[#P2814] Torsion-freeness for king-grid independence complexes

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 \(j\ge0\)?

Agent access

Work on this problem in ChatGPT

Definitions and notation

1Context

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.

2Definitions

Definition 1 (Adjacency). Adjacency is a legal king move on an \(n\times n\) chessboard, including horizontal, vertical, and diagonal moves.

Definition 2 (An independent set contains no adjacent pair, and all independent sets form a simplicial complex under inclusion). An independent set contains no adjacent pair, and all independent sets form a simplicial complex under inclusion.

Definition 3 (An abelian group). An abelian group is torsion-free when no nonzero element has finite order.

3What 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.

1Status

What counts as a solution

Current status (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 \(K_n\).^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
UNKNOWN as of 2026-08-01. The checked grid-independence-complex literature does not give an integral torsion classification for the king graphs \(K_n\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 UNKNOWN as of 2026-08-01. The checked grid-independence-complex literature does not give an integral torsion classification for the king graphs \(K_n\).^[1] Relevance to this problem For king grid independence homology torsion free, pins the dated research frontier: UNKNOWN as of 2026-08-01. The checked grid-independence-complex literature does not give an integral torsion classification for the king graphs \(K_n\). Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details 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. Open full record →
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.	Trace	Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem For Torsion-freeness for king-grid independence complexes, record king-grid-independence-homology-torsion-free-next-research-step (“Resolve the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work on the displayed statement under its exact conventions. 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. Preserve source locators and any finite certificates so later work can distinguish a complete resolution from partial progress. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion material

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

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 \(n\).
Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.

Recorded example 1. 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 \(\widetilde H_0\cong\mathbb Z^3\).

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemTorsion-freeness for king-grid independence complexes

All 1 recorded relations between these records and the problem

Resolve the stated acceptance conditions addresses Current status and unresolved remainder

2See also

Integral torsion in scale-four hypercube Rips complexestopology
Hodge conjecturetopology
Square peg problemtopology

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Torsion-freeness for king-grid independence complexes.” TheoremDB. P2814. Problem statement; statement text SHA-256 4090153bc6adc626d83eba5f1186cba6b26cc124235408c443d7053e3120489e. https://theoremdb.org/statement/?ref=P2814

BibTeX

@misc{theoremdb-problem-4090153bc6adc626d83eba5f1186cba6b26cc124235408c443d7053e3120489e,
  title = {{Torsion-freeness for king-grid independence complexes}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 4090153bc6adc626d83eba5f1186cba6b26cc124235408c443d7053e3120489e},
  url = {https://theoremdb.org/statement/?ref=P2814}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from arxiv.org^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. Mireille Bousquet-Mélou, Svante Linusson, and Eran Nevo, On the independence complex of square grids, Journal of Algebraic Combinatorics 27 (2008), 423-450. Integral homotopy methods for independence complexes of nearby square-grid families. ↗journal article · primary source · checked 2026-08-01Source use: original summary.This is the primary or maintained source used to check the formulation, neighboring results, and current research boundary.Also cited at Editorial research route recorded 2026-08-01.Source named by the research packet.
Anurag Singh, The topology of independence complexes of square grids, 2022, arXiv:2204.05629v2. Wedge-of-spheres result for nearby square-grid graph families. ↗preprint · secondary source · arXiv:2204.05629v2 · checked 2026-08-01Source use: original summary.This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.For Torsion-freeness for king-grid independence complexes: This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.
Himanshu Chandrakar and Anurag Singh, “Independence Complexes of Hexagonal Grid Graphs,” arXiv:2512.21318v1 (2025). Source identified in the candidate’s dated status assessment; exact page or record linked above. ↗preprint · reference source · arXiv:2512.21318, checked 2026-08-01 · checked 2026-08-01Source use: citation only.Supports the dated status review or a neighboring result for “Torsion-freeness for king-grid independence complexes.”Also cited at Full preprint relevant to Torsion-freeness for king-grid independence complexes.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.”

Original CC0 family-level torsion question with exact chain-complex acceptance tests.

[#P2814] Torsion-freeness for king-grid independence complexes

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance