Exceptional surgery and boundary slopes

Let X be a norm curve in the \mathit{SL}(2,\mathbb{C})-character variety of a knot exterior M. Let t = \|\beta\| / \|\alpha\| be the ratio of the Culler-Shalen norms of two distinct non-zero classes \alpha, \beta \in H_1(\partial M,\mathbb{Z}). We demonstrate that either X has exactly two associated...

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Bibliographic Details
Published in:Osaka journal of mathematics 2006-12, Vol.43 (no. 4), p.807-821
Main Authors: Ishikawa, Masaharu, Mattman, Thomas W, Shimokawa, Koya
Format: Article
Language:English
Subjects:
57M25
57N10
Online Access:Get full text
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Summary:Let X be a norm curve in the \mathit{SL}(2,\mathbb{C})-character variety of a knot exterior M. Let t = \|\beta\| / \|\alpha\| be the ratio of the Culler-Shalen norms of two distinct non-zero classes \alpha, \beta \in H_1(\partial M,\mathbb{Z}). We demonstrate that either X has exactly two associated strict boundary slopes \pm t, or else there are strict boundary slopes r_1 and r_2 with |r_1| > t and |r_2| < t. As a consequence, we show that there are strict boundary slopes near cyclic, finite, and Seifert slopes. We also prove that the diameter of the set of strict boundary slopes can be bounded below using the Culler-Shalen norm of those slopes.