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The universal $s l_{2}$ invariant and Milnor invariants

Jean-Baptiste Meilhan

Institut Fourier, Université Grenoble Alpes, F-38000 Grenoble, France

E-mail Address: jean-baptiste.meilhan@ujf-grenoble.fr

Corresponding author.

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and

Sakie Suzuki

The Hakubi Center for Advanced Research, Kyoto University, Yoshida-honmachi, Sakyo-Ku, Kyoto 606-8501, Japan

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

E-mail Address: sakie@kurims.kyoto-u.ac.jp

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https://doi.org/10.1142/S0129167X16500907Cited by:2 (Source: Crossref)

Abstract

The universal $s l_{2}$ invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the $ℏ$ -adic completed tensor powers of the quantized enveloping algebra of $s l_{2}$ . In this paper, we exhibit explicit relationships between the universal $s l_{2}$ 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 $s l_{2}$ 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 $s l_{2}$ invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal $s l_{2}$ invariant.

Keywords:

AMSC: 57M25, 57M27

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