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On the universal $sl_2$ invariant of ribbon bottom tangles

Sakie Suzuki

Algebraic & Geometric Topology 10 (2010) 1027–1061

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 sl2 invariant of n–component bottom tangles takes values in the n–fold completed tensor power of the quantized enveloping algebra Uh(sl2), 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 sl2 invariant of T is contained in a certain small subalgebra of the completed tensor power of Uh(sl2). 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.

bottom tangle, boundary bottom tangle, boundary link, universal $sl_2$ invariant, colored Jones polynomial
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25
Received: 29 May 2009
Accepted: 12 January 2010
Published: 26 April 2010
Sakie Suzuki
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502