Fox's Trapezoidal Conjecture for Four-Strand Turk's Head Knots and Links

We prove Fox’s trapezoidal conjecture for the four-strand Turk’s head knots and links $Th(4,q)$, for all $q\ge 1$. Equivalently, we show that the absolute values of the coefficients of the one-variable Alexander polynomial of $Th(4,q)$ form a trapezoidal sequence. The proof begins with a uniform Burau factorization for the closures of $({\sigma }_1{\sigma }_2^{-1}{\sigma }_3{)}^q$, which expresses the Alexander polynomial in terms of reciprocal quadratic factors indexed by the $q$-th roots of unity. The odd and even exponent cases then follow from a common log-concavity argument based on a four-block smoothing theorem for reciprocal quartic factors.