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Free quandles and knot quandles are residually finite


Authors: Valeriy G. Bardakov, Mahender Singh and Manpreet Singh
Journal: Proc. Amer. Math. Soc. 147 (2019), 3621-3633
MSC (2010): Primary 57M25; Secondary 20E26, 57M05, 20N05
DOI: https://doi.org/10.1090/proc/14488
Published electronically: April 8, 2019
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Abstract: In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core, and Alexander quandles) is established. Further, residual finiteness of automorphism groups of some residually finite quandles is also discussed.


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Valeriy G. Bardakov
Affiliation: Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, 630090, Novosibirsk, Russia; Novosibirsk State University, 2 Pirogova Street, 630090, Novosibirsk, Russia; Novosibirsk State Agrarian University, Dobrolyubova Street, 160, Novosibirsk, 630039, Russia
Email: bardakov@math.nsc.ru

Mahender Singh
Affiliation: Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India
Email: mahender@iisermohali.ac.in

Manpreet Singh
Affiliation: Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India
Email: manpreetsingh@iisermohali.ac.in

DOI: https://doi.org/10.1090/proc/14488
Keywords: 3-manifold group, enveloping group, free quandle, knot group, knot quandle, peripheral subgroup, residually finite quandle
Received by editor(s): May 20, 2018
Received by editor(s) in revised form: November 22, 2018
Published electronically: April 8, 2019
Additional Notes: The first author acknowledges support from the Russian Science Foundation project N 16-41-02006.
The second author acknowledges support from INT/RUS/RSF/P-02 grant and SERB Matrics Grant.
The third author thanks IISER Mohali for the PhD Research Fellowship.
Communicated by: David Futer
Article copyright: © Copyright 2019 American Mathematical Society