The universal sl_2 invariant and Milnor invariants

TL;DR

This paper explores the relationship between the universal sl_2 invariant of string links and classical Milnor invariants, revealing new topological insights and invariance criteria related to quantum invariants and link concordance.

Contribution

It establishes explicit connections between the universal sl_2 invariant and Milnor invariants, including invariance criteria and partial constructions for the weight system.

Findings

Universal sl_2 invariant relates to Milnor invariants.

Criteria for invariance under concordance and link-homotopy.

Partial construction of the universal sl_2 invariant's weight system.

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.