{
  "abstract": "We study oriented $t_3$- and $\\bar t_6$-moves through the problem of lifting Fox $3$-colorings to colorings by a nine-element Alexander quandle. We define a nonnegative integer lifting defect $δ$ for oriented links and prove that it is invariant under these moves. For knots, the defect equals the rank of an alternating bilinear form obtained from a Seifert matrix, so it is even. We also prove that it is additive under connected sum. Let $K_s$ be the connected sum of $s\\geq 1$ copies of $9_{46}$. Then $δ(K_s)=2s$, so these knots represent infinitely many distinct equivalence classes under the oriented moves, and none is equivalent to an unlink. After forgetting orientation, each $K_s$ can be transformed into an unlink with $2s+1$ components by $3s$ ordinary $3$-moves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34831",
  "author_total": 1,
  "authors": [
    {
      "name": "Wenbiao Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99d22425qmkyz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34831",
  "fulltext_url": null,
  "id": "99d22425qmkyz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34831",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7223938174",
  "pdf_url": "https://arxiv.org/pdf/2610.34831",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99d22425qmkyz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34831",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34831"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A lifting defect for oriented $t_3$ and $\\bar t_6$ moves",
  "title_text": "A lifting defect for oriented t_3 and t̅_6 moves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/99d22425qmkyz",
  "venue": "arXiv",
  "versions": []
}