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Left-orderable Fundamental Group and Dehn Surgery on the Knot 52

Published online by Cambridge University Press:  20 November 2018

Ryoto Hakamata  and
Masakazu Teragaito
Ryoto Hakamata
Affiliation:
Graduate School of Education, Hiroshima University, Higashi-hiroshima, Japan 739-8524
Masakazu Teragaito
Affiliation:
Department of Mathematics and Mathematics Education, Hiroshima University, Higashi-hiroshima, Japan 739-8524 e-mail: [email protected]
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Abstract

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We show that the manifold resulting from $r$-surgery on the knot ${{5}_{2}}$, which is the two-bridge knot corresponding to the rational number $3/7$, has a left-orderable fundamental group if the slope $r$ satisfies $0\,\le \,r\,\le \,4$.

Keywords

57M2506F15left-orderingDehn surgery
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 2 , 14 June 2014 , pp. 310 - 317
Copyright
Copyright © Canadian Mathematical Society 2014

Footnotes

The second author is partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 22540088.

References

[1] Bargmann, V., Irreducible unitary representations of the Lorentz group. Ann. of Math. 48 (1947, 568–640.http://dx.doi.org/10.2307/1969129 Google Scholar
[2] Bergman, G. M., Right orderable groups that are not locally indicable. Pacific J. Math. 147 (1991, no. 2, 243–248.http://dx.doi.org/10.2140/pjm.1991.147.243 Google Scholar
[3] Boyer, S., McA. Gordon, C., 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] Boyer, S., Rolfsen, D., and Wiest, B., Orderable 3-manifold groups. Ann. Inst. Fourier (Grenoble) 55 (2005, no. 1, 243–288. http://dx.doi.org/10.5802/aif.2098 Google Scholar
[5] Brittenham, M. and Wu, Y.-Q., The classification of exceptional Dehn surgeries on 2-bridge knots. Comm. Anal. Geom. 9 (2001, no. 1, 97–113.CrossRefGoogle Scholar
[6] Clay, A., Lidman, T., and Watson, L., Graph manifolds, left-orderability and amalgamation. Algebr. Geom. Topol. 13 (2013, no. 4, 2347–2368.http://dx.doi.org/10.2140/agt.2013.13.2347 Google Scholar
[7] Ghys, É., Groups acting on the circle. Enseign. Math. 47 (2001, no. 3–4, 329–407.Google Scholar
[8] Hatcher, A. and Thurston, W., Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79 (1985, no. 2, 225–246.http://dx.doi.org/10.1007/BF01388971 CrossRefGoogle Scholar
[9] Khoi, V. T., A cut-and-paste method for computing the Seifert volumes.Math. Ann. 326 (2003, no. 4, 759–801.http://dx.doi.org/10.1007/s00208-003-0438-5 Google Scholar
[10] Lion, G. and Vergne, M., The Weil representation, Maslov index and theta series. Progress in Mathematics, 6, Birkhäuser, Boston, Mass., 1980.Google Scholar
[11] Ni, Y., Knot Floer homology detects fibred knots. Invent. Math. 170 (2007, no. 3, 577–608.http://dx.doi.org/10.1007/s00222-007-0075-9 Google Scholar
[12] Ozsv´ath, P. and Szabó, Z., On knot Floer homology and lens space surgeries. Topology 44 (2005, no. 6, 1281–1300.http://dx.doi.org/10.1016/j.top.2005.05.001 Google Scholar
[13] Riley, R., Nonabelian representations of 2-bridge knot groups. Quart. J. Math. Oxford Ser. (2) 35 (1984, no. 138, 191–208.http://dx.doi.org/10.1093/qmath/35.2.191 CrossRefGoogle Scholar
[14] Rolfsen, D., Knots and links. Mathematics Lecture Series, 7, Publish or Perish, Inc., Berkeley, Calif., 1976.Google Scholar
[15] Schubert, H., Knoten mit zwei Br¨ucken. Math. Z. 65 (1956, 133–170.http://dx.doi.org/10.1007/BF01473875 Google Scholar
[16] Teragaito, M., Left-orderability and exceptional Dehn surgery on twist knots. Canad. Math. Bull., to appear.http://dx.doi.org/10.4153/CMB-2012-011-0 Google Scholar
[17] Wood, J.W., Bundles with totally disconnected structure group. Comment. Math. Helv. 46 (1971, 257–273.http://dx.doi.org/10.1007/BF02566843 Google Scholar