On the universal $sl_2$ invariant of ribbon bottom tangles

Abstract

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are placed on the bottom, and every link can be represented as the closure of a bottom tangle. The universal $s{l}_2$ invariant of $n$–component bottom tangles takes values in the $n$–fold completed tensor power of the quantized enveloping algebra $U_h\left(s{l}_2\right)$, and has a universality property for the colored Jones polynomials of $n$–component links via quantum traces in finite dimensional representations. In the present paper, we prove that if the closure of a bottom tangle $T$ is a ribbon link, then the universal $s{l}_2$ invariant of $T$ is contained in a certain small subalgebra of the completed tensor power of $U_h\left(s{l}_2\right)$. As an application, we prove that ribbon links have stronger divisibility by cyclotomic polynomials than algebraically split links for Habiro’s reduced version of the colored Jones polynomials.