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 $q$/$q^{-1}$ parameter assignment is used in the recursive formula, alternating at each level of depth.

This $q$-deformation is characterized by:

• Modular invariance: The assignment $[x]_q$ is uniquely determined by requiring $[A(x)]_q = A_q([x]_q)$ for all $A \in PSL_2(\mathbb{Z})$, where $A_q$ denotes the $q$-deformed modular action (e.g., $T_q(X) = qX + 1$, $S_q(X) = -1/(qX)$), and $[0]_q = 0$ (Morier-Genoud et al., 31 Mar 2025).

The polynomials $R(q)$ and $S(q)$ in $[r/s]_q = R(q)/S(q)$ are constructed recursively, matching the classical recurrence for continuants with $q$-integers and $q$-weighted terms (Morier-Genoud et al., 2018).

2. Algebraic, Combinatorial, and Geometric Properties

Total Positivity and Unimodality

• For any $x = r/s > y = r'/s'$, $[x]_q = N(q)/M(q)$, $[y]_q = N'(q)/M'(q)$, the difference $N(q)M'(q) - M(q)N'(q)$ is a polynomial in $q$ 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 $q$-binomial and Gaussian polynomials (Morier-Genoud et al., 31 Mar 2025).

Combinatorial Models

• The coefficients of $R(q)$, $S(q)$ 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 $q$ recording the “area” or number of boxes under a given path (Ovenhouse, 2021).

Table: Combinatorial Models for $q$-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, $R(q)$ and $S(q)$ naturally emerge as weighted generating functions.

3. Modular Group, Matrix, and Farey Actions

• $q$-rational numbers are modules for $PSL_2(\mathbb{Z})$ via their $q$-deformation, with group actions realized through $q$-deformed matrices:

$$R(q) = \begin{pmatrix} q & 1 \ 0 & 1 \end{pmatrix}, \ L(q) = \begin{pmatrix} q & 0 \ q & 1 \end{pmatrix}$$

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

Left and Right $q$-rationals

• There are distinct “right” and “left” $q$-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 $0/1$ under modular action.
  • Left: base point $1/(1-q)$; polynomials satisfy distinct palindromicity and congruence properties in their coefficients.

4. Stabilization, $q$-Irrationals, and Analytic Aspects

• For any irrational $x$, write $x_k$ as its sequence of rational approximants. Then for each monomial $q^n$ in the Taylor expansion of $[x_k]_q$, the coefficient $\kappa_n$ stabilizes for large $k$, yielding

$$[x]_q = \lim_{k \to \infty} [x_k]_q = \sum_{n=0}^\infty \kappa_n q^n$$

where $\kappa_n \in \mathbb{Z}_{\geq 0}$.

• For quadratic irrationals with periodic continued fractions (e.g., metallic numbers), associated $q$-irrationals solve $q$-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 $q$-rationals to $q$-irrational numbers, crucial for applications to $q$-deformed dynamical systems and quantum invariants.

5. Further Algebraic Structures and Applications

Knot Invariants and Jones Polynomials

• For every rational knot $E$, there is a correspondence:

$J_E(q) = q R(q) + (1-q) S(q)$

where $J_E(q)$ is the Jones polynomial, and $R(q)$, $S(q)$ arise from the $q$-rational associated to $E$ (Morier-Genoud et al., 2018, Ren et al., 5 Feb 2025).

• The palindromicity and positivity of $R(q)$, $S(q)$ connect to the unimodality and normalization properties of knot invariants.
• The trace of the $q$-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).

$q$-Binomial Coefficients and Special Functions

• $q$-rational numbers provide a basis for $q$-binomial and $q$-Chu-Vandermonde identities over non-integer (even $q$-real) arguments, extending the scope of $q$-analogues beyond the integer case.
• $q$-Gamma functions with new shift properties are defined using $q$-rationals:

$\Gamma_q(\alpha+1) = [\alpha]_q \Gamma_q(\alpha)$

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

6. Extensions: Higher $q$-Continued Fractions, $q$-Catalan Numbers, and Quantum Invariants

• The theory generalizes to higher $q$-continued fractions, constructed as ratios of $q$-weighted generating functions of $P$-partitions on certain posets (e.g., chain, snake, or fence graphs), with corresponding matrix formulas generalizing classical recurrences (Burcroff et al., 13 Aug 2024).
• Rational $q$-Catalan numbers $\mathrm{Cat}(a, b)_q$ can be interpreted as $q$-binomial expressions divided by a $q$-integer, conjecturally possessing positive coefficients, and relate to lattice point statistics and ribbon decompositions of posets (Armstrong, 10 Mar 2024).
• The behavior of $q$-rationals under modular transformations, $q$-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
$q$-rational	Continued fraction $\to$ $R(q)/S(q)$	Modular invariance, positivity, unimodality	(Morier-Genoud et al., 2018Morier-Genoud et al., 31 Mar 2025)
Combinatorial model	Fence poset, snake graph, quiver	Closure counting, order ideals, area stat	(Ovenhouse, 2021Morier-Genoud et al., 2018)
Matrix realization	$R(q), L(q)$ products in $PSL_{q}(2, \mathbb{Z})$	Farey graph deformation, modular action	(Morier-Genoud et al., 2018Ren et al., 5 Feb 2025)
Knot invariant	$J_E(q) = q R(q) + (1-q) S(q)$	Polynomials from $q$-rational, palindromic traces	(Morier-Genoud et al., 2018Ren et al., 5 Feb 2025)
$q$-irrational	Stabilized Taylor expansion limit	Analytic properties, convergence radius	(Ren, 2021Morier-Genoud et al., 31 Mar 2025)
Higher analogs	$q$-Catalan, higher CF, Gamma, binomial	Positivity, symmetry, $q$-analogue theory	(Burcroff et al., 13 Aug 2024Machacek et al., 2023)

8. Interconnections and Mathematical Impact

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

• Coherent $q$-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 $q$-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 $q$-rational numbers.