Conjecture 12.16 — (X.-S.

Definitions and mathematical statement

Let $G$ denote the object in the source, and let $\operatorname{Aut}(G)$ mean its structure-preserving automorphisms. Write $\mu$ for the source's mirror map. The supplied excerpt does not define $G$, its structure, the mirror operation, or the ambient space.

Conjecture (X.-S. Lin [262]). Every automorphism of $G$ is either the identity or the mirror map: \[ \forall\varphi\in\operatorname{Aut}(G),\qquad \varphi=\operatorname{id}_G\ \text{or}\ \varphi=\mu. \] Equivalently in the intended geometric setup, the source says that every automorphism is induced by a diffeomorphism of the ambient space. Once that space $X$ and its action are defined, this latter assertion would say that the induced action map \[ \operatorname{Diff}(X)\longrightarrow\operatorname{Aut}(G) \] is surjective. The excerpt does not provide the hypotheses identifying the image of this map with the identity and mirror operations; that identification must be checked rather than assumed for an unspecified $G$.

The draft retains the universal two-operation rigidity claim without replacing $G$ by an inferred knot graph, a group, or another object. A precise definition of the ambient structures and induced action is required for a fully determined conjecture.

Short English statement

Lin conjectures that the only automorphisms of the source's object $G$ are the identity and mirror map, or equivalently in its intended setup that all are induced by ambient diffeomorphisms. The object $G$ and the ambient action are not defined in this excerpt.

Sources

• Local UnsolvedMath registry, permanent ID 10400221, source code catalogue entry; attribution X.-S. Lin [262]. The stored title is truncated after "Conjecture 12.16 --- (X.-S.".
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Conjecture 12.16, PDF page 166. Supplied URL: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; recorded access 2026-07-29; extraction pdf-font-structure. Internal reference [262] has no bibliographic entry in the record. The collection distinguishes 231 numbered bold problem, conjecture, and question headings.
• Supplied status open and OPEN-TRIAGE. The research summary describes Lin's ambient-diffeomorphism conjecture. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.