|XXXIX Workshop on Geometric Methods in Physics||27.06-3.07.2021|
|IX School on Geometry and Physics||21-25.06.2021|
On integrability, geometrization and knots
The talk is a review of the relation between Thurston’s geometrization and Liouville integrability. Using as the main example the geodesic flows on the $3$-folds with $SL(2,\mathbb R)$-geometry in Thurston's sense, I will show that the corresponding phase space contains two open regions with integrable and chaotic behaviour respectively.
A particular case of such $3$-folds the modular quotient $SL(2,\mathbb R)/SL(2,\mathbb Z)$, which is known to be equivalent to the complement in $3$-sphere of the trefoil knot. I will show that the remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these hyperbolic knots are replaced by the cable knots of trefoil.
The talk is partly based on a recent joint work with Alexey Bolsinov and Yiru Ye.
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