#### Volume 10, issue 2 (2010)

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 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
On the universal $sl_2$ invariant of ribbon bottom tangles

### Sakie Suzuki

Algebraic & Geometric Topology 10 (2010) 1027–1061
##### 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.

##### Keywords
bottom tangle, boundary bottom tangle, boundary link, universal $sl_2$ invariant, colored Jones polynomial
Primary: 57M27
Secondary: 57M25