Problem 12.11 — (D.

Definitions and mathematical statement

A shadow diagram in the source's sense encodes a $3$-manifold or link using a shadow surface and gleam data; the required local models, gleam conventions and reconstruction are not included in this extract. An alternating knot has a diagram whose over- and under-crossings alternate along the knot. For a hyperbolic knot $K\subset S^3$, its hyperbolic volume is the volume of the complete finite-volume metric on $S^3\setminus K$.

D. Thurston's Problem 12.11 asks for a condition on shadow diagrams that is satisfied by shadow diagrams coming from alternating knots and yields a lower bound on hyperbolic volume. A concrete solution must specify the relevant shadow class, a checkable condition and the numerical lower-bound expression. The record does not give any of these, does not say which shadow diagrams of an alternating knot are intended, and does not specify whether volume refers to a knot complement or another reconstructed manifold. Not every alternating knot is hyperbolic, so the volume claim needs a domain convention or an explicit restriction; none is silently inserted here.

The following context states that Reshetikhin--Turaev and Turaev--Viro--Ocneanu invariants can be expressed through the KTG algebra, using $I$-bundles and $S^1$-bundles over shadow surfaces, respectively. It says their relation comes from two $3$-manifold constructions in Figure 23. The surviving diagram starts with closed shadow surfaces of zero gleams: making $I$-bundles and capping boundary yields closed $3$-manifolds, while making $S^1$-bundles yields certain closed $3$-manifolds. Its surviving map is written schematically as \[ M\longmapsto M\# M\#\left(\text{copies of }S^2\times S^1\right). \] The number of copies and the orientation or any lost marking on the second $M$ cannot be recovered from the corrupted extraction. The full figure, bundle/capping conventions and KTG operations must be checked. This contextual relation supplies neither the requested shadow condition nor a specific volume inequality.

Short English statement

Find a shadow-diagram condition covering the intended alternating-knot examples and giving a useful hyperbolic-volume lower bound. The diagram class, volume domain and bound are missing. The accompanying shadow-bundle discussion is preserved with its damaged figure data explicitly unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $10400216$, exact source code catalogue entry, Problem 12.11, attributed to D. Thurston. The title is truncated after "(D."
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.11 and Figure 23 context, PDF page 164: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold headings distinguish 231 numbered items from surrounding references.
• Figure 23 is supplied as damaged text with null/control artifacts and missing arrows or markings. Its caption concerns two ways to obtain $3$-manifolds from shadow surfaces; no independent reconstruction has been attempted.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied classification OPEN-TRIAGE identifies Thurston's shadow-diagram volume-bound question without a further citation 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.