Knots, black holes, databases, and birthdays: Collision entropy of knot invariants
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 Dn and Dn' are independent n-crossing diagrams, then Pr{ΔK(Dn)=ΔK(Dn')}=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, supa≥ 1Pr{det K(Dn)=a}=O(n−1/2). For a discrete invariant I, let αn(I)=Pr{I(Dn)=I(Dn')}. Its collision entropy is H2(I(Dn))=−logαn(I). An independent sample of size M has binomM2αn(I) colliding pairs on average and birthday scale αn(I)−1/2. If Mn2αn(I)→∞, a repeated value occurs with probability tending to one even when αn(I)→0. For the determinant, M=o(n1/4) 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.
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BibTeX
@misc{olivaressanchez2026knots,
title = {{Knots, black holes, databases, and birthdays: Collision entropy of knot invariants}},
author = {Pedro Olivares-Sánchez and Edison Jessie Vázquez Gordillo and Radmila Sazdanović and Carlos Alfonso Ruiz Guido and Aldo Guzmán-Sáenz and Renato Osvaldo Salmerón-García and Ramiro López-Vázquez and Ernesto Lupercio},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08298},
eprint = {2609.08298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08298},
}
RIS
TY - UNPB
TI - Knots, black holes, databases, and birthdays: Collision entropy of knot invariants
AU - Pedro Olivares-Sánchez
AU - Edison Jessie Vázquez Gordillo
AU - Radmila Sazdanović
AU - Carlos Alfonso Ruiz Guido
AU - Aldo Guzmán-Sáenz
AU - Renato Osvaldo Salmerón-García
AU - Ramiro López-Vázquez
AU - Ernesto Lupercio
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08298
UR - https://arxiv.org/abs/2609.08298
AB - 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≥ 1Pr{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 binomM2α_n(I) colliding pairs on average and birthday scale α_n(I)^−1/2. If M_n^2α_n(I)→∞, a repeated value occurs with probability tending to one even when α_n(I)→0. 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.
ER -