Fox's trapezoidal conjecture for the four-strand Turk's head knots

Abstract

Let $Th(4,2n+1)=\widehat{β_{2n+1}}$ be the four-strand Turk's head knot, where $β_{2n+1}=σ_1σ_2^{-1}σ_3)^{2n+1}$. In our earlier work the Alexander polynomial of this family was reduced, after the substitution $t=-z$, to the factorization \[ A_{2n+1}(z)=(1+z+\cdots+z^{2n})D_n(z)^2, \qquad D_n(z)=\prod_{r=1}^n \left(z^2+4\sin^2\frac{πr}{2n+1}\,z+1\right). \] The remaining difficulty was to prove log-concavity of the coefficient sequence of $D_n(z)$. In this work we prove that the coefficient sequence of $D_n(z)$ is log-concave. The main new ingredient is a four-block smoothing theorem for products of reciprocal quartics $(1+az+z^2)(1+bz+z^2), \: 0\le a,b\le 4,\: a+b\ge 4.$ The smoothing theorem is proved by a finite exact positivity certificate using only integer arithmetic. Combining this smoothing theorem with the trigonometric pairing $r\leftrightarrow n+1-r$ proves that $D_n(z)$ is log-concave for all $n\ge1.$ It follows that the absolute values of the coefficients of the Alexander polynomial of $Th(4,2n+1)$ form a trapezoidal sequence. Thus Fox's trapezoidal conjecture holds for the entire family of four-strand Turk's head knots.