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KNOT GROUPS WITH MANY KILLERS

Part of: Low-dimensional topology

Published online by Cambridge University Press:  13 April 2010

DANIEL S. SILVER ,
WILBUR WHITTEN  and
SUSAN G. WILLIAMS
DANIEL S. SILVER*
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA (email: silver@jaguar1.usouthal.edu)
WILBUR WHITTEN
Affiliation:
1620 Cottontown Road, Forest, VA 24551, USA (email: bjwcw@aol.com)
SUSAN G. WILLIAMS
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA (email: swilliam@jaguar1.usouthal.edu)
*
✉For correspondence; e-mail: silver@jaguar1.usouthal.edu
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Abstract

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The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none of which is the automorphic image of another, such that each normally generates the group.

Keywords

knot groupmeridian2-bridgeunknotting number

MSC classification

Secondary: 57M25: Knots and links in $S^3$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 3 , June 2010 , pp. 507 - 513
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first and third authors are partially supported by NSF grant DMS-0706798.

References

[1]Boileau, M., Boyer, S., Reid, A. and Wang, S., ‘Simon’s conjecture for 2-bridge knots’, Comm. Anal. Geom., to appear, arXiv:0903.2898.Google Scholar
[2]Burde, G. and Zieschang, H., Knots, 2nd edn de Gruyter Studies in Mathematics, 5 (de Gruyter, Berlin, 2003).Google Scholar
[3]Jaco, W., Lectures on Three-Manifold Topology, CBMS Regional Conference Series in Mathematics, 43 (American Mathematical Society, Providence, RI, 1980).Google Scholar
[4]Johannson, K., Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, 761 (Springer, Berlin, 1979).Google Scholar
[5]Johnson, D. and Livingston, C., ‘Peripherally specified homomorphs of knot groups’, Trans. Amer. Math. Soc. 311 (1989), 135–146.CrossRefGoogle Scholar
[6]Kawauchi, A., A Survey of Knot Theory (Birkhäuser, Basel, 1996).Google Scholar
[7]Kirby, R., ‘Problems in low-dimensional topology’, in: Geometric Topology (Athens, GA, 1993). (American Mathematical Society Publications, Providence, RI, 1997), pp. 35–473.Google Scholar
[8]Kronheimer, P. B. and Mrowka, T. S., ‘Witten’s conjecture and property P’, Geom. Topol. 8 (2004), 295–310.CrossRefGoogle Scholar
[9]Maclachlan, C. and Reid, A. W., The Arithmetic of Hyperbolic 3-Manifolds (Springer, New York, 2003).CrossRefGoogle Scholar
[10]Riley, R., ‘Parabolic representations of knot groups, I’, Proc. London Math. Soc. 24 (1972), 217–242.CrossRefGoogle Scholar
[11]Schönet, M.et al., GAP—Groups, Algorithms, and Programming — version 3 release 4 patchlevel 4. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1997.Google Scholar
[12]Silver, D. S. and Whitten, W., ‘Knot group epimorphisms’, J. Knot Theory Ramifications 15 (2006), 153–166.CrossRefGoogle Scholar
[13]Simon, J., ‘Wirtinger approximations and the knot groups of F n in §n+2’, Pacific J. Math. 90 (1980), 177–189.CrossRefGoogle Scholar
[14]Thurston, W. P., The Geometry and Topology of Three-Manifolds, Volume 1 (Princeton University Press, Princeton, NJ, 1997).Google Scholar
[15]Tsau, C. M., ‘Nonalgebraic killers of knot groups’, Proc. Amer. Math. Soc. 95 (1985), 139–146.CrossRefGoogle Scholar