Exceptional surgery and boundary slopes

Abstract

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.