Proof-critical dependence graph

On $q$ -real and $q$ -complex numbers

A statement-restricted proof-dependence graph for arXiv:2508.08440v2. It covers the paper's main analytic continuation and convergence theorems, exact $q$-Bessel formulas, monotonicity and almost-everywhere singularity, the unit-circle Jones/Burau bounds, the mod-$m$ classification results, and the hypergeometric construction of $q$-complex numbers. Expansion follows only claims whose truth can affect one of those marked results and stops at published books. Historical citations, comparisons, conjectures, and alternate routes are documented separately as non-edges.

Graph scope15 nodes15 proof-critical linksChecked August 27, 2026

Oriented proof graph

Dependence map

Arrows point from a prerequisite toward the paper whose marked statement uses it.

Solid arrow: headline proof lineage
Dashed arrow: a separately marked side or appendix claim
Dashed square: a terminal book

Hover over, or focus, a square to see its full citation.

Every visible arrow is documented in the evidence ledger below. A non-book leaf means that no earlier proof-critical source was identified for the marked statement—not that the paper has no other citations.

Evidence ledger

Proof-critical links

Basic Hypergeometric Series→Three families of $q$ -Lommel polynomialsHeadline lineageSource-chain endpoint

Kim-Stanton derives the continued fraction through a Heine contiguous relation for basic hypergeometric series.

Citation location: Kim-Stanton arXiv:2105.10096v3, proof of Theorem 5.3, PDF pp. 14-15.

Theorem 5.3 and Equation (6.4)Supplies the exact basic-Bessel continued fraction specialized in focal Lemma 3.1.

Three families of $q$ -Lommel polynomials→On $q$ -real and $q$ -complex numbersHeadline lineageSource link checked

Focal Lemma 3.1 imports Kim-Stanton Equation (6.4); the focal continued fraction is obtained by $y_{KS}=y/q$ and $x_{KS}=x/q$ , after which Proposition 3.2 specializes the parameters.

Citation location: Focal arXiv:2508.08440v2 PDF p. 7, Lemma 3.1; Kim-Stanton v3 PDF p. 19, Equation (6.4).

Lemma 3.1 and Proposition 3.2Identifies arithmetic negative continued fractions with ratios of renormalized $q$ -Bessel functions and gives explicit $q$ -cotangent formulas.

Cluster algebras and Jones polynomials→ $q$ -deformed rationals and $q$ -continued fractionsHeadline lineageSource link checked

MO1 Appendix A invokes Lee-Schiffler's rational-link Jones numerator formula to identify the normalized Jones polynomial with a $q$ -rational numerator.

Citation location: MO1 VOR Appendix A, pp. 43-45; Lee-Schiffler arXiv:1710.08063v2 Theorem 7.4, PDF p. 29.

Appendix A, Propositions A.1-A.2Identifies the rational-link Jones polynomial with the $q$ -rational numerator and supplies the explicit Burau matrices used at the endpoint.

$q$ -deformed rationals and $q$ -continued fractions→On $q$ -real and $q$ -complex numbersHeadline lineageSource link partially checked

The focal monotonicity extension begins with strict monotonicity for rational inputs. The unit-circle section also imports MO1 Appendix A's rational-link Jones identification and explicit matrices.

Citation location: Focal v2 PDF pp. 29 and 40-41, Proposition 5.1 and Section 7.2, citations [MO1].Verification note: The Appendix A left/right rescaling typo is harmless and uniquely repairable.

Proposition 5.1Strict monotonicity of $x\mapsto[x]_q$ for $0<q<1$ .
Proposition 7.3Bounds the normalized four-plat Jones polynomial by the positive-semidefinite Temperley-Lieb/Burau form.
Proposition 7.4Proves the exact endpoint equality at primitive sixth roots.

Kleinian Groups→Bounding deformation spaces of Kleinian groups with two generatorsHeadline lineageSource-chain endpoint

EGMS uses the Klein-Maskit combination theorem to obtain the free-product geometry needed for the faithful representation conclusion.

Citation location: EGMS arXiv:2405.15970v3 proof of Corollary 6.5, PDF pp. 16-17.

Lemma 5.1, Theorem 6.3, Lemma 6.4, and Corollary 6.5Bounds the nonfaithful Burau specialization locus by the closed annulus $R_*\le |q|\le R_*^{-1}$ .

Combinatorial Group Theory→Bounding deformation spaces of Kleinian groups with two generatorsHeadline lineageSource-chain endpoint

EGMS uses the Hopfian property of the finitely generated residually finite free product to upgrade a surjection to an isomorphism.

Citation location: EGMS arXiv:2405.15970v3 proof of Corollary 6.5, PDF p. 17.

Lemma 5.1, Theorem 6.3, Lemma 6.4, and Corollary 6.5Bounds the nonfaithful Burau specialization locus by the closed annulus $R_*\le |q|\le R_*^{-1}$ .

Bounding deformation spaces of Kleinian groups with two generators→On $q$ -real and $q$ -complex numbersHeadline lineageSource link partially checked

The focal proof applies the EGMS annulus bound to the numerator polynomial of every rational convergent and then derives uniform lower bounds inside $|q|<2-\sqrt{3}$ .

Citation location: Focal v2 PDF pp. 18-19, citations to EGMS Corollary 6.4; current EGMS v3 locator is Corollary 6.5, PDF pp. 16-17.Verification note: Numbering drift is version-exact and documented; the result is mathematically the same.

Theorem 1.2Extends convergence and nonvanishing to $|q|<2-\sqrt{3}$ using the uniform zero-free annulus for $q$ -rational numerator polynomials.

$q$ -deformed rationals and $q$ -continued fractions→Burau representation of braid groups and $q$ -rationalsHeadline lineageSource link checked

MOV uses the $q$ -rational singular set and positivity/coprimality properties developed in MO1 to connect polynomial zeros with specialized Burau kernels.

Citation location: MOV IMRN VOR pp. 8620-8624; citations to Morier-Genoud-Ovsienko's $q$-rational theory.

Theorem 1Equates a zero of a $q$ -rational numerator polynomial with nonfaithfulness of the specialized reduced Burau representation.

The Theory of Groups→Burau representation of braid groups and $q$ -rationalsHeadline lineageSource-chain endpoint

The MOV proof invokes the Kurosh subgroup theorem to identify the relevant subgroup of the modular/free-product presentation as free.

Citation location: MOV IMRN VOR proof of Theorem 1, pp. 8622-8624.

Theorem 1Equates a zero of a $q$ -rational numerator polynomial with nonfaithfulness of the specialized reduced Burau representation.

Burau representation of braid groups and $q$ -rationals→On $q$ -real and $q$ -complex numbersHeadline lineageSource link checked

MOV Theorem 1 identifies each numerator zero with a nonfaithful Burau specialization, making the EGMS nonfaithful-locus annulus applicable.

Citation location: Focal v2 PDF p. 18, proof of Theorem 1.2, citation [MOV, Theorem 1].

Theorem 1.2Extends convergence and nonvanishing to $|q|<2-\sqrt{3}$ using the uniform zero-free annulus for $q$ -rational numerator polynomials.

Continued Fractions→Diophantine approximation by negative continued fractionHeadline lineageSource-chain endpoint

Ito recodes negative continued fractions into regular continued fractions and applies the classical metric approximation theorem.

Citation location: Ito arXiv:2005.04371v1 proof of Theorem 3.1, PDF pp. 3-5.

Theorem 3.1, first limitGives the almost-everywhere growth/approximation limit used to make the focal difference quotient vanish.

Diophantine approximation by negative continued fraction→On $q$ -real and $q$ -complex numbersHeadline lineageSource link checked

Focal Proposition 5.4 quotes the first limit in Ito Theorem 3.1 to control the denominator growth for almost every real number.

Citation location: Focal v2 PDF p. 30, proof of Proposition 5.4; Ito fallback v1 PDF p. 3.

Proposition 5.4Shows that $d[x]_q/dx=0$ almost everywhere.

The Burau representation is unitary→On $q$ -real and $q$ -complex numbersHeadline lineageSource link checked

The invariant Hermitian form for the reduced Burau representation yields the positive-semidefinite form used in the focal Jones-polynomial norm estimate.

Citation location: Focal v2 PDF p. 40, proof of Proposition 7.3, citation [Squier]; Squier VOR pp. 199-202.

Proposition 7.3Bounds the normalized four-plat Jones polynomial by the positive-semidefinite Temperley-Lieb/Burau form.

Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds→On $q$ -real and $q$ -complex numbersHeadline lineageSource-chain endpoint

The standard two-dimensional Temperley-Lieb four-plat state space, plat vector, matrix coefficient, and loop value are needed to identify the Jones value with a matrix coefficient whose norm is controlled by the invariant form.

Citation location: Focal v2 PDF p. 40: MO1 Appendix A and Jones/Kauffman are cited immediately before Proposition 7.3, and Squier is cited in its proof, but no exact locator pins the combined state-space, plat-vector, matrix-coefficient, and loop-normalization conventions.Verification note: Added as a canonical terminal normalization reference; the exact book locator remains under-localized.

Proposition 7.3Bounds the normalized four-plat Jones polynomial by the positive-semidefinite Temperley-Lieb/Burau form.

A Course of Modern Analysis→On $q$ -real and $q$ -complex numbersHeadline lineageSource-chain endpoint

The $q$ -complex construction repeatedly uses the classical Gauss connection formula, Euler transformations, hypergeometric parameter continuation, and the modular $\lambda$ uniformization summarized in the cited monograph.

Citation location: Focal v2 PDF pp. 48-52, Section 9.1 and citation [WW].

Theorem 9.1Constructs and uniquely characterizes the meromorphic $q$ -complex deformation in terms of hypergeometric and modular functions.