Mathematics Colloquia and Seminars

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(Joint with Ichihara) For a hyperbolic knot $K$ in $S^3$, we say a non-meridional slope is exceptional if Dehn surgery on that slope results in a non-hyperbolic manifold. We provide evidence in support of two conjectures. The first (inspired by a question of Motegi) states that any exceptional surgery slope occurs in the interval ${[}b_m, b_M {]}$,
where $b_m$ (respectively, $b_M$) is the least (greatest) finite boundary slope. We say a boundary slope is NIT if it is non-integral or toroidal.
Secondly, when there are exceptional surgeries, we conjecture
there are (possibly equal) NIT boundary slopes $b_1 \leq b_2$ so that the exceptional slopes lie in $[\lfloor b_1 \rfloor, \lceil b_2 \rceil]$.
Moreover, when $\lceil b_1 \rceil \leq \lfloor b_2 \rfloor$, then every integer in ${[}\lceil b_1 \rceil, \lfloor b_2 \rfloor{]}$ is an exceptional slope.