Invariants of Colored Links and a Property of the Clebsch-Gordan Coefficients of $U_q(g)$

TL;DR

This paper explores how multivariable colored link invariants originate from roots of unity representations of quantum groups and introduces a key property of Clebsch-Gordan coefficients crucial for defining these invariants, with explicit proofs for $U_q(sl_2)$.

Contribution

It proposes a new property of Clebsch-Gordan coefficients of quantum groups and constructs generalized link invariants based on this property.

Findings

Derived multivariable colored link invariants from roots of unity representations.

Proved a key property of Clebsch-Gordan coefficients for $U_q(sl_2)$.

Constructed invariants that generalize the multivariable Alexander polynomial.

Abstract

We show that multivariable colored link invariants are derived from the roots of unity representations of $U_q(g)$. We propose a property of the Clebsch-Gordan coefficients of $U_q(g)$, which is important for defining the invariants of colored links. For $U_q(sl_2) we explicitly prove the property, and then construct invariants of colored links and colored ribbon graphs, which generalize the multivariable Alexander polynomial.