Problem 2.28 — (M.-J.

Definitions and mathematical statement

Problem 2.28, attributed to M.-J. Jeong and C.-Y. Park, asks for a minimal finite subset $A_n$ of $V_n$ satisfying \[ A_n\subset V_n,\qquad |A_n|<\infty,\qquad \operatorname{span}(A_n)=V_n. \] Neither $V_n$, the coefficient system for the span, nor the range of $n$ is defined in the supplied record. Thus the object whose generators are requested cannot be identified precisely from this extract.

If $V_n$ is intended as a vector space over a field $F$, then \[ \operatorname{span}_F(A_n)= \left\{\sum_{j=1}^{r}a_jv_j:r\ge0,\ a_j\in F,\ v_j\in A_n\right\}. \] In that setting an inclusion-minimal spanning set satisfies $\operatorname{span}_F(B)\ne V_n$ for every proper subset $B\subsetneq A_n$; equivalently it is a basis. A minimum-cardinality spanning set would have size $\dim_F V_n$. These conditional interpretations explain "minimal" but do not identify $V_n$ or provide the requested set. If the source instead uses a module or a constrained family of invariants, its spanning and minimality conventions must be stated explicitly. No particular finite-type invariant space, coefficient field, or uniform construction in $n$ is inferred from the notation alone.

Short English statement

Find a smallest or irredundant finite spanning family for the source's space $V_n$. Its definition and coefficients are missing, so a source-specific generating family cannot yet be specified. The supplied open-triage summary calls the question not obviously resolved.

Sources

• Local UnsolvedMath registry, permanent ID $10400050$, exact source code catalogue entry, Problem 2.28, attributed in the statement to M.-J. Jeong and C.-Y. Park. The supplied title is truncated after "(M.-J."
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 2.28, PDF page 51: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold printed headings distinguish 231 numbered items from references in surrounding exposition.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Research classification OPEN-TRIAGE; supplied summary: minimal generating set for $V_n$, not obviously resolved. No further authoritative reference defining $V_n$ is supplied.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.