Conway polynomials of $Th(p,q)$ and $Th(q,p)$: a proof of Takemura's conjecture

Abstract

Takemura conjectured that the Conway polynomials $\sum a_\ell z^\ell$ of the Turk's head link $Th(p,q)$ and $\sum b_\ell z^\ell$ of $Th(q,p)$ satisfy $a_\ell=b_\ell$ for $\ell\equiv0,1\pmod 4$ and $a_\ell=-b_\ell$ for $\ell\equiv2,3\pmod4$, and proved it for $q=2,3$. We prove it for all $p,q\ge1$, knots and links. Equivalently, $\nabla_{Th(q,p)}(z)=c\,\nabla_{Th(p,q)}(iz)$ with $c=1$ if $\gcd(p,q)$ is odd and $c=-i$ if it is even. The proof combines a product formula $\nabla_{Th(p,q)}(z)^2=\prod_{j=1}^{p-1}\prod_{k=1}^{q-1}(z^2+2\cos\frac{2πk}{q}-2\cos\frac{2πj}{p})$, which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid.