PreprintCC BY 4.0arXiv
Conway polynomials of Th(p,q) and Th(q,p): a proof of Takemura's conjecture
Abstract
Takemura conjectured that the Conway polynomials ∑ aℓ zℓ of the Turk's head link Th(p,q) and ∑ bℓ zℓ of Th(q,p) satisfy aℓ=bℓ for ℓ≡0,1pmod 4 and aℓ=−bℓ for ℓ≡2,3pmod4, and proved it for q=2,3. We prove it for all p,q≥1, knots and links. Equivalently, ∇Th(q,p)(z)=c ∇Th(p,q)(iz) with c=1 if gcd(p,q) is odd and c=−i if it is even. The proof combines a product formula ∇Th(p,q)(z)2=∏j=1p−1∏k=1q−1(z2+2cos2πk/q−2cos2πj/p), which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid.
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BibTeX
@misc{stanggassinger2026conway,
title = {{Conway polynomials of $Th(p,q)$ and $Th(q,p)$: a proof of Takemura's conjecture}},
author = {Werner Alois Stanggassinger},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32269},
eprint = {2610.32269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Conway polynomials of Th(p,q) and Th(q,p): a proof of Takemura's conjecture
AU - Werner Alois Stanggassinger
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32269
UR - https://arxiv.org/abs/2610.32269
AB - Takemura conjectured that the Conway polynomials ∑ a_ℓ z^ℓ of the Turk's head link Th(p,q) and ∑ b_ℓ z^ℓ of Th(q,p) satisfy a_ℓ=b_ℓ for ℓ≡0,1pmod 4 and a_ℓ=−b_ℓ for ℓ≡2,3pmod4, and proved it for q=2,3. We prove it for all p,q≥1, knots and links. Equivalently, ∇_Th(q,p)(z)=c ∇_Th(p,q)(iz) with c=1 if gcd(p,q) is odd and c=−i if it is even. The proof combines a product formula ∇_Th(p,q)(z)^2=∏_j=1^p−1∏_k=1^q−1(z^2+2cos2πk/q−2cos2πj/p), which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid.
ER -