RademacherSymbol

The Rademacher symbol $\Psi$ 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 $R$ 's minus the number of $L$ 's in its word.

WikipediaDedekind sumMathWorldDedekindSumWikidataQ2463775

RademacherSymbol(matrix)

\Psi(M) = \Phi(M) - 3\,\mathrm{sign}(c(a+d))

, 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)

s(h,k)

, exactly

Typed signatures

Type	Package	Applies
`(expression<ModularMatrix> \| list \| string) -> integer`	`modular`	always

Domain: The modular group

Details

$\mathrm{SL}(2,\mathbb{Z})\backslash\mathrm{SL}(2,\mathbb{R})$ is the complement of a trefoil in $S^3$ , so a closed orbit of the modular flow is a knot in that complement
Ghys: the modular knots are exactly the LORENZ knots, and $\mathrm{lk}(k_\gamma,\text{trefoil}) = \Psi(\gamma)$
$\Phi(M) = (a+d)/c - 12\,\mathrm{sign}(c)\,s(d,|c|)$ is a quasimorphism, NOT a class function; only the corrected $\Psi$ is
Dedekind sums are checked against reciprocity, $s(h,k)+s(k,h) = -\frac14 + \frac{1}{12}(h/k + k/h + 1/hk)$
$\Psi$ is undefined on elliptic and parabolic elements — they have no closed geodesic