  We investigate Fox's trapezoidal conjecture for alternating links. We show
that it holds for diagrammatic Murasugi sums of special alternating links,
where all sums involved have length less than three (which includes
diagrammatic plumbing). It also holds for links containing a large twist
region, which we call twist-concentrated. Furthermore, we show some weaker
inequalities between consecutive coefficients of the Alexander polynomial of an
alternating 3-braid closure, and extend this to arbitrary alternating links. We
then study an extension of the trapezoidal conjecture due to Hirasawa and
Murasugi, which states that the stable length of the Alexander polynomial of an
alternating link can be bounded from above using the signature. We estabilish
this and determine when equality holds for diagrammatic Murasugi sums of
special alternating knots where each sum has length less than three, and also
for twist-concentrated 3-braids. Finally, we study the behavior of the
Hirasawa-Murasugi inequality under concordance.
