We prove that an iterated torus knot type in fails the uniform thickness property (UTP) if and only if it is formed from repeated positive cablings, which is precisely when an iterated torus knot supports the standard contact structure. This is the first complete UTP classification for a large class of knots. We also show that all iterated torus knots that fail the UTP support cabling knot types that are transversely nonsimple.
"Studying uniform thickness II: Transversely nonsimple iterated torus knots." Algebr. Geom. Topol. 11 (5) 2741 - 2774, 2011. https://doi.org/10.2140/agt.2011.11.2741