  In 2013, Adams introduced the $n$-crossing number of a knot $K$, denoted by
$c_n(K)$. Inequalities between the $2$-, $3$-, $4$-, and $5$-crossing numbers
have been previously established. We prove $c_9(K)\leq c_3(K)-2$ for all knots
$K$ that are not the trivial, trefoil, or figure-eight knot. We show this
inequality is optimal and obtain previously unknown values of $c_9(K)$. We
generalize this inequality to prove that $c_{13}(K) < c_{5}(K)$ for a certain
set of classes of knots.
