Problem 12.17 — (X.-S.

Definitions and mathematical statement

Fix a knot type $K$. X.-S. Lin's Problem 12.17 asks for the homotopy type of the space $L(K)$ of "long ropes," with knot type $K$, as defined by a picture in the source. That picture is absent from the supplied record.

If $\mathcal L$ denotes the source's full space of admissible ropes and $\kappa$ its knot-type assignment, the intended subspace is \[ L(K)=\{r\in\mathcal L:\kappa(r)=K\}. \] This identity expresses the fixed-type restriction without defining the missing ambient space. Determining its homotopy type requires an explicit topological model for $\mathcal L$, the subspace topology, and the construction of $\kappa$. The record gives no rope domain, endpoint or behavior-at-infinity conditions, smoothness, embedding conditions, length restrictions, parametrization quotient, or closure convention for obtaining $K$. Consequently it does not justify identifying $L(K)$ with a standard component of a long-knot embedding space. A solution should identify the actual source-defined space up to the intended homotopy equivalence, not merely describe its set of knot types.

Short English statement

Determine the homotopy type of the space of long ropes realizing a fixed knot type. The source's defining picture and topology are missing, so the rope space cannot be silently replaced by a familiar long-knot space.

Sources

• Local UnsolvedMath registry, permanent ID $10400222$, exact source code catalogue entry, Problem 12.17, attributed to X.-S. Lin [262]. The bibliography entry [262] and the defining picture are absent; the title is truncated.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.17, PDF page 166: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold 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 calls for a current primary-literature audit. Supplied classification OPEN-TRIAGE identifies Lin's fixed-knot-type rope-space question without further details or resolution.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.