{
  "abstract": "A knot invariant is a fingerprint shared by equivalent knots: unequal values certify inequivalence, while equal values may conceal different knots. Under two rooted random-diagram models, we prove that the incomplete normalized Alexander polynomial separates random pairs but collides in a growing database. If $D_n$ and $D_n'$ are independent $n$-crossing diagrams, then $\\Pr\\{Δ_{K(D_n)}=Δ_{K(D_n')}\\}=O(n^{-1/2})$, and every fixed normalized Alexander polynomial is exponentially rare. With exponentially high probability, the diagram shadow contains linearly many disjoint opened-trefoil slots. Conditional on the shadow and exterior crossing signs, their indicators are independent Bernoulli$(1/4)$ variables whose sum is a binomial coordinate in the $3$-adic determinant valuation. Consequently, $\\sup_{a\\geq 1}\\Pr\\{\\det K(D_n)=a\\}=O(n^{-1/2})$. For a discrete invariant $I$, let $α_n(I)=\\Pr\\{I(D_n)=I(D_n')\\}$. Its collision entropy is $H_2(I(D_n))=-\\logα_n(I)$. An independent sample of size $M$ has $\\binom{M}{2}α_n(I)$ colliding pairs on average and birthday scale $α_n(I)^{-1/2}$. If $M_n^2α_n(I)\\to\\infty$, a repeated value occurs with probability tending to one even when $α_n(I)\\to0$. For the determinant, $M=o(n^{1/4})$ suffices for collision freedom with high probability; a matching lower bound is open. We also give exact finite-population formulas for fixed censuses, calculate the expected cost of invariant cascades, and relate collision probability to the frequency of calls to a complete equivalence procedure. On a balanced pair-classification benchmark, the normalized-Alexander rule has balanced accuracy $1-O(n^{-1/2})$ but cannot distinguish inequivalent pairs in one fiber.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08298",
  "author_total": 8,
  "authors": [
    {
      "name": "Pedro Olivares-Sánchez"
    },
    {
      "name": "Edison Jessie Vázquez Gordillo"
    },
    {
      "name": "Radmila Sazdanović"
    },
    {
      "name": "Carlos Alfonso Ruiz Guido"
    },
    {
      "name": "Aldo Guzmán-Sáenz"
    },
    {
      "name": "Renato Osvaldo Salmerón-García"
    },
    {
      "name": "Ramiro López-Vázquez"
    },
    {
      "name": "Ernesto Lupercio"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3bkmn6024npgj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08298",
  "fulltext_url": null,
  "id": "3bkmn6024npgj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08298",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212022026",
  "pdf_url": "https://arxiv.org/pdf/2609.08298",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3bkmn6024npgj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08298",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08298"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Knots, black holes, databases, and birthdays: Collision entropy of knot invariants",
  "title_text": "Knots, black holes, databases, and birthdays: Collision entropy of knot invariants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/3bkmn6024npgj",
  "venue": "arXiv",
  "versions": []
}