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Testing Bi-orderability of Knot Groups

Published online by Cambridge University Press:  20 November 2018

Adam Clay ,
Colin Desmarais  and
Patrick Naylor
Adam Clay
Affiliation:
Department of Mathematics, 420 Machray Hall, University of Manitoba, Winnipeg, MB, R3T 2N2 e-mail: adam.clay@umanitoba.ca e-mail: umdesmac@myumanitoba.ca
Colin Desmarais
Affiliation:
Department of Mathematics, 420 Machray Hall, University of Manitoba, Winnipeg, MB, R3T 2N2 e-mail: adam.clay@umanitoba.ca e-mail: umdesmac@myumanitoba.ca
Patrick Naylor
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1 e-mail: naylorp@myumanitoba.ca
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Abstract

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We investigate the bi-orderability of two-bridge knot groups and the groups of knots with 12 or fewer crossings by applying recent theorems of Chiswell, Glass and Wilson. Amongst all knots with 12 or fewer crossings (of which there are 2977), previous theorems were only able to determine bi-orderability of 499 of the corresponding knot groups. With our methods we are able to deal with 191 more.

Keywords

57M2557M2706F15knotsfundamental groupsorderable groups
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 472 - 482
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Akhmedovand, A. Martin, C., Non-bi-orderability of 62 and76- arxiv:abs/1512.00372.Google Scholar
[2] Boyer, S., Rolfsen, D., and Wiest, B., Orderable 3-manifold groups. Ann. Institut Fourier (Grenoble) 55(2005), 243–288. http://dx.doi.org/10.5802/aif.2098 Google Scholar
[3] Boyer, S., Gordon, C. M., and Watson, L., On L-spaces and left-orderable fundamental groups. Math. Ann. 356(2013), no. 4, 1213–1245. http://dx.doi.org/10.1007/s00208-012-0852-7 Google Scholar
[4] Cha, J. C. and Livingston, C., Knotinfo: Table of knot invariants, Http://Www.Indiana.Edu/Knotinfo.Google Scholar
[5] Chiswell, I. M., Glass, A. M. W., and Wilson, J. S., Residual nilpotence and ordering in one-relator groups and knot groups. Math. Proc. Cambridge Philos. Soc. 158(2015), no. 2, 275–288. http://dx.doi.org/10.1017/S0305004114000644 Google Scholar
[6] Clay, A. and Rolfsen, D., Ordered groups, eigenvalues, knots, surgery and L-spaces. Math.Proc. Cambridge Philos. Soc. 152(2012), no. 1,115-129.http://dx.doi.org/10.1017/S0305004111000557 Google Scholar
[7] Culler, M., Dunfield, N. M., and Weeks, J. R., SnapPy, a computer program for studying the topology ofi-manifolds. http://snappy.computop.orgGoogle Scholar
[8] Dorrie, H., 100 great problems of elementary mathematics: their history and solution. Dover, New York, 1965.Google Scholar
[9] Hoste, J. and Shanahan, P. D., Twisted Alexander polynomials of2-bridge knots. J. Knot Theory Ramifications 22(2013), no. 1, 1250138, 29.http://dx.doi.org/10.1142/S0218216512501386 Google Scholar
[10] Ito, T.. A remark on the Alexander polynomial criterion for the bi-orderabilityoffibered 3-manifold groups. Int. Math. Res. Not. IMRN (1):156-169, 2013.Google Scholar
[11] Linnell, P. A., Rhemtulla, A. H., and Rolfsen, D. P. O., Invariant group orderings and Galois conjugates. J. Algebra 319(2008), no. 12, 4891–4898. http://dx.doi.org/10.1016/j.jalgebra.2008.03.002 Google Scholar
[12] Lyndon, R. C. and Schupp, P. E., Combinatorial group theory. Ergebnisse der Mathematik und ihrerGrenzgebiete 89, Springer-Verlag, Berlin, 1977.Google Scholar
[13] Murasugi, K., Remarks on knots with two bridges. Proc. Japan Acad. 37(1961), 294–297. http://dx.doi.org/10.3792/pja/11 95523675 Google Scholar
[14] Naylor, G. and Rolfsen, D., Generalized torsion in knot groups. Canad.Math. Bull. 59(2016), no. 1, 182–189. http://dx.doi.org/10.4153/CMB-2015-004-4 Google Scholar
[15] Perron, B. and Rolfsen, D., On orderability of fibred knot groups. Math. Proc. Cambridge Philos. Soc. 135(2003), no. 1, 147–153. http://dx.doi.org/1 0.101 7/S0305004103006674 Google Scholar
[16] Perron, B. and Rolfsen, D., Invariant ordering of surface groups and 3-manifolds which fibre over S'.Math. Proc. Cambridge Philos. Soc. 141(2006), no. 2, 273–280. http://dx.doi.org/10.1017/S0305004106009558 Google Scholar
[17] Saveliev, N., Lectures on the topology ofS-manifolds. deGruyter Textbook. Walter de Gruyter, Berlin, 1999.Google Scholar
[18] Schubert, H., Knotenmit zweiBrucken. Math. Z. 65(1956), 133–170. http://dx.doi.org/!0.1007/BF01473875 Google Scholar