ModularClasses

Every hyperbolic conjugacy class whose $LR$ word has the given length — equivalently, every closed geodesic of that symbolic period on the modular surface. Conjugation rotates the word, so the classes are binary NECKLACES.

WikipediaModular group↗MathWorldModularGroupGamma↗WikidataQ1834342↗

ModularClasses(length)the classes, each named by the least rotation of its word

ModularClasses(length, True)only the primitive ones — not a repeat of a shorter geodesic

ModularClass(word)the canonical name of one class

Typed signatures

Type	Package	Applies
(integer, boolean?) -> list	modular	always

Domain: The modular group

Details

• Conjugating by the first letter rotates the word: $x^{-1}(xw)x = wx$
• The count is $\frac1n\sum_{d\mid n}\varphi(d)2^{n/d} - 2$ — binary necklaces, less the two constant ones, which are parabolic
• The primitive classes are the aperiodic necklaces, counted by $\frac1n\sum_{d\mid n}\mu(d)2^{n/d}$ — Lyndon words
• Word length is the SYMBOLIC period; the geodesic's length is $2\,\mathrm{arccosh}(|\mathrm{tr}|/2)$, and the two orderings differ

Examples

Basic #

four closed geodesics of symbolic length 4

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In$\mathrm{ModularClasses}(4)$Out$\bigl\lbrack\text{LLLR}, \text{LLRR}, \text{LRLR}, \text{LRRR}\bigr\rbrack$

Scope #

same word length as $LLLR$, different geodesic length

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In$\mathrm{ModularTrace}(\text{LRLR})$Out$7$

Properties #

the same geodesic, entered elsewhere

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In$\mathrm{ModularClass}(\text{LRL})$Out$\text{LLR}$

$LRLR$ is $LR$ traversed twice

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In$\mathrm{IsPrimitiveClass}(\text{LRLR})$Out$\mathrm{False}$

the same geodesic, entered yet another way

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In$\mathrm{ModularClass}(\text{RLL})$Out$\text{LLR}$

$LRR$ is primitive

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In$\mathrm{IsPrimitiveClass}(\text{LRR})$Out$\mathrm{True}$

About

Examples	6
Crosswalk	WikipediaMathWorldWikidata

See also: RademacherSymbol, ModularWord