From colored Jones invariants to logarithmic invariants

TL;DR

This paper introduces a formula linking the logarithmic invariant of knots to derivatives of the colored Jones invariant, exploring its relation to logarithmic conformal field theory and hyperbolic volume of cone manifolds.

Contribution

It provides a new explicit formula for the logarithmic invariant in terms of derivatives of the colored Jones invariant and investigates its geometric and theoretical significance.

Findings

Derived a formula connecting logarithmic invariants and colored Jones derivatives

Established a relation between logarithmic invariants and hyperbolic cone manifold volume

Linked the invariant to logarithmic conformal field theory

Abstract

In this work, we give a formula for the logarithmic invariant of knots in terms of certain derivatives of the colored Jones invariant. This invariant is related to the logarithmic conformal field theory, and was defined by using the centers in the radical of the restricted quantum group at root of unity. A relation between logarithmic invariant and the hyperbolic volume of a cone manifold is also investigated.