# P2804: Flip-graph diameter for triangulations of C(10,4)

- ID: `P2804`
- Reference: `cyclic-polytope-c4-10-flip-diameter`
- Page: https://theoremdb.org/statements/P2804
- Record maturity: Reviewed problem with recorded work

## Problem

Let C(10,4) be the convex hull in \(\mathbb R^4\) of \((t,t^2,t^3,t^4)\) for \(t=1,\ldots,10\). Form the graph whose vertices are triangulations of this point configuration without added vertices, with two triangulations adjacent when they differ by one bistellar flip. Determine its connected components and the diameter of each component.

### Definitions

- **Definition.** A triangulation is a face-to-face subdivision of the polytope into 4-simplices whose vertices are among the ten given points.
- **Definition.** A bistellar flip replaces one triangulation of the convex hull of a circuit by the other triangulation supported on that circuit.
- **Definition.** The diameter of a connected graph is the largest shortest-path distance between two vertices.

### What counts as a solution

- Enumerate the triangulations with independently checkable canonical hashes, replay all bistellar-flip adjacencies, and certify the component partition and every reported diameter.

## Status

UNKNOWN as of 2026-08-01: The 2026-08-01 search found a published count of triangulations but no component or diameter result for this exact flip graph. [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-08-01: The 2026-08-01 search found a published count of triangulations but no component or diameter result for this exact flip graph.

UNKNOWN as of 2026-08-01: The 2026-08-01 search found a published count of triangulations but no component or diameter result for this exact flip graph.

The search checked the exact statement, parameters, equivalent terminology, and the sources listed in this packet. Database silence is treated only as bounded status evidence. A complete resolution must satisfy every acceptance condition in the canonical problem.

### Background and intake notes

The known finite census makes this a focused reconfiguration problem. Its graph can support later questions about regular triangulations, monotone orientations, and higher Stasheff-Tamari orders.

- Original intake status: UNKNOWN: The 2026-07-27 search found a published count of triangulations but no component or diameter result for this exact flip graph.
- 2026-07-27: Searches for C(10,4) triangulations found a reverse-search paper reporting 4824 triangulations. Searches pairing the exact polytope with flip-graph diameter returned no value.
- 2026-07-27: The target was checked against earlier candidate files and all live prospecting records with no duplicate.
- A complete artifact should include canonical simplex lists, circuit-supported flip edges, component labels, and shortest-path or eccentricity certificates.

- Recorded example: Pulling the vertices in the order 1 through 10 gives a triangulation and therefore at least one vertex of the flip graph.

### Open directions

- **Route 1** (reported): Enumerate the triangulations with independently checkable canonical hashes, replay all bistellar-flip adjacencies, and certify the component partition and every reported diameter. [2](#reference-2)

### Computational notes

- The cyclic 4-polytope has 35 tetrahedral facets by the formula \(n(n-3)/2\). The count of 4824 triangulations is attributed to the published reverse-search comparison and was not independently reproduced.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `cyclic-polytope-c4-10-flip-diameter`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1 entry has incomplete source metadata. Each affected row names the fields that still need editorial review.

1. <a id="reference-1"></a>Francisco Santos, The number of triangulations of the cyclic polytope C(n,n-4), 2000. Published reverse-search census containing the 4,824 count for C(10,4) https://www.wm.uni-bayreuth.de/de/team/rambau_joerg/TOPCOM/SymLexSubsetRS-2.pdf
   - website; reference source; author manuscript PDF checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Flip-graph diameter for triangulations of C(10,4): This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.
2. <a id="reference-2"></a>Status assessment source Triangulations and bistellar flips for cyclic polytopes https://doi.org/10.1112/S0025579300012055
   - Also cited at Editorial research route recorded 2026-08-01
   - scholarly_publication; reference source
   - Source metadata incomplete: publication-style citation.
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - Source named by the research packet.