On the volume conjecture of the colored Jones invariants with arbitrary colors

TL;DR

This paper explores the volume conjecture for colored Jones invariants, analyzing limits related to hyperbolic structures of link complements, specifically for the figure-eight knot and Borromean rings.

Contribution

It investigates the limits of colored Jones invariants for specific links and connects these limits to the volumes of hyperbolic cone manifolds.

Findings

Limits of colored Jones invariants relate to hyperbolic cone manifold volumes.

Studied links include the figure-eight knot and Borromean rings.

Results support the volume conjecture in the context of hyperbolic structures.

Abstract

We study the volume conjecture of the colored Jones invariants with sequences of colors corresponding to the deformation of the hyperbolic structure of a link complement. In particular, we investigate certain limits of the colored Jones invariants of the figure-eight knot and the Borromean rings and show that the limits are related to the volumes of hyperbolic cone manifolds whose singular sets are the links.