Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On the universal sl 2 invariant of Brunnian bottom tangles

SAKIE SUZUKI

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan. e-mail: sakie@kurims.kyoto-u.ac.jp

Abstract

The universal sl 2 invariant is an invariant of bottom tangles from which one can recover the colored Jones polynomial of links. We are interested in the relationship between topological properties of bottom tangles and algebraic properties of the universal sl 2 invariant. A bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we prove that the universal sl 2 invariant of n-component Brunnian bottom tangles takes values in a small subalgebra of the n-fold completed tensor power of the quantized enveloping algebra U h (sl 2). As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.

(Received February 27 2012)

(Revised June 12 2012)

(Online publication October 01 2012)