Volume conjecture, geometric decomposition and deformation of hyperbolic structures (1912.11779v3)

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.

• The paper introduces an infinite family of prime links with two hyperbolic pieces via cabling on the figure eight knot.
• It employs quantum dilogarithms and asymptotic evaluations of colored Jones polynomials to rigorously test the generalized volume conjecture.
• The study links exponential growth rates in quantum invariants to the simplicial volume of link complements, broadening applications in knot theory.

Analysis of "Volume Conjecture, Geometric Decomposition and Deformation of Hyperbolic Structures"

This paper, authored by Ka Ho Wong, explores the profound exploration of the generalized volume conjecture specifically focusing on the colored Jones polynomials. It centers on links whose complements comprise more than one hyperbolic piece, expanding the conventional understanding of knot theory in three-dimensional spaces. The paper makes substantial use of the figure eight knot as its basis, exploring its cabling by Whitehead chains to construct an infinite family of links with notable complexity.

Summary of Key Contributions

1. Construction of Links with Hyperbolic Components: Wong presents an infinite collection of prime links obtained via cabling on the figure eight knot using Whitehead chains. These links exhibit two hyperbolic pieces within their complement's JSJ decomposition. The research successfully demonstrates that for these links, the exponential growth rates of the colored Jones polynomials at the $(N+\frac{1}{2})$-th root of unity link to the simplicial volume of the link complements.
2. Proving Volume Conjecture: The paper advances the volume conjecture for Turaev-Viro invariants, strengthening the correlation between quantum invariants and geometric properties of link complements.
3. Generalized Volume Conjecture Extension: Besides formulating the volume conjecture for links with multiple hyperbolic components, the paper introduces new insights by considering the iterated Whitehead double and Hopf link. Two distinct infinite families of prime links emerge from this approach. By assigning natural incomplete hyperbolic structures to the hyperbolic pieces, a novel result reveals that the sum of these volumes equates to the exponential growth rate of sequences of colored Jones polynomials.

Technical Structure and Results

Wong's approach includes a detailed examination of the asymptotic behavior of the $\vec{M}$-th colored Jones polynomials evaluated at specialized roots of unity (i.e., $(N+\frac{1}{2})$-th roots). The focus is on understanding how these polynomials capture intricacies of the simplicial volume for complex link complements. This investigation extends classical understanding by accommodating non-hyperbolic link scenarios, offering a broader applicability of the volume conjecture.

Several crucial mathematical tools are employed throughout the paper. The use of quantum dilogarithms plays a significant role in examining the functional behavior of these invariants under various geometric deformations. The rigorous methodology and extensive numerical computations provide evidence supporting the proposed conjecture extensions.

Implications and Future Outlook

The implications of the results presented in this paper are both theoretical and practicable. Theoretically, they push the frontiers of knot theory and quantum topology, broadening the scenarios where the volume conjecture holds. Practically, they suggest pathways for more expansive applications of quantum invariants in evaluating complex topological structures, potentially impacting areas such as quantum computing or materials science where knot theoretic concepts find application.

Wong's results also open several avenues for future exploration. The confirmation of generalized volume conjectures in broader classes of links can lead to deeper understanding of hyperbolic 3-manifolds and their invariants. Further research could focus on validating the proposed conjectures for other knot structures or exploring the interplay between geometric decomposition and quantum invariants in more intricate topological settings.

The work by Ka Ho Wong stands as a crucial contribution to the paper of knot invariants and their relation to hyperbolic geometry, enhancing both the scope and depth of the volume conjecture in mathematical physics and geometric topology.