Volume conjecture, geometric decomposition and deformation of hyperbolic structures

TL;DR

This paper explores the generalized volume conjecture for links with multiple hyperbolic pieces, constructing infinite families of prime links and connecting colored Jones polynomial growth to hyperbolic volume and Turaev-Viro invariants.

Contribution

It introduces new prime links with multiple hyperbolic components and proves the volume conjecture for their Turaev-Viro invariants, extending the conjecture to more complex link complements.

Findings

Exponential growth rates of colored Jones polynomials match the simplicial volume.

Volume conjecture holds for Turaev-Viro invariants of constructed links.

Sum of hyperbolic volumes equals growth rates of polynomial sequences.

Abstract

In this paper, we study the generalized volume conjecture for the colored Jones polynomials of links with complements containing more than one hyperbolic piece. First of all, we construct an infinite family of prime links by considering the cabling on the figure eight knot by the Whitehead chains. The complement of these links consist of two hyperbolic pieces in their JSJ decompositions. We show that at the $(N+\frac{1}{2})$-th root of unity, the exponential growth rates for the $\vec{N}$-th colored Jones polynomials for these links capture the simplicial volume of the link complements. As an application, we prove the volume conjecture for the Turaev-Viro invariants for these links complements. We also generalize the volume conjecture for links whose complement have more than one hyperbolic piece in another direction. By considering the iterated Whitehead double on the figure eight knot and the Hopf link, we construct two infinite families of prime links. Furthermore, we assign certain "natural" incomplete hyperbolic structures on the hyperbolic pieces of the complements of these links and prove that the sum of their volume coincides with the exponential growth rate of certain sequences of values of colored Jones polynomials of the links.