#### Volume 12, issue 2 (2012)

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On the universal $sl_2$ invariant of boundary bottom tangles

### Sakie Suzuki

Algebraic & Geometric Topology 12 (2012) 997–1057
##### 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.

##### Keywords
quantum invariant, universal invariant, colored Jones polynomial, boundary link, bottom tangle
Primary: 57M27
Secondary: 57M25
##### Publication
Received: 2 April 2011
Revised: 1 February 2012
Accepted: 29 November 2011
Published: 7 May 2012
##### Authors
 Sakie Suzuki Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan http://www.kurims.kyoto-u.ac.jp/~sakie/