Commensurability classes of (−2,3,n) pretzel knot complements

Macasieb, Melissa; Mattman, Thomas

doi:10.2140/agt.2008.8.1833

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Commensurability classes of $(-2,3,n)$ pretzel knot complements

Melissa L Macasieb and Thomas W Mattman

Algebraic & Geometric Topology 8 (2008) 1833–1853

DOI: 10.2140/agt.2008.8.1833

Abstract

Let $K$ be a hyperbolic $(- 2, 3, n)$ pretzel knot and $M = S^{3} ∖ K$ its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of $M$ . Indeed, if $n \neq 7$ , we show that $M$ is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots.

Keywords

commensurability class, pretzel knot, trace field

Mathematical Subject Classification 2000

Primary: 57M25

References

Publication

Received: 2 April 2008

Revised: 17 July 2008

Accepted: 22 August 2008

Published: 20 October 2008

Authors

Melissa L Macasieb
Department of Mathematics The University of British Columbia Room 121, 1984 Mathematics Road Vancouver, BC V6T 1Z2 Canada

Thomas W Mattman
Department of Mathematics and Statistics California State University, Chico Chico, CA 95929-0525 USA

Volume 8, issue 3 (2008)