Stable Khovanov homology and Volume

Abstract

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 full twists of each twist region tends toward infinity, proving a quantum version of Thurston's hyperbolic Dehn surgery theorem implicit in Rozansky's work, and giving a categorical version of 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 volumes of regular ideal octahedra that arise naturally from the evaluation of the colored Jones polynomials of the link.