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Published online by Cambridge University Press:  30 October 2019

School of Mathematics, University of Leeds, Leeds LS2 9JT, UK email
28 Archuleta Road, Ranchos de Taos, NM 87557, USA email


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.

Research Article
© 2019 Australian Mathematical Publishing Association Inc.

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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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