The Kashaev and quantum hyperbolic link invariants

TL;DR

This paper establishes a connection between quantum hyperbolic link invariants and Kashaev's invariants, introduces explicit enhanced Yang-Baxter operators, and disproves a conjecture about semi-classical limits conflicting with hyperbolic link properties.

Contribution

It defines quantum hyperbolic invariants via planar state sums, explicitly computes new Yang-Baxter operators, and relates these invariants to Kashaev's specializations of colored Jones polynomials.

Findings

Quantum hyperbolic invariants coincide with Kashaev invariants.

Explicit form of enhanced Yang-Baxter operators is provided.

Disproves a conjecture about semi-classical limits conflicting with hyperbolic links.

Abstract

We show that the link invariants derived from 3-dimensional quantum hyperbolic geometry can be defined by means of planar state sums based on link diagrams and a new family of enhanced Yang-Baxteroperators (YBO) that we compute explicitly. By a local comparison of the respective YBO's we show that these invariants coincide with the Kashaev specializations of the colored Jones polynomials. As a further application we disprove a conjecture about the semi-classical limits of quantum hyperbolic partition functions, by showing that it conflicts with the existence of hyperbolic links that verify the volume conjecture.