MS Seminar (Mathematics - String Theory)

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
Place: Zoom
Related File: 2649.pdf
Abstract: 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'' )

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.