Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-12T19:45:15.615Z Has data issue: false hasContentIssue false
  • English
  • Français

THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT

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

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 [email protected]
SHEILA K. MILLER
Affiliation:
28 Archuleta Road, Ranchos de Taos, NM 87557, USA email [email protected]
*
✉[email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Bertram, W., The Geometry of Jordan and Lie Structures, Lecture Notes in Mathematics, 1754 (Springer, Berlin, 2000).Google Scholar
Brieskorn, E., ‘Automorphic sets and braids and singularities’, in: Braids (Santa Cruz, CA, 1986), Contemporary Mathematics, 78 (American Mathematical Society, Providence, RI, 1988), 45–115.Google Scholar
Camerlo, R. and Gao, S., ‘The completeness of the isomorphism relation for countable Boolean algebras’, Trans. Amer. Math. Soc. 353(2) (2001), 491–518.Google Scholar
Dehornoy, P., ‘Braid groups and self-distributive operations’, Trans. Amer. Math. Soc. 345(1) (1994), 115–151.Google Scholar
Dehornoy, P., Braids and Self Distributivity, Progress in Mathematics, 192 (Birkhäuser, Basel, 2000).Google Scholar
Dehornoy, P., ‘Elementary embeddings and algebra’, in: The Handbook of Set Theory (eds. Foreman, M. and Kanamori, A.) (Springer, The Netherlands, 2010), 737–774.Google Scholar
Elhamdadi, M. and Nelson, S., Quandles: An Introduction to the Algebra of Knots, Student Mathematical Library, 74 (AMS, Providence, RI, 2015).Google Scholar
Farah, I., Toms, A. and Törnquist, A., ‘Turbulence, orbit equivalence, and the classification of nuclear C ∗ -algebras’, J. reine angew. Math. (Crelle’s Journal) 688 (2014), 101–146.Google Scholar
Foreman, M., Rudolph, D. and Weiss, B., ‘The conjugacy problem in ergodic theory’, Ann. Math. 173 (2011), 1529–1586.Google Scholar
Friedman, H. and Stanley, L., ‘A Borel reducibility theory for classes of countable structures’, J. Symbolic Logic 54(3) (1989), 894–914.Google Scholar
Gao, S., Invariant Descriptive Set Theory, Pure and Applied Mathematics (Chapman & Hall/CRC Press, an imprint of Taylor & Francis Group, Boca Raton, 2009).Google Scholar
Hjorth, G., Classification and Orbit Equivalence Relations, Mathematical Surveys and Monographs, 75 (American Mathematical Society, Providence, RI, 2002).Google Scholar
Joyce, D., ‘A classifying invariant of knots, the knot quandle’, J. Pure Appl. Algebra 23 (1982), 37–65.Google Scholar
Kamada, S., ‘Knot invariants derived from quandles and racks’, in: Invariants of Knots and 3-Manifolds (Kyoto, 2001), Geometry and Topology Monographs, 4 (eds. Ohtsuki, T., Kohno, T., Le, T., Murakami, J., Roberts, J. and Turaev, V.) (Geometry and Topology Publications, Coventry, 2004).Google Scholar
Kamada, S., ‘Quandles derived from dynamical systems and subsets which are closed under quandle operations’, Topol. Appl. 157 (2010), 298–301.Google Scholar
Kulikov, V., ‘A non-classification result for wild knots’, Trans. Amer. Math. Soc. 369 (2017), 5829–5853.Google Scholar
Laver, R., ‘The left distributive law and the freeness of an algebra of elementary embeddings’, Adv. Math. 91 (1992), 209–231.Google Scholar
Laver, R., ‘A division algorithm for the free left distributive algebra’, in: Logic Colloquium ’90 (Helsinki, 1990), Lecture Notes in Logic, 2 (Springer, Berlin, 1993), 155–162.Google Scholar
Laver, R. and Miller, S. K., ‘The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm’, Cent. Eur. J. Math. 11(12) (2013), 2150–2175.Google Scholar
Mekler, A. H., ‘Stability of nilpotent groups of class 2 and prime exponent’, J. Symbolic Logic 46(4) (1981), 781–788.Google Scholar
Przeździecki, A. J., ‘An almost full embedding of the category of graphs into the category of abelian groups’, Adv. Math. 257 (2014), 527–545.Google Scholar
Pultr, A. and Trnková, V., Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories (Academia, Prague, 1980).Google Scholar
Takasaki, M., ‘Abstraction of symmetric transformations’, Tohoku Math. J. 49 (1943), 145–207.Google Scholar