Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-01T03:23:30.837Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected], [email protected], [email protected])
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 ([email protected], [email protected], [email protected])
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 ([email protected], [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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.)

References

Andruskiewitsch, N. and Graña, M., From racks to pointed Hopf algebras, Adv. Math. 178(2) (2003), 177–243.CrossRefGoogle Scholar
Asaeda, Marta M., Przytycki, Jozef H. and Sikora, Adam S., Kauffman-Harary conjecture holds for Montesinos knots, J. Knot Theory Ramifications 13(4) (2004), 467–477.CrossRefGoogle Scholar
Bardakov, V. and Nasybullov, T., Embeddings of quandles into groups, J. Algebra Appl. 19(7) (2020), 2050136.CrossRefGoogle Scholar
Bardakov, V. G., Dey, P. and Singh, M., Automorphism groups of quandles arising from groups, Monatsh. Math. 184 (2017), 519–530.CrossRefGoogle Scholar
Bardakov, V., Nasybullov, T. and Singh, M., Automorphism groups of quandles and related groups, Monatsh. Math. 189(1) (2019), 1–21.CrossRefGoogle Scholar
Bardakov, V. G., Passi, I. B. S. and Singh, M., Zero-divisors and idempotents in quandle rings, Osaka J. Math (2021), to appear, arXiv:2001.06843v2Google Scholar
Boyer, S., Rolfsen, D. and Wiest, B., Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble) 55(1) (2005), 243–288.CrossRefGoogle Scholar
Carter, J. S., Kamada, S. and Saito, M., Diagrammatic computations for quandles and cocycle knot invariants, Diagrammatic morphisms and applications (San Francisco, CA, 2000), pp. 51–74, Contemp. Math., Volume 318 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
Clay, A. and Rolfsen, D., Ordered groups and topology, Graduate Studies in Mathematics, Volume 176, pp. x+154 (American Mathematical Society, Providence, RI, 2016).CrossRefGoogle Scholar
Dabkowska, M. A., Dabkowski, M. K., Harizanov, V. S., Przytycki, J. H. and Veve, M. A., Compactness of the space of left orders, J. Knot Theory Ramifications 16(7) (2007), 257–266.CrossRefGoogle Scholar
Dehornoy, P., Braid groups and left distributive operations, Trans. Am. Math. Soc. 345 (1994), 115–150.CrossRefGoogle Scholar
Dehornoy, P., Dynnikov, I., Rolfsen, D. and Wiest, B., Ordering braids, Mathematical Surveys and Monographs, Volume 148, x+323 (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar
Dowdall, N. E., Mattman, T. W., Meek, K. and Solis, P. R., On the Harary-Kauffman conjecture and Turk's head knots, Kobe J. Math. 27(1-2) (2010), 1–20.Google Scholar
Falk, M. and Randell, R., Pure braid groups and products of free groups, Contemp. Math. 78 (1988), 217–228.CrossRefGoogle Scholar
Fenn, R. and Rourke, C., Racks and links in codimension two, J. Knot Theory Ramifications 1(4) (1992), 343–406.CrossRefGoogle Scholar
Ha, T., On algorithmic properties of computable magmas, Thesis (Ph.D.) (The George Washington University, 2018), 80.Google Scholar
Ha, T. and Harizanov, V., Orders on magmas and computability theory, J. Knot Theory Ramifications 27(7) (2018), 1841001. 13 pp.10.1142/S0218216518410018CrossRefGoogle Scholar
Joyce, D., An algebraic approach to symmetry with applications to knot theory, Ph.D. Thesis (University of Pennsylvania, 1979).Google Scholar
Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), 37–65.CrossRefGoogle Scholar
Kamada, S., Kyokumen musubime riron (Surface-knot theory) (in Japanese), Springer Gendai Sugaku Series, Volume 16 (Maruzen Publishing Co. Ltd, 2012).Google Scholar
Kamada, S., Surface-knots in 4-space. An introduction, Springer Monographs in Mathematics, xi+212 (Springer, Singapore, 2017).CrossRefGoogle Scholar
Kauffman, L. H. and Lambropoulou, S., On the classification of rational tangles, Adv. Appl. Math. 33 (2004), 199–237.10.1016/j.aam.2003.06.002CrossRefGoogle Scholar
Linnell, P. A., Rhemtulla, A. H. and Rolfsen, D., Invariant group orderings and Galois conjugates, J. Algebra 319 (2008), 4891–4898.CrossRefGoogle Scholar
Mattman, T. W. and Solis, P., A proof of the Kauffman-Harary conjecture, Algebr. Geom. Topol. 9 (2009), 2027–2039.CrossRefGoogle Scholar
Matveev, S. V., Distributive groupoids in knot theory, Russian: Mat. Sb. (N.S.) 119(1) (1982), 78–88. translated in Math. USSR Sb. 47.Google Scholar
Nakamura, T., Positive alternating links are positively alternating, J. Knot Theory Ramifications 9(1) (2000), 107–112.CrossRefGoogle Scholar
Nakamura, T., On a positive knot without positive minimal diagrams, Proceedings of the Winter Workshop of Topology/Workshop of Topology and Computer (Sendai, 2002/Nara, 2001), Interdiscip. Inf. Sci. 9 (1) (2003), 61–75.CrossRefGoogle Scholar
Perron, B. and Rolfsen, D., On orderability of fibred knot groups, Math. Proc. Cambridge Philos. Soc. 135(1) (2003), 147–153.CrossRefGoogle Scholar
Rost, M. and Zieschang, H., Meridional generators and plat presentations of torus links, J. London Math. Soc. (2) 35(3) (1987), 551–562.CrossRefGoogle Scholar
Rourke, C. and Wiest, B., Order automatic mapping class groups, Pacific J. Math. 194 (2000), 209–227.CrossRefGoogle Scholar
Ryder, H., An algebraic condition to determine whether a knot is prime, Math. Proc. Camb. Philos. Soc. 120 (1996), 385–389.CrossRefGoogle Scholar
Stoimenow, A., On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks, Trans. Am. Math. Soc. 354(10) (2002), 3927–3954.10.1090/S0002-9947-02-03022-2CrossRefGoogle Scholar
Vinogradov, A. A., On the free product of ordered groups, (Russian) Mat. Sbornik N.S. 25(67) (1949), 163–168.Google Scholar
Winker, S. K., Quandles, knots invariants and the n-fold branched cover, Thesis (Ph.D.) (University of Illinois at Chicago, 1984), 198.Google Scholar