Timeline for What is known about exceptional slopes of hyperbolic knots?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2022 at 15:59 | comment | added | Ian Agol | @CalvinMcPhail-Snyder there’s two issues there: can the complete hyperbolic metric be deformed to the metric on the filling through cone metrics (or generalized Dehn fillings)? Or as you say, is the core curve a hyperbolic geodesic? The second is not known to imply the first. However the best results on both still come from the method of Hodgson-Kerckhoff. The upshot is that we don’t have very precise understanding of the exceptional slopes in this sense. | |
| Mar 28, 2022 at 14:06 | comment | added | Calvin McPhail-Snyder | In this case, do we know that the hyperbolic metric on $M_{p/q}$ actually comes from that of $K$? (Here I'm thinking of this question: mathoverflow.net/questions/121774/…). I think this is equivalent to asking whether the slope $p/q$ is a geodesic in $M_{p/q}$ whenever $M_{p/q}$ is hyperbolic. | |
| Mar 28, 2022 at 13:56 | vote | accept | Calvin McPhail-Snyder | ||
| Mar 27, 2022 at 20:33 | comment | added | Ian Agol | @CalvinMcPhail-Snyder yes, exactly. | |
| Mar 27, 2022 at 20:09 | comment | added | Calvin McPhail-Snyder | If I have a hyperbolic knot $K$ in $S^3$, then $1/0$ surgery gives $S^3$, and from this table we have $\Delta(1/0, p/q) = |q| \le 2$ for any exceptional slope $p/q$. In particular this gives the denominator bound in your comment. Is that correct? | |
| Mar 26, 2022 at 18:03 | history | answered | Ian Agol | CC BY-SA 4.0 |