Jones Polynomial at 5th Roots Links Knots to Fibonacci Anyons — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Jones polynomial at 5th roots of unity connects to Fibonacci anyons and golden-ratio braiding, linking knot invariants to topological quantum computing. | MATH: Jones polynomial VL(t) satisfies skein relation t−1VL+ − tVL- = (t1/2 − t−1/2)VL0. At t = e2π i/5, the quantum dimension d = ϕ = (1+√5)/2 ≈ 1.618 (Fibonacci anyon fusion rule ϕ ⊗ ϕ = 1 ⊕ ϕ). R-matrix eigenvalues at 5th root: λ1 = e4π i/5, λ2 = −e−2π i/5, satisfying λ1/λ2 = −ϕ2. Fibonacci numbers appear via Fn+1 = ϕ Fn + (−ϕ)−n (Binet form). | CONNECTION: Golden ratio ϕ = 1.618 and its inverse ϕ−1 = 0.618 are the quantum dimensions of the Fibonacci anyon — the simplest non-Abelian anyon model. The 5th root of unity e2π i/5 generates the dihedral symmetry D5 (pentagonal, crystallographic point group 5/m in 2D), and the Jones polynomial at this
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026jones,
title = {{Jones Polynomial at 5th Roots Links Knots to Fibonacci Anyons — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23616600},
url = {https://zenodo.org/records/23616600},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Jones Polynomial at 5th Roots Links Knots to Fibonacci Anyons — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23616600
UR - https://zenodo.org/records/23616600
AB - FINDING: Jones polynomial at 5th roots of unity connects to Fibonacci anyons and golden-ratio braiding, linking knot invariants to topological quantum computing. | MATH: Jones polynomial V_L(t) satisfies skein relation t^−1V_L_+ − tV_L_- = (t^1/2 − t^−1/2)V_L_0. At t = e^2π i/5, the quantum dimension d = ϕ = (1+√5)/2 ≈ 1.618 (Fibonacci anyon fusion rule ϕ ⊗ ϕ = 1 ⊕ ϕ). R-matrix eigenvalues at 5th root: λ_1 = e^4π i/5, λ_2 = −e^−2π i/5, satisfying λ_1/λ_2 = −ϕ^2. Fibonacci numbers appear via F_n+1 = ϕ F_n + (−ϕ)^−n (Binet form). | CONNECTION: Golden ratio ϕ = 1.618 and its inverse ϕ^−1 = 0.618 are the quantum dimensions of the Fibonacci anyon — the simplest non-Abelian anyon model. The 5th root of unity e^2π i/5 generates the dihedral symmetry D_5 (pentagonal, crystallographic point group 5/m in 2D), and the Jones polynomial at this Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -