Stabilization of divisors in high-dimensional contact manifolds

A stabilization operation is defined for codimension $2$ contact submanifolds in $\dim \ge 5$ contact manifolds $(M,\xi )$. The definition is such that (1) a given $(M,\xi )$ is overtwisted iff its standard transverse unknot is stabilized and (2) transverse stabilization preserves the formal contact isotopy class and intrinsic contact structure of a link. We prove that many transverse links are non-simple.