeddability in the 3-Sphere Is Decidable
1. Mathematics of computing
  1. Continuous mathematics
    1. Topology
      1. Geometric topology
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 65, Issue 1

February 2018

209 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3155102

Editor:
Éva Tardos
Cornell University

Issue’s Table of Contents

Copyright © 2018 ACM.

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Association for Computing Machinery

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Publication History

Published: 23 January 2018

Accepted: 01 April 2017

Revised: 01 February 2017

Received: 01 March 2014

Published in JACM Volume 65, Issue 1

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Cited By

Arseneva EKleist LKlemz BLöffler MSchulz AVogtenhuber BWolff A(2023)Adjacency Graphs of Polyhedral SurfacesDiscrete & Computational Geometry10.1007/s00454-023-00537-671:4(1429-1455)Online publication date: 18-Oct-2023
https://doi.org/10.1007/s00454-023-00537-6
Fulek RTóth C(2022)Atomic Embeddability, Clustered Planarity, and ThickenabilityJournal of the ACM10.1145/350226469:2(1-34)Online publication date: 31-Jan-2022
https://dl.acm.org/doi/10.1145/3502264
Fulek RTóth CChawla S(2020)Atomic embeddability, clustered planarity, and thickenabilityProceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3381089.3381264(2876-2895)Online publication date: 5-Jan-2020
https://dl.acm.org/doi/10.5555/3381089.3381264
Mesmay ARieck YSedgwick ETancer M(2020)Embeddability in R3 is NP-hardJournal of the ACM10.1145/339659367:4(1-29)Online publication date: 4-Jun-2020
https://dl.acm.org/doi/10.1145/3396593
Skopenkov A(2020)Invariants of Graph Drawings in the PlaneArnold Mathematical Journal10.1007/s40598-019-00128-5Online publication date: 13-Feb-2020
https://doi.org/10.1007/s40598-019-00128-5

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