Problem 2.19 — (Y.

Definitions and mathematical statement

For an integer $\mu>2$, a $\mu$-component link is an embedding of $\mu$ disjoint circles in $S^3$, considered up to ambient isotopy. Component ordering and orientation must be fixed when stating a classification; the excerpt does not specify them.

A $\Delta$-move is the standard local three-strand replacement associated with a Borromean-type tangle, supported in a ball and preserving its boundary endpoints. The source uses the technical term "$\Delta$ link homotopic" without reproducing the allowed component pattern for the three strands or the equivalence-generating moves. Unrestricted $\Delta$-equivalence, moves entirely on one component, other component restrictions, and ordinary link homotopy by self-crossing changes must not be identified silently.

With the source's equivalence relation $\sim_\Delta$ fixed, find necessary and sufficient conditions for \[ L\sim_\Delta L' \] for arbitrary $\mu$-component links with $\mu\ge3$. Equivalently, provide a complete criterion or invariant $I$ satisfying $L\sim_\Delta L'$ if and only if $I(L)=I(L')$, with the intended label/orientation conventions. The source does not require a finite list or a particular algorithm, and neither is imposed.

The supplied summary says the 2002 question was not obviously resolved but gives no classification theorem or modern citation. That open/triage wording is a qualified source assessment, not independently verified current status. The missing move convention is essential before any candidate criterion can be evaluated.

Short English statement

Classify links with more than two components up to the source's $\Delta$-link-homotopy relation. The precise component restrictions on the moves, and orientation/label conventions, are missing and must be recovered before stating a complete criterion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400041$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 2.19, attributed to Y. Ohyama, PDF page 43: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items.
• No exact modern classification or separate move-definition reference is supplied.