On the universal sl2 invariant of ribbon bottom tangles

Suzuki, Sakie

doi:10.2140/agt.2010.10.1027

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On the universal $sl_2$ invariant of ribbon bottom tangles

Sakie Suzuki

Algebraic & Geometric Topology 10 (2010) 1027–1061

DOI: 10.2140/agt.2010.10.1027

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 $U_{h} (s l_{2})$ , 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 $U_{h} (s l_{2})$ . 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

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57M25

References

Publication

Received: 29 May 2009

Accepted: 12 January 2010

Published: 26 April 2010

Authors

Sakie Suzuki
Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan
http://www.kurims.kyoto-u.ac.jp/~sakie/

Volume 10, issue 2 (2010)