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FINITENESS OF CANONICAL QUOTIENTS OF DEHN QUANDLES OF SURFACES

Published online by Cambridge University Press:  11 March 2024

NEERAJ K. DHANWANI  and
MAHENDER SINGH
NEERAJ K. DHANWANI*
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
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 e-mail: mahender@iisermohali.ac.in
*
e-mail: neerajk.dhanwani@gmail.com
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Abstract

The Dehn quandle of a closed orientable surface is the set of isotopy classes of nonseparating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider the finiteness of some canonical quotients of these quandles. For a surface of positive genus, we give a precise description of the 2-quandle of its Dehn quandle. Further, with some exceptions for genus more than 2, we determine all values of n for which the n-quandle of its Dehn quandle is finite. The result can be thought of as the Dehn quandle analogue of a similar result of Hoste and Shanahan for link quandles [‘Links with finite n-quandles’, Algebr. Geom. Topol. 17(5) (2017), 2807–2823]. We also compute the size of the smallest nontrivial quandle quotient of the Dehn quandle of a surface. Along the way, we prove that the involutory quotient of an Artin quandle is precisely the corresponding Coxeter quandle, and also determine the smallest nontrivial quotient of a braid quandle.

Keywords

Artin groupCoxeter quandleDehn quandleinvolutory quandleorientable surfacemapping class group
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 18
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

N.K.D. acknowledges support from the NBHM via grant number 0204/1/2023/R &D-II/606. M.S. is supported by the Swarna Jayanti Fellowship grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20/04.

Communicated by Ben Martin

References

Agarwal, N., Dey, S., Dhanwani, N. K. and Rajeevsarathy, K., ‘Liftable mapping class groups of regular cyclic covers’, Houston J. Math. 47(1) (2021), 223–243.Google Scholar
Akita, T., ‘The adjoint group of a Coxeter quandle’, Kyoto J. Math. 60 (2020), 1245–1260.CrossRefGoogle Scholar
Akita, T., Hasegawa, A. and Tanno, M., ‘Structure of the associated groups of quandles’, Kodai Math. J. 45 (2022), 270–281.CrossRefGoogle Scholar
Allcock, D., ‘Reflection centralizers in Coxeter groups’, Transform. Groups 18(3) (2013), 599–613.CrossRefGoogle Scholar
Birman, J. S., Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, 82 (Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974).Google Scholar
Brink, B., ‘On centralizers of reflections in Coxeter groups’, Bull. Lond. Math. Soc. 28(5) (1996), 465–470.CrossRefGoogle Scholar
Brink, B. and Howlett, R. B., ‘Normalizers of parabolic subgroups in Coxeter groups’, Invent. Math. 136(2) (1999), 323–351.CrossRefGoogle Scholar
Dey, S., Dhanwani, N. K., Patil, H. and Rajeevsarathy, K., ‘Generating the liftable mapping class groups of regular cyclic covers’, Preprint, 2021, arXiv:2111.01626v1.Google Scholar
Dhanwani, N. K., Raundal, H. and Singh, M., ‘Dehn quandles of groups and orientable surfaces’, Fund. Math. 263 (2023), 167–201.CrossRefGoogle Scholar
Dhanwani, N. K., Raundal, H. and Singh, M., ‘Presentations of Dehn quandles’, J. Algebra 636 (2023), 207–247.CrossRefGoogle Scholar
Dunbar, W. D., ‘Geometric orbifolds’, Rev. Mat. Univ. Complut. Madrid 1(1–3) (1988), 67–99.Google Scholar
Farb, B. and Margalit, D., A Primer on Mapping Class Groups, Princeton Mathematical Series, 49 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Funar, L., ‘On the TQFT representations of the mapping class groups’, Pacific J. Math. 188(2) (1999), 251–274.CrossRefGoogle Scholar
Hoste, J. and Shanahan, P. D., ‘Links with finite $n$ -quandles’, Algebr. Geom. Topol. 17(5) (2017), 2807–2823.CrossRefGoogle Scholar
Howlett, R. B., ‘Normalizers of parabolic subgroups of reflection groups’, J. Lond. Math. Soc. (2) 21(1) (1980), 62–80.CrossRefGoogle Scholar
Humphries, S., ‘Normal closures of powers of Dehn twists in mapping class groups’, Glasg. Math. J. 34(3) (1992), 314–317.CrossRefGoogle Scholar
Joyce, D., ‘An algebraic approach to symmetry with applications to knot theory’, PhD 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. and Matsumoto, Y., ‘Certain racks associated with the braid groups’, in: Knots in Hellas ’98 (Delphi), Series on Knots and Everything, 24 (eds C. McA. Gordon, V. F. R. Jones, L. Kauffman, S. Lambropoulou and J. H. Przytycki) (World Scientific Publishing, River Edge, NJ, 2000), 118–130.CrossRefGoogle Scholar
Kolay, S., ‘Smallest non-cyclic quotients of braid and mapping class groups’, Geom. Topol. 27 (2023), 2479–2496.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(1) (1984), 73–83.Google Scholar
Mennicke, J., ‘Zur theorie der Siegelschen Modulgruppe’, Math. Ann. 159 (1965), 115–129.CrossRefGoogle Scholar
Niebrzydowski, M. and Przytycki, J. H., ‘The quandle of the trefoil knot as the Dehn quandle of the torus’, Osaka J. Math. 46 (2009), 645–659.Google Scholar
Nosaka, T., ‘Central extensions of groups and adjoint groups of quandles’, in: Geometry and Analysis of Discrete Groups and Hyperbolic Spaces, RIMS Kokyuroku Bessatsu, B66 (eds M. Fujii, N. Kawazumi and K. Ohshika) (Research Institute for Mathematical Sciences (RIMS), Kyoto, 2017), 167–184.Google Scholar
Putman, A., ‘Lectures on the Torelli group’. www3.nd.edu/~andyp/teaching/2014SpringMath541/TorelliBook.pdf.Google Scholar
Winker, S. K., ‘Quandles, knots invariants and the $n$ -fold branched cover’, PhD Thesis, University of Illinois at Chicago, 1984.Google Scholar
Yetter, D. N., ‘Quandles and Lefschetz fibrations’, Preprint, 2002, arXiv:math/0201270.Google Scholar
Yetter, D. N., ‘Quandles and monodromy’, J. Knot Theory Ramifications 12(4) (2003), 523–541.CrossRefGoogle Scholar
Zablow, J., ‘Loops, waves, and an “algebra” for Heegaard splittings’, PhD Thesis, City University of New York, 1999.Google Scholar
Zablow, J., ‘Loops and disks in surfaces and handlebodies’, J. Knot Theory Ramifications 12(2) (2003), 203–223.CrossRefGoogle Scholar
Zablow, J., ‘On relations and homology of the Dehn quandle’, Algebr. Geom. Topol. 8(1) (2008), 19–51.CrossRefGoogle Scholar