Stable Khovanov homology and Volume

TL;DR

This paper demonstrates that the categorifications of colored Jones polynomials of highly twisted links converge to the categorification of the Kauffman bracket of a skein element, linking quantum invariants to hyperbolic volume via a limit process.

Contribution

It establishes a categorical limit of colored Jones polynomials approaching the Kauffman bracket, providing a quantum analogue of Thurston's hyperbolic Dehn surgery theorem.

Findings

Categorifications of colored Jones polynomials approach the Kauffman bracket in a limit.

Asymptotic growth rate of the Kauffman bracket relates to hyperbolic volumes.

Provides a categorical perspective on the volume conjecture.

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.