The universal sl2 invariant and Milnor invariants

Abstract

The universal [Formula: see text] invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the [Formula: see text]-adic completed tensor powers of the quantized enveloping algebra of [Formula: see text]. In this paper, we exhibit explicit relationships between the universal [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal [Formula: see text] invariant.