q-Rational Numbers

• q-Rational numbers are q-deformations of classical rationals defined via continued fraction expansions, preserving modular invariance and exhibiting total positivity.
• They bridge combinatorial models such as lattice paths, fence posets, and snake graphs, yielding unimodal polynomials with practical applications in quantum invariants.
• Their algebraic structure extends to q-irrationals and higher q-analogues, impacting knot invariants, q-binomial identities, and representation theory.

A $q$-rational number is a $q$-deformation of an ordinary rational, defined as a rational function in $q$ whose combinatorial, algebraic, and geometric structures emerge from continued fraction expansions, actions of (deformed) modular groups, lattice path combinatorics, and connections to knot invariants. These $q$-rational constructions unify several themes: modular invariance, positivity, and the intricate interaction of continued fraction combinatorics, representation theory, and algebraic geometry.

1. Definitions and Foundational Constructions

A $q$-rational number is most naturally defined via the continued fraction expansion of a rational $x = r/s \in \mathbb{Q}$:

• For a (regular or negative) continued fraction $x = [c_0, c_1, ..., c_\ell]$, the $q$-rational is

$$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$

where $[n]_q = 1 + q + ... + q^{n-1}$.

• Alternatively, for regular continued fractions, a mixed $q$0/$q$1 parameter assignment is used in the recursive formula, alternating at each level of depth.

This $q$2-deformation is characterized by:

• Modular invariance: The assignment $q$3 is uniquely determined by requiring $q$4 for all $q$5, where $q$6 denotes the $q$7-deformed modular action (e.g., $q$8, $q$9), and $q$0 (Morier-Genoud et al., 31 Mar 2025).

The polynomials $q$1 and $q$2 in $q$3 are constructed recursively, matching the classical recurrence for continuants with $q$4-integers and $q$5-weighted terms (Morier-Genoud et al., 2018).

2. Algebraic, Combinatorial, and Geometric Properties

Total Positivity and Unimodality

• For any $q$6, $q$7, $q$8, the difference $q$9 is a polynomial in $q$0 with strictly positive integer coefficients (“total positivity”) (Morier-Genoud et al., 2018, Morier-Genoud et al., 31 Mar 2025).
• The numerator and denominator polynomials are unimodal: coefficients first increase then decrease monotonically, a property aligned with $q$1-binomial and Gaussian polynomials (Morier-Genoud et al., 31 Mar 2025).

Combinatorial Models

• The coefficients of $q$2, $q$3 count combinatorial objects:
  • Closures in the dual quiver of a polygonal triangulation (Morier-Genoud et al., 2018).
  • Order ideals in a “fence” poset associated to the continued fraction (Ovenhouse, 2021).
  • Lattice paths or Young diagrams inside a snake graph, with $q$4 recording the “area” or number of boxes under a given path (Ovenhouse, 2021).

Table: Combinatorial Models for $q$5-Rationals

Model	Polynomial Coefficient Interpreted As	Source
Triangulation/Quiver	Subrepresentation count of maximal indecomposable	(Morier-Genoud et al., 2018)
Fence Poset	Order ideal rank generating function	(Ovenhouse, 2021)
Snake Graph	Area of lattice paths, Young diagrams	(Ovenhouse, 2021, Morier-Genoud et al., 31 Mar 2025)

In each model, $q$6 and $q$7 naturally emerge as weighted generating functions.

3. Modular Group, Matrix, and Farey Actions

• $q$8-rational numbers are modules for $q$9 via their $q$0-deformation, with group actions realized through $q$1-deformed matrices:

$q$2

• The $q$3-Farey graph is the $q$4-analogue of the classical Farey tessellation, with mediant (Farey sum) and recursions encoded by $q$5-weighted edges and triangles (Morier-Genoud et al., 2018, Morier-Genoud et al., 31 Mar 2025).
• The construction ensures that $q$6 agrees with conventions for $q$7-integers and that $q$8 is compatible with both the Farey arithmetic and modular group symmetries.

Left and Right $q$9-rationals

• There are distinct “right” and “left” $x = r/s \in \mathbb{Q}$0-rational numbers, arising from different normalization choices in matrix products and continued fractions (Ren et al., 5 Feb 2025, Bapat et al., 2022):
  • Right: base point $x = r/s \in \mathbb{Q}$1 under modular action.
  • Left: base point $x = r/s \in \mathbb{Q}$2; polynomials satisfy distinct palindromicity and congruence properties in their coefficients.

4. Stabilization, $x = r/s \in \mathbb{Q}$3-Irrationals, and Analytic Aspects

• For any irrational $x = r/s \in \mathbb{Q}$4, write $x = r/s \in \mathbb{Q}$5 as its sequence of rational approximants. Then for each monomial $x = r/s \in \mathbb{Q}$6 in the Taylor expansion of $x = r/s \in \mathbb{Q}$7, the coefficient $x = r/s \in \mathbb{Q}$8 stabilizes for large $x = r/s \in \mathbb{Q}$9, yielding

$x = [c_0, c_1, ..., c_\ell]$0

where $x = [c_0, c_1, ..., c_\ell]$1.

• For quadratic irrationals with periodic continued fractions (e.g., metallic numbers), associated $x = [c_0, c_1, ..., c_\ell]$2-irrationals solve $x = [c_0, c_1, ..., c_\ell]$3-deformed algebraic equations, and their radii of convergence are determined by palindromic polynomials arising from the combinatorics of the continued fraction expansion (Ren, 2021).
• The stabilization property allows the extension of $x = [c_0, c_1, ..., c_\ell]$4-rationals to $x = [c_0, c_1, ..., c_\ell]$5-irrational numbers, crucial for applications to $x = [c_0, c_1, ..., c_\ell]$6-deformed dynamical systems and quantum invariants.

5. Further Algebraic Structures and Applications

Knot Invariants and Jones Polynomials

• For every rational knot $x = [c_0, c_1, ..., c_\ell]$7, there is a correspondence:

$x = [c_0, c_1, ..., c_\ell]$8

where $x = [c_0, c_1, ..., c_\ell]$9 is the Jones polynomial, and $q$0, $q$1 arise from the $q$2-rational associated to $q$3 (Morier-Genoud et al., 2018, Ren et al., 5 Feb 2025).

• The palindromicity and positivity of $q$4, $q$5 connect to the unimodality and normalization properties of knot invariants.
• The trace of the $q$6-deformed matrix encodes deviation from palindromicity in the Jones polynomial, with direct combinatorial implications for fence posets and cluster algebras (Ren et al., 5 Feb 2025).

$q$7-Binomial Coefficients and Special Functions

• $q$8-rational numbers provide a basis for $q$9-binomial and $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$0-Chu-Vandermonde identities over non-integer (even $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$1-real) arguments, extending the scope of $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$2-analogues beyond the integer case.
• $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$3-Gamma functions with new shift properties are defined using $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$4-rationals:

$$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$5

facilitating reflection and expansion formulas with integer coefficients (Machacek et al., 2023).

6. Extensions: Higher $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$6-Continued Fractions, $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$7-Catalan Numbers, and Quantum Invariants

• The theory generalizes to higher $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$8-continued fractions, constructed as ratios of $$[x]_q = [c_0]_q - \cfrac{q^{c_0-1}}{[c_1]_q - \cfrac{q^{c_1-1}}{[c_2]_q - \cdots - \cfrac{q^{c_{\ell-1}-1}}{[c_\ell]_q}}}$$9-weighted generating functions of $[n]_q = 1 + q + ... + q^{n-1}$0-partitions on certain posets (e.g., chain, snake, or fence graphs), with corresponding matrix formulas generalizing classical recurrences (Burcroff et al., 2024).
• Rational $[n]_q = 1 + q + ... + q^{n-1}$1-Catalan numbers $[n]_q = 1 + q + ... + q^{n-1}$2 can be interpreted as $[n]_q = 1 + q + ... + q^{n-1}$3-binomial expressions divided by a $[n]_q = 1 + q + ... + q^{n-1}$4-integer, conjecturally possessing positive coefficients, and relate to lattice point statistics and ribbon decompositions of posets (Armstrong, 2024).
• The behavior of $[n]_q = 1 + q + ... + q^{n-1}$5-rationals under modular transformations, $[n]_q = 1 + q + ... + q^{n-1}$6-transposes, and congruences facilitates connections to unimodality conjectures in polynomial combinatorics and classifications of cluster variables in representation theory.

7. Summary Table: Core Mathematical Constructs and Interactions

Structure	Construct	Key Features / Properties	Paper Reference
$[n]_q = 1 + q + ... + q^{n-1}$7-rational	Continued fraction $[n]_q = 1 + q + ... + q^{n-1}$8 $[n]_q = 1 + q + ... + q^{n-1}$9	Modular invariance, positivity, unimodality	(Morier-Genoud et al., 2018, Morier-Genoud et al., 31 Mar 2025)
Combinatorial model	Fence poset, snake graph, quiver	Closure counting, order ideals, area stat	(Ovenhouse, 2021, Morier-Genoud et al., 2018)
Matrix realization	$q$00 products in $q$01	Farey graph deformation, modular action	(Morier-Genoud et al., 2018, Ren et al., 5 Feb 2025)
Knot invariant	$q$02	Polynomials from $q$03-rational, palindromic traces	(Morier-Genoud et al., 2018, Ren et al., 5 Feb 2025)
$q$04-irrational	Stabilized Taylor expansion limit	Analytic properties, convergence radius	(Ren, 2021, Morier-Genoud et al., 31 Mar 2025)
Higher analogs	$q$05-Catalan, higher CF, Gamma, binomial	Positivity, symmetry, $q$06-analogue theory	(Burcroff et al., 2024, Machacek et al., 2023)

8. Interconnections and Mathematical Impact

The $q$07-rational framework bridges continued fractions, combinatorial representation theory, and quantum knot invariants via:

• Coherent $q$08-deformations respecting modular and arithmetic symmetries.
• Universal positivity and combinatorial interpretations for all coefficients.
• Deep connections to cluster algebras, Schubert cell geometry, knot theory, and the total positivity phenomenon.

The extension to $q$09-irrationals and higher order constructions provides new tools for analytic number theory, quantum algebra, and algebraic combinatorics, with broad ongoing research motivated by the structural richness and universality of $q$10-rational numbers.