Brooke-Taylor, A orcid.org/0000-0003-3734-0933 (2020) The Quandary of Quandles: a Borel Complete Knot Invariant. Journal of the Australian Mathematical Society, 108 (2). pp. 262-277. ISSN 1446-7887
Abstract
We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019 Australian Mathematical Publishing Association Inc. This is an author produced version of a paper published in Journal of the Australian Mathematical Society. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | quandle; left distributive; Borel complete; knot invariant |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number EPSRC (Engineering and Physical Sciences Research Council) EP/K035703/2 |
Depositing User: | Symplectic Publications |
Date Deposited: | 26 Jul 2019 09:33 |
Last Modified: | 27 Jan 2022 17:50 |
Status: | Published |
Publisher: | Cambridge University Press |
Identification Number: | 10.1017/S1446788719000399 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:149046 |