Planar Disk Filters and Real Chromatic Roots at Beraha Numbers
Abstract
We construct simple plane triangulations whose chromatic polynomials have real roots accumulating at every noninteger Beraha number Bℓ=4cos2(π/ℓ); the integer values $0,1,2,3$ are roots of K4. The construction rests on the observation that a triangulated disk with four boundary vertices acts, inside a larger triangulation, as a transfer operator lying in a commutative three-parameter algebra. Rotating such disks by a quarter turn and composing them in series moves their eigenvalue triples around this algebra in a controllable way, and for every fixed q∈(B10,4) one can manufacture a finite disk whose two competing local eigenvalues are strictly dominated by the third -- a physical filter. Closing a grid of such disks between two poles produces a sphere triangulation, and a fused Temperley-Lieb trace identity accounts for its chromatic polynomial completely, sector by sector, with no appeal to semisimplicity at the root of unity itself -- generic semisimplicity enters only through polynomial identities continued from δ>2 -- and no singular higher Jones-Wenzl projectors. In the one-cap sector a Krein-type argument shows that every sweep word has exactly one expanding eigenvalue; Gram-form inertia controls every other sector uniformly in the width. At odd Beraha indices the trace coefficient of the expanding eigenvalue vanishes; at even indices two copies of that eigenvalue cancel between adjacent sectors, and a short real-variable lemma converts either situation into genuine real roots converging to the target. Indices below eleven are treated by elementary families, exact outward-rounded integer certificates, and one explicit disk at B10 combined with Tutte's golden identity. All graphs are ordinary, simple, unweighted plane triangulations.
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BibTeX
@misc{mactabi2026planar,
title = {{Planar Disk Filters and Real Chromatic Roots at Beraha Numbers}},
author = {Reza Mactabi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23989},
eprint = {2610.23989},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23989},
}
RIS
TY - UNPB
TI - Planar Disk Filters and Real Chromatic Roots at Beraha Numbers
AU - Reza Mactabi
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.23989
UR - https://arxiv.org/abs/2610.23989
AB - We construct simple plane triangulations whose chromatic polynomials have real roots accumulating at every noninteger Beraha number B_ℓ=4cos^2(π/ℓ); the integer values $0,1,2,3$ are roots of K_4. The construction rests on the observation that a triangulated disk with four boundary vertices acts, inside a larger triangulation, as a transfer operator lying in a commutative three-parameter algebra. Rotating such disks by a quarter turn and composing them in series moves their eigenvalue triples around this algebra in a controllable way, and for every fixed q∈(B_10,4) one can manufacture a finite disk whose two competing local eigenvalues are strictly dominated by the third -- a physical filter. Closing a grid of such disks between two poles produces a sphere triangulation, and a fused Temperley-Lieb trace identity accounts for its chromatic polynomial completely, sector by sector, with no appeal to semisimplicity at the root of unity itself -- generic semisimplicity enters only through polynomial identities continued from δ>2 -- and no singular higher Jones-Wenzl projectors. In the one-cap sector a Krein-type argument shows that every sweep word has exactly one expanding eigenvalue; Gram-form inertia controls every other sector uniformly in the width. At odd Beraha indices the trace coefficient of the expanding eigenvalue vanishes; at even indices two copies of that eigenvalue cancel between adjacent sectors, and a short real-variable lemma converts either situation into genuine real roots converging to the target. Indices below eleven are treated by elementary families, exact outward-rounded integer certificates, and one explicit disk at B_10 combined with Tutte's golden identity. All graphs are ordinary, simple, unweighted plane triangulations.
ER -