Strict Inequalities for the $n$-crossing Number

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 $c_9(K)\le c_3(K)-2$ 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.