The universal 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 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra of the Lie algebra . 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.
"On the universal $sl_2$ invariant of boundary bottom tangles." Algebr. Geom. Topol. 12 (2) 997 - 1057, 2012. https://doi.org/10.2140/agt.2012.12.997