1.4 (Danciger) — Convex projective structures on glued figure-eight complements

Definitions and mathematical statement

Let $E=S^3\setminus\operatorname{int}\nu K$ be the compact exterior of the figure-eight knot $K$, so $\partial E\cong T^2$. Taking the compact exterior is necessary to interpret the source's "complement" with torus boundary. For an arbitrary homeomorphism $f:\partial E_1\to\partial E_2$ between two copies, form the closed $3$-manifold \[ N_f=E_1\cup_f E_2. \] Danciger's Problem 1.4 asks whether $N_f$ admits a convex real-projective structure for every such gluing $f$.

In the usual properly convex interpretation, this asks for a properly convex domain $\Omega\subset{\mathbb{R}}\mathrm P^3$, convex and bounded in some affine chart, and a discrete subgroup $\Gamma\subset\mathrm{PGL}_4({\mathbb{R}})$ preserving $\Omega$ and acting freely, properly discontinuously and cocompactly, such that \[ N_f\cong\Omega/\Gamma. \] The source says convex, not strictly convex; no strict convexity or smooth-boundary hypothesis is imposed. Its precise convex-projective convention should be confirmed against the source.

The supplied source cites Ballas--Danciger--Lee for a positive answer for identity gluing, namely the double of $E$. The research summary also reports general convex-gluing machinery subject to a holonomy-matching condition, while saying arbitrary gluing maps remain unresolved. The special case and conditional gluing framework do not assert a solution for every $f$.

Short English statement

Does every closed three-manifold obtained by gluing two figure-eight knot exteriors along their boundary tori admit a convex projective structure? The supplied source reports a positive identity-gluing case and the summary notes conditional gluing tools; the general map remains the stated question.

Sources

• Local UnsolvedMath registry, permanent ID $10900004$, exact source code catalogue entry, subsection 1.4, attributed to Danciger; no proposed year supplied.
• Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), 1.4, PDF page 1: https://arxiv.org/pdf/1512.04620#page=1; supplied status-source link https://arxiv.org/abs/1512.04620.
• The source cites Ballas--Danciger--Lee for the double of the figure-eight knot exterior. No complete bibliographic entry is supplied. The research summary mentions holonomy-matching convex-gluing machinery without a separate citation.
• Source-TeX extraction checked against rendered PDF text, accessed 2026-07-29. Source evidence NEEDS_REVIEW: open in 2015 with no complete modern primary-literature audit recorded. Registry status partially_solved and classification PARTIAL-PROGRESS retain the identity case and unresolved general gluing question.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author-submitted source TeX and PDF, redistribution terms requiring release review.