On the universal $sl_2$ invariant of boundary bottom tangles

Abstract

The universal $s{l}_2$ invariant of bottom tangles has a universality property for the colored Jones polynomial of links. A bottom tangle is called boundary if its components admit mutually disjoint Seifert surfaces. Habiro conjectured that the universal $s{l}_2$ invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra $U_h\left(s{l}_2\right)$ of the Lie algebra $s{l}_2$. In the present paper, we prove an improved version of Habiro’s conjecture. As an application, we prove a divisibility property of the colored Jones polynomial of boundary links.