|Speaker:||Jun Ueki (Tokyo Denki U)|
|Title:||Modular knots of triangle groups, Rademacher symbols, and 2-cocycles (joint with T.Matsusaka)|
|Date (JST):||Tue, Mar 30, 2021, 13:00 - 14:00|
The exterior of the trefoil is homeomorphic to the unit tangent bundle of the modular orbifold PSL2Z╲H2.
Corresponding to conjugacy classes of primitive hyperbolic elements in PSL2Z, there defined the “modular knots” as the closed orbits of so-called the geodesic flow, which is topologically equivalent to the Lorenz flow. (cf. ``Lorenz and modular flows: a visual introduction'' http://www.josleys.com/articles/ams_article/Lorenz3.htm )
Ghys’s theorem [Ghys2007ICM] asserts that the linking number of modular knots and the missing trefoil is given by a highly ubiquitous function called the Rademacher symbol $╲Psi$. Indeed, Atiyah [Atiyah1987] proved the equivalence of seven very distinct definitions of $╲Psi$ and Ghys went further.
Matsusaka and I generalize this situation to the triangle groups $╲Gamma(p,q,╲infty)$ and the knots around the torus knots with excitement;
This work gives a new connection between automorphic forms and arithmetic topology.