Boundary slopes (nearly) bound exceptional slopes

TL;DR

This paper investigates the distribution of exceptional Dehn surgeries on hyperbolic knots in $S^3$, proposing conjectures that boundary slopes, especially non-integral or toroidal ones, bound the set of all such surgeries.

Contribution

It introduces conjectures relating boundary slopes to the location of exceptional surgeries, providing new insights into their bounded nature and distribution.

Findings

Provides evidence supporting the conjectures about boundary slopes bounding exceptional surgeries.

Suggests that all non-trivial exceptional surgeries occur within specific intervals defined by boundary slopes.

Proposes that integers within certain boundary slope intervals are always exceptional surgeries.

Abstract

For a hyperbolic knot in $S^3$, Dehn surgery along slope $r \in \Q \cup \{\frac10\}$ is {\em exceptional} if it results in a non-hyperbolic manifold. We say meridional surgery, $r = \frac10$, is {\em trivial} as it recovers the manifold $S^3$. We provide evidence in support of two conjectures. The first (inspired by a question of Professor Motegi) states that there are boundary slopes $b_1 < b_2$ such that all non-trivial exceptional surgeries occur, as rational numbers, in the interval $[b_1,b_2]$. We say a boundary slope is {\em NIT} if it is non-integral or toroidal. Second, when there are non-trivial exceptional surgeries, we conjecture there are NIT boundary slopes $b_1 \leq b_2$ so that the exceptional surgeries lie in $[\floor{b_1},\ceil{b_2}]$. Moreover, if $\ceil{b_1} \leq \floor{b_2}$, the integers in the interval $[ \ceil{b_1}, \floor{b_2} ]$ are all exceptional surgeries.