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Bulletin of the London Mathematical Society Advance Access originally published online on April 7, 2009
Bulletin of the London Mathematical Society 2009 41(3):535-540; doi:10.1112/blms/bdp029
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© 2009 London Mathematical Society

Twisted Alexander polynomials and representation shifts

Daniel S. Silver and Susan G. Williams

Department of Mathematics and Statistics
University of South Alabama
ILB 325
Mobile, AL 36688
USA
swilliam@jaguar1.usouthal.edu

Received 19 October 2007. Revision received 17 September 2008.

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.

2000 Mathematics Subject Classification 57M25 (primary), 37B40 (secondary).

The authors are partially supported by NSF grant DMS-0304971 and DMS-0706798.


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