{
  "abstract": "Let $K$ be the trefoil or figure-eight knot and let $M_K$ denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover \\[ M_K\\times S^1\\longrightarrow T^2\\times S^2 \\] whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-$g$ fibered knot with hyperelliptic monodromy, the same construction gives a braided $(2g+2)$-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over $T^2$. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of $Y_K$ and its twisted double $X_K$, as studied by the author in \\cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to $\\CP^2\\#k\\CPb^2$, $k=3,5$ \\cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a $k=5$ example \\cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00409",
  "author_total": 1,
  "authors": [
    {
      "name": "Anar Akhmedov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dyp5ze4z4fr2n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00409",
  "fulltext_url": null,
  "id": "dyp5ze4z4fr2n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00409",
  "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.GT"
    ],
    [
      "arxiv",
      "math.SG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.00409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dyp5ze4z4fr2n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00409"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Braided Surfaces and Small Exotic 4-Manifolds",
  "title_text": "Braided Surfaces and Small Exotic 4-Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/dyp5ze4z4fr2n",
  "venue": "arXiv",
  "versions": []
}