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 sl2 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 sl2 invariant of boundary bottom tangles takes values in certain subalgebras of the completed tensor powers of the quantized enveloping algebra Uh(sl2) of the Lie algebra sl2. 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
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M25
References
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/