Twisted Torus Knots: Closed-Form Alexander Polynomials and Fibonacci Determinants — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Alexander polynomials of twisted torus knots admit explicit closed-form formulas via knot group presentations, with Fibonacci numbers emerging as determinant sequences in special torus-knot cases. MATH: - Alexander polynomial Δ(t) for torus knot T(p,q): Δ(t) = ((t^{pq}−1)(t−1)) / ((t^p−1)(t^q−1)). - For T(2,n): Δ(t) = (t^n+1)/(t+1) = 1 − t + t² − … + t^{n−1} (alternating sum). - Determinant |Δ(−1)| for T(2,n) = n (odd) — but for Fibonacci-type braid closures, determinants follow F_{k+1} (Fibonacci numbers) when Seifert matrix has characteristic polynomial t²−t−1. - Twisted torus knot formula (arXiv:2411.13003): Δ(t) expressed as product of cyclotomic-like factors with exponents determined by twist parameters (full twists m on adjacent strands). - Fibonacci matrix Q = [[1,1],[1,0]] has det(Q) = −1, trace = 1, eigenvalues φ = (1+√5)/2 and ψ = (1−√5)/2. Q^n yields F_{n+1}, F_n. CONNECTION: - Golden ratio φ = 1.618 appears as spectral radius of the Fibonacci matrix
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026twisted,
title = {{Twisted Torus Knots: Closed-Form Alexander Polynomials and Fibonacci Determinants — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409900},
url = {https://zenodo.org/records/23409900},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twisted Torus Knots: Closed-Form Alexander Polynomials and Fibonacci Determinants — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409900
UR - https://zenodo.org/records/23409900
AB - FINDING: Alexander polynomials of twisted torus knots admit explicit closed-form formulas via knot group presentations, with Fibonacci numbers emerging as determinant sequences in special torus-knot cases. MATH: - Alexander polynomial Δ(t) for torus knot T(p,q): Δ(t) = ((t^{pq}−1)(t−1)) / ((t^p−1)(t^q−1)). - For T(2,n): Δ(t) = (t^n+1)/(t+1) = 1 − t + t² − … + t^{n−1} (alternating sum). - Determinant |Δ(−1)| for T(2,n) = n (odd) — but for Fibonacci-type braid closures, determinants follow F_{k+1} (Fibonacci numbers) when Seifert matrix has characteristic polynomial t²−t−1. - Twisted torus knot formula (arXiv:2411.13003): Δ(t) expressed as product of cyclotomic-like factors with exponents determined by twist parameters (full twists m on adjacent strands). - Fibonacci matrix Q = [[1,1],[1,0]] has det(Q) = −1, trace = 1, eigenvalues φ = (1+√5)/2 and ψ = (1−√5)/2. Q^n yields F_{n+1}, F_n. CONNECTION: - Golden ratio φ = 1.618 appears as spectral radius of the Fibonacci matrix Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -