{
  "abstract": "We study a cut-and-paste operation in which torus neighborhoods in the author's symplectic building blocks $Y_K$ and $X_K$, associated to a genus-one fibered knot $K$, are replaced by marked exteriors of tori in $S^4$ with infinite cyclic complement group and two null peripheral slopes. We also recall the exact Luttinger-surgery realization of $M_K\\times S^1$ from $Σ_g\\times T^2$, keeping the knot-surgery/fiber-sum and Luttinger-surgery viewpoints in the same framework. For the trefoil block $Y_K$, two marked replacements give a simply connected manifold with intersection form $H$, hence a manifold homeomorphic to $S^2\\times S^2$. A one-exterior gluing gives a smooth homotopy $4$-sphere. For the rank-six construction we use the identity double of two copies of $Y_K\\setminusνΣ_2$, rather than the involutive gluing defining the original $X_K$. Two marked replacements along the surviving rim tori give a simply connected manifold with $e=8$ and $σ=0$. An explicit geometric basis has intersection form $3H$, so the resulting manifold is homeomorphic to $\\#_3(S^2\\times S^2)$. The Case II small-perturbation Seiberg--Witten invariant vanishes. The Seiberg--Witten invariant of the identity-glued rank-six family also vanishes; in particular, these manifolds are nonsymplectic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17603",
  "author_total": 1,
  "authors": [
    {
      "name": "Anar Akhmedov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4wh6vy3xxf5vb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17603",
  "fulltext_url": null,
  "id": "4wh6vy3xxf5vb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17603",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.SG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.17603",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4wh6vy3xxf5vb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.17603",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17603"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "$\\mathbb{Z}$-Torus Exteriors and Small Knot-Surgery Four-Manifolds",
  "title_text": "ℤ-Torus Exteriors and Small Knot-Surgery Four-Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/4wh6vy3xxf5vb",
  "venue": "arXiv",
  "versions": []
}