The universal sl_2 invariant and Milnor invariants

Abstract

The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invariants, which are classical invariants generalizing the linking number, providing some new topological insight into quantum invariants. More precisely, we define a reduction of the universal sl_2 invariant, and show how it is captured by Milnor concordance invariants. We also show how a stronger reduction corresponds to Milnor link-homotopy invariants. As a byproduct, we give explicit criterions for invariance under concordance and link-homotopy of the universal sl_2 invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal sl_2 invariant.Comment: 30 pages ; final version, to appear in Int. J. Mat