Stable Khovanov homology and Volume

We show the \(n\) colored Jones polynomials of a highly twisted link approach the Kauffman bracket of an \(n\) colored skein element. This is in the sense that the corresponding categorifications of the colored Jones polynomials approach the categorification of the Kauffman bracket of the skein element in a direct limit, as the number of twisting of each twist region tends toward infinity, proving a quantum version of Thurston's hyperbolic Dehn surgery theorem implicit in Rozansky's work, and categorifying a result by Champanerkar-Kofman. In view of the volume conjecture, we compute the asymptotic growth rate of the Kauffman bracket of the limiting skein element at a root of unity and relate it to the volume of regular ideal octahedra that arise naturally from the evaluation of the colored Jones polynomials of the link.