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THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT

Part of: Set theory Other generalizations of groups Low-dimensional topology

Published online by Cambridge University Press:  30 October 2019

ANDREW D. BROOKE-TAYLOR Open the ORCID record for ANDREW D. BROOKE-TAYLOR [Opens in a new window]  and
SHEILA K. MILLER
ANDREW D. BROOKE-TAYLOR*
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK email andrewbt@gmail.com
SHEILA K. MILLER
Affiliation:
28 Archuleta Road, Ranchos de Taos, NM 87557, USA email sheila.miller@colorado.edu
*
✉andrewbt@gmail.com
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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.

Keywords

quandleleft distributiveBorel completeknot invariant

MSC classification

Primary: 20N99: None of the above, but in this section
Secondary: 03E15: Descriptive set theory 57M27: Invariants of knots and 3-manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 2 , April 2020 , pp. 262 - 277
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Much of this work was carried out while both authors were Visiting Fellows at the Isaac Newton Institute for Mathematical Sciences in the programme ‘Mathematical, Foundational and Computational Aspects of the Higher Infinite’ (HIF) funded by EPSRC grant EP/K032208/1. The first author was supported by the UK Engineering and Physical Sciences Research Council Early Career Fellowship EP/K035703/1 and EP/K035703/2, Bringing set theory and algebraic topology together, and undertook some of the work whilst visiting the Centre de Recerca Matemàtica for the programme Large cardinals and strong logics. The second author was supported by grants from PSC-CUNY and the City Tech PDAC.

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