# P2814: Torsion-freeness for king-grid independence complexes

- ID: `P2814`
- Reference: `king-grid-independence-homology-torsion-free`
- Page: https://theoremdb.org/statements/P2814
- Record maturity: Reviewed problem with recorded work

## Problem

For \(n\ge1\), let \(K_n\) have vertex set \([n]\times[n]\), with two distinct vertices adjacent when their coordinate differences are both at most one. Let \(I(K_n)\) be the simplicial complex whose faces are the independent vertex sets of \(K_n\). Is every integral homology group \(H_j(I(K_n);\mathbb Z)\) torsion-free for every \(n\) and every \(j\ge0\)?

### Problem setup

- **Remark.** Adjacency is a legal king move on an \(n\times n\) chessboard, including horizontal, vertical, and diagonal moves.
- **Definition.** An independent set contains no adjacent pair, and all independent sets form a simplicial complex under inclusion.
- **Definition.** An abelian group is torsion-free when no nonzero element has finite order.

### What counts as a solution

- Prove integral torsion-freeness for all \(n,j\), or give specific \(n,j\) together with a cycle and Smith-normal-form or equivalent certificate establishing nonzero torsion.

## Status

UNKNOWN as of 2026-08-01. The checked grid-independence-complex literature does not give an integral torsion classification for the king graphs \(K_n\). [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-08-01. The checked grid-independence-complex literature does not give an integral torsion classification for the king graphs \(K_n\).

UNKNOWN as of 2026-08-01. The checked grid-independence-complex literature does not give an integral torsion classification for the king graphs \(K_n\).

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 hard-square constraint gives transfer recurrences for face counts, while integral homology retains information those recurrences miss. Chain reductions, Morse matchings, and minimal torsion witnesses would remain reusable across nearby lattice complexes.

- Original intake status: UNKNOWN as of 2026-07-27. The checked grid-independence-complex literature does not give an integral torsion classification for the king graphs \(K_n\).
- 2026-07-27 prior-art search checked independence complexes of rectangular grids, king graphs, hard-square complexes, matching complexes, and homology torsion; no theorem covering this family was found.
- Euler characteristics and Betti numbers over one field cannot certify torsion-freeness. Smith normal forms or integral discrete Morse matchings are needed.
- A counterexample consists of one \(n,j\) and an integral boundary-matrix certificate with a nonunit Smith invariant; a positive proof could build a wedge decomposition or a torsion-free Morse complex uniformly in \(n\).

- Recorded example: For \(n=1\), the complex is a point. For \(n=2\), \(K_2\) is complete on four vertices, so \(I(K_2)\) is four isolated points and \(\widetilde H_0\cong\mathbb Z^3\).

### Open directions

- **Route 1** (reported): Prove integral torsion-freeness for all \(n,j\), or give specific \(n,j\) together with a cycle and Smith-normal-form or equivalent certificate establishing nonzero torsion. [3](#reference-3)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `king-grid-independence-homology-torsion-free`, 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>Himanshu Chandrakar and Anurag Singh, “Independence Complexes of Hexagonal Grid Graphs,” arXiv:2512.21318v1 (2025). Full preprint relevant to Torsion-freeness for king-grid independence complexes. https://arxiv.org/abs/2512.21318
   - preprint; reference source; arXiv:2512.21318, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Torsion-freeness for king-grid independence complexes: Supports the dated status review or a neighboring result for “Torsion-freeness for king-grid independence complexes.”
2. <a id="reference-2"></a>Anurag Singh, “The topology of independence complexes of square grids”. arXiv:2204.05629 (2022). Wedge-of-spheres result for nearby square-grid graph families https://arxiv.org/abs/2204.05629
   - preprint; secondary source; arXiv:2204.05629v2; checked 2026-08-01
   - Source use: original_summary
   - For Torsion-freeness for king-grid independence complexes: This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.
3. <a id="reference-3"></a>Integral homotopy methods for independence complexes of nearby square-grid families https://arxiv.org/abs/math/0701890
   - Also cited at Editorial research route recorded 2026-08-01
   - preprint; reference source; checked 2026-08-01
   - Source metadata incomplete: publication-style citation, source version.
   - Source use: citation_only
   - Source named by the research packet.