Milnor’s invariants for knots and links in closed orientable 3-manifolds

Stees, Ryan

doi:10.2140/gt.2026.30.983

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Milnor's invariants for knots and links in closed orientable $3$-manifolds

Ryan Stees

Geometry & Topology 30 (2026) 983–1050

DOI: 10.2140/gt.2026.30.983

Abstract

In his 1957 paper, John Milnor introduced a collection of invariants for links in $S^{3}$ detecting higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants, now known as Milnor’s $\bar{μ}$ -invariants, were later shown to be topological link concordance invariants and have since inspired decades of consequential research. Milnor’s invariants have many interpretations, and there have been numerous attempts to extend them to other settings. In this paper, we extend Milnor’s invariants to topological concordance invariants of knots and links in general closed orientable 3-manifolds. These invariants unify and generalize all previous versions of Milnor’s invariants in dimension 3, including Milnor’s original invariants for links in $S^{3}$ .

Keywords

knot, link, 3-manifold, concordance, Milnor's invariants, homology cobordism

Mathematical Subject Classification

Primary: 57K10, 57K31

Secondary: 57K30, 57K40, 57N65

References

Publication

Received: 12 March 2024

Revised: 21 February 2025

Accepted: 29 August 2025

Published: 20 April 2026

Proposed: Cameron Gordon

Seconded: Robion Kirby, David Gabai

Authors

Ryan Stees
Department of Mathematics University of Virginia Charlottesville, VA United States

Open Access made possible by participating institutions via Subscribe to Open.

Volume 30, issue 3 (2026)