New York Journal of Mathematics
Volume 27 (2021), 1494-1523


Nicholas Rouse

Arithmetic of the canonical component of the knot 74

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Published: October 30, 2021.
Keywords: Dehn surgery, hyperbolic knots, Azumaya algebras, character varieties, elliptic curves.
Subject: 57K32, 57K10, 11G05, 11R52.

We prove two arithmetic properties of Dehn surgery points on the canonical component of the SL2C-character variety of the knot 74. The first is that the residue characteristics of the ramified places of the Dehn surgery points form an infinite set, providing evidence for a conjecture of Chinburg, Reid, and Stover. The second is that the Dehn surgery points have infinite order in the Mordell-Weil group of the elliptic curve obtained by a simple birational transformation of the canonical component into Weierstrass form.


The author wishes to thank his advisor, Alan Reid, for suggesting the problems in this paper as well as his support and guidance in both the mathematical and writing phases of this paper's preparation. The author would also like to acknowledge the anonymous referees for their helpful comments and suggestions, with special thanks to the one who pointed out Theorem 5.11, which simplified the original argument.

Author information

Nicholas Rouse:
Department of Mathematics
Rice University
Houston, TX 77005, USA