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Orderability of link quandles

Part of: Other generalizations of groups

Published online by Cambridge University Press:  16 August 2021

Hitesh Raundal ,
Mahender Singh  and
Manpreet Singh
Hitesh Raundal
Affiliation:
Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Mohali, Punjab140306, India (hiteshrndl@gmail.com, mahender@iisermohali.ac.in, mp15009@iisermohali.ac.in)
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, Mohali, Punjab140306, India (hiteshrndl@gmail.com, mahender@iisermohali.ac.in, mp15009@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, Mohali, Punjab140306, India (hiteshrndl@gmail.com, mahender@iisermohali.ac.in, mp15009@iisermohali.ac.in)
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Abstract

The paper develops a general theory of orderability of quandles with a focus on link quandles of tame links and gives some general constructions of orderable quandles. We prove that knot quandles of many fibred prime knots are right-orderable, whereas link quandles of most non-trivial torus links are not right-orderable. As a consequence, we deduce that the knot quandle of the trefoil is neither left nor right-orderable. Further, it is proved that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The paper also explores interconnections between orderability of quandles and that of their enveloping groups. The results establish that orderability of link quandles behaves quite differently than that of corresponding link groups.

Keywords

link quandlelink groupMontesinos linkorderable grouporderable quandletorus linkquandle 2-cocycle

MSC classification

Secondary: 20N02: Sets with a single binary operation (groupoids)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 3 , August 2021 , pp. 620 - 649
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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