Problem 12.20 — (J.

Definitions and mathematical statement

Roberts asks for the generating function for $q$-spin-network evaluations. The source does not specify the network class, label conventions, evaluation normalization, or variables.

For a fixed finite graph $\Gamma$ with edge set $E$, a customary candidate interpretation is as follows. Let $\mathcal C(\Gamma)\subseteq{\mathbb{Z}}_{\geq0}^{E}$ be the admissible edge colorings for the chosen quantum theory, and let $\operatorname{ev}_q(\Gamma,\mathbf c)$ be its spin-network evaluation in a specified coefficient ring $A$. With one formal variable $z_e$ per edge, set \[G_\Gamma(q;\mathbf z)=\sum_{\mathbf c\in\mathcal C(\Gamma)}\operatorname{ev}_q(\Gamma,\mathbf c)\prod_{e\in E}z_e^{c_e}.\] Once the input conventions are fixed this is a formal power series; the desired result would give a useful explicit description or formula for it beyond restating this coefficient definition. The excerpt does not assert rationality, a product form or a differential equation, and none is added here. Nor does it say that this particular multivariate coloring series, as opposed to a one-variable weighting or a series summing over graphs, is intended.

The label admissibility, choice of quantum group, treatment of vertices, graph embedding/framing and normalization of evaluations must be supplied. Whether $q$ is an indeterminate, a complex parameter or a root of unity affects the coefficient ring and admissible labels; no such choice is made silently. Any analytic rather than formal interpretation would also need a convergence domain.

Short English statement

Find an explicit generating function packaging quantum spin-network evaluations. A natural coloring-series formulation is given conditionally because the source does not identify the graphs, labels, normalization or formal variables.

Sources

• Local UnsolvedMath registry, permanent ID $10400225$, exact source code catalogue entry.
• J. Roberts, attribution in Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.20, PDF page 168: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Recorded access 2026-07-29, pdf-font-structure extraction.