RademacherSymbol

The Rademacher symbol of a hyperbolic element. By Ghys's theorem it is the linking number of the element's modular knot with the trefoil — and it is also just the number of 's minus the number of 's in its word.

WikipediaDedekind sum↗MathWorldDedekindSum↗WikidataQ2463775↗

RademacherSymbol(matrix), via Dedekind sums

WordSymbol(word)the same number by counting letters

LinkingWithTrefoil(matrix)the same number again, named for what it measures

DedekindSum(h, k), exactly

Domain: The modular group

Details

• is the complement of a trefoil in , so a closed orbit of the modular flow is a knot in that complement
• Ghys: the modular knots are exactly the LORENZ knots, and
• is a quasimorphism, NOT a class function; only the corrected is
• Dedekind sums are checked against reciprocity,
• is undefined on elliptic and parabolic elements — they have no closed geodesic

Examples

See also: ModularClasses, ModularWord