{
  "abstract": "We introduce Mayer path homology, a new homology theory for directed path complexes obtained by equipping path complexes with an $N$-nilpotent differential. The main novelty of this work is the introduction of an $N$-differential on path complexes, giving rise to $N$-chain complexes of $\\partial$-invariant paths and Mayer path homology groups $H_n^{N,q}(P)$. We prove that this construction defines a canonical invariant of directed graphs and is more sensitive than standard path homology, distinguishing directed network motifs that ordinary path homology cannot separate. We further establish a complete classification of generators of $Ω_2^N$ and $Ω_3^N$, determining all admissible combinatorial types. Finally, we characterize elements of the first Mayer path cycles group $Z_1^{N,q}$ in terms of weighted directed cycles arising from spanning-tree constructions. These results provide the first systematic structural theory for Mayer path complexes and reveal new higher-order algebraic structures in directed graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.16525",
  "author_total": 2,
  "authors": [
    {
      "name": "Dilan Karaguler"
    },
    {
      "name": "Guo-Wei Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ee02zc9cq26pq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.16525",
  "fulltext_url": null,
  "id": "ee02zc9cq26pq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.16525",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2605.16525",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ee02zc9cq26pq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.16525",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.16525"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Mayer Path Homology",
  "title_text": "Mayer Path Homology",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ee02zc9cq26pq",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.3934/fods.2026025",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}