Cisco Webex, Online seminar
(線上演講 Cisco Webex)
Modular Knots, Automorphic Forms, and the Rademacher Symbols for Triangle Groups
Toshiki Matsusaka (Nagoya University)
Abstract:
In a celebrated paper “Knots and dynamics”, Étienne Ghys proved that the linking numbers of modular knots and the missing trefoil in
![](https://chart.googleapis.com/chart?cht=tx&chl=%24S%5E3%24&chf=bg,s,333333&chco=ffffff)
coincide with the values of a highly ubiquitous function called the Rademacher symbol for
![](https://chart.googleapis.com/chart?cht=tx&chl=%24SL(2%2CZ)%24&chf=bg,s,333333&chco=ffffff)
. In this talk, we replace
![](https://chart.googleapis.com/chart?cht=tx&chl=%24SL(2%2CZ)%20%3D%20%5CGamma(2%2C3)%24&chf=bg,s,333333&chco=ffffff)
by the triangle group
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5CGamma(p%2Cq)%24&chf=bg,s,333333&chco=ffffff)
for any coprime pair $(p,q)$ of integers with
![](https://chart.googleapis.com/chart?cht=tx&chl=%242%20%5Cleq%20p%20%3C%20q%24&chf=bg,s,333333&chco=ffffff)
. We invoke the theory of harmonic Maass forms for
![](https://chart.googleapis.com/chart?cht=tx&chl=%24%5CGamma(p%2Cq)%24&chf=bg,s,333333&chco=ffffff)
to introduce the notion of the Rademacher symbol and provide several characterizations. Among other things, we generalize Ghys’s theorem for modular knots around any missing torus knot in
![](https://chart.googleapis.com/chart?cht=tx&chl=%24S%5E3%24&chf=bg,s,333333&chco=ffffff)
and in a lens space. This is joint work with Jun Ueki (Tokyo Denki University).
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