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Low complexity algorithms in knot theory

Lookup NU author(s): Dr Alina Vdovina



This is the authors' accepted manuscript of an article that has been published in its final definitive form by World Scientific Publishing Co. Pte Ltd, 2019.

For re-use rights please refer to the publisher's terms and conditions.


© 2018 World Scientific Publishing Company. Agol, Haas and Thurston showed that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. This shows that (unless P=NP) the genus problem has high computational complexity even for knots in a 3-manifold. We initiate the study of classes of knots where the genus problem and even the equivalence problem have very low computational complexity. We show that the genus problem for alternating knots with n crossings has linear time complexity and is in Logspace(n). Alternating knots with some additional combinatorial structure will be referred to as standard. As expected, almost all alternating knots of a given genus are standard. We show that the genus problem for these knots belongs to TC0 circuit complexity class. We also show, that the equivalence problem for such knots with n crossings has time complexity nlog(n) and is in Logspace(n) and TC0 complexity classes.

Publication metadata

Author(s): Kharlampovich O, Vdovina A

Publication type: Article

Publication status: Published

Journal: International Journal of Algebra and Computation

Year: 2019

Volume: 29

Issue: 2

Pages: 245-262

Print publication date: 01/03/2019

Online publication date: 12/10/2018

Acceptance date: 09/09/2018

Date deposited: 21/11/2018

ISSN (print): 0218-1967

ISSN (electronic): 1793-6500

Publisher: World Scientific Publishing Co. Pte Ltd


DOI: 10.1142/S0218196718500698


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