Universal Sequences of Plane Curves for Knot Diagrams

Definitions and mathematical statement

A generic plane immersion is understood here as a smooth immersion $\gamma:S^1\to{\mathbb{R}}^2$ whose only multiple points are finitely many transverse double points. A crossing assignment chooses, at each double point, which of the two branches passes over the other. This produces a knot diagram; it does not mean smoothing the crossing.

For a specified collection $\mathscr K$ of knot types, let \[ \mathcal D(\gamma)=\{K\in\mathscr K: \text{some crossing assignment on }\gamma\text{ gives a diagram of }K\}. \] The usual interpretation takes $\mathscr K$ to be the ambient isotopy classes of tame, unoriented knots in ${\mathbb{R}}^3$. This convention, as well as the smoothness convention above, is not expressly specified in the supplied record and requires confirmation.

Let $(\gamma_n)_{n\geq 1}$ be a sequence of generic plane immersions of $S^1$. Under the eventual-tail interpretation of "eventually carries every knot," the requested problem is to find necessary and sufficient conditions on this sequence for \[ \forall K\in\mathscr K\;\exists N_K\geq 1\;\forall n\geq N_K, \qquad K\in\mathcal D(\gamma_n). \tag{1} \] The crossing assignment may depend on both $K$ and $n$.

Unresolved quantifier qualification. The supplied statement does not explicitly distinguish (1) from the weaker occurrence property \[ \forall K\in\mathscr K\;\exists n\geq 1, \qquad K\in\mathcal D(\gamma_n). \tag{2} \] No nesting or monotonicity hypothesis on the sequence is supplied, so these formulations must not be identified. A definitive characterization problem requires confirmation of the intended quantifiers. The supplied record labels the characterization problem open; this label is not an independently verified mathematical or literature conclusion.

Short English statement

Characterize sequences of immersed plane circles whose crossings can be assigned over/under information to represent every knot. The source leaves unclear whether each knot must occur on every sufficiently late curve or merely on some curve. The knot category and regularity conventions also require confirmation.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30002958$, title Universal Sequences of Plane Curves for Knot Diagrams, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified in the supplied record; the proposed year is recorded as 2015.
• Supplied source citation: Computational Geometric and Algebraic Topology (2016), Oberwolfach Reports, pp. 2637--2699, DOI 10.4171/owr/2015/45. Supplied source URL: https://doi.org/10.4171/owr/2015/45. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports. The citation's 2016 label and the report's supplied 2015 identification require bibliographic reconciliation.
• Supplied literature source: Oberwolfach Report 45, 2015, https://doi.org/10.4171/owr/2015/45. Literature checked at: 2026-08-21. This is supplied dated curation, not independently verified here. Its status is recorded as open, and its assessment states that no necessary-and-sufficient characterization is known and that the report records only examples and necessary conditions. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement as exact and describes it as retained byte-for-byte and publication-ready. These are supplied verification labels; they do not resolve the formulation issues identified here. No report excerpt or problem-specific page locator is supplied.