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Some Conditionally Hard Problems on Links and 3-Manifolds

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Abstract

We show that three natural decision problems about links and 3-manifolds are computationally hard, assuming some conjectures in complexity theory. The first problem is determining whether a link in the 3-sphere bounds a Seifert surface with Thurston norm at most a given integer; this is shown to be NP-complete. The second problem is the homeomorphism problem for closed 3-manifolds; this is shown to be at least as hard as the graph isomorphism problem. The third problem is determining whether a given link in the 3-sphere is a sublink of another given link; this is shown to be NP-hard.

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Acknowledgements

The author would like to thank the referee for their suggestions which have undoubtedly improved this paper.

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  1. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Marc Lackenby

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  1. Marc Lackenby
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Correspondence to Marc Lackenby.

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Lackenby, M. Some Conditionally Hard Problems on Links and 3-Manifolds. Discrete Comput Geom 58, 580–595 (2017). https://doi.org/10.1007/s00454-017-9905-8

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