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Abstract
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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
invariant of
–component bottom tangles
takes values in the –fold
completed tensor power of the quantized enveloping algebra
,
and has a universality property for the colored Jones polynomials of
–component
links via quantum traces in finite dimensional representations. In
the present paper, we prove that if the closure of a bottom tangle
is a ribbon link,
then the universal
invariant of
is contained in a certain small subalgebra of the completed tensor power of
. 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.
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Keywords
bottom tangle, boundary bottom tangle, boundary link,
universal $sl_2$ invariant, colored Jones polynomial
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Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57M25
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Publication
Received: 29 May 2009
Accepted: 12 January 2010
Published: 26 April 2010
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