An analogue of Milnor’s invariants for knots in 3–manifolds

Kuzbary, Miriam

doi:10.2140/agt.2024.24.3043

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An analogue of Milnor's invariants for knots in $3$–manifolds

Miriam Kuzbary

Algebraic & Geometric Topology 24 (2024) 3043–3067

DOI: 10.2140/agt.2024.24.3043

Abstract

Milnor’s invariants are some of the more fundamental oriented link concordance invariants; they behave as higher-order linking numbers and can be computed using combinatorial group theory (due to Milnor), Massey products (due to Turaev and Porter) and higher-order intersections (due to Cochran). We generalize the first nonvanishing Milnor’s invariants to oriented knots inside a closed oriented $3$ –manifold $M$ . We call this the Dwyer number of a knot and show methods to compute it for nullhomologous knots inside connected sums of $S^{1} \times S^{2}$ . We further show in this case that the Dwyer number provides the weight of the first nonvanishing Massey product in the knot complement in the ambient manifold. Additionally, we prove that the Dwyer number detects a family of nullhomotopic knots $K$ in $#^{ℓ} S^{1} \times S^{2}$ bounding smoothly embedded disks in $♮^{ℓ} D^{2} \times S^{2}$ which are not concordant to the unknot.

Keywords

knot concordance, link concordance, Milnor's invariants, Milnor numbers, links in $3$–manifolds, $S^1 \times S^2$, lower central series, knots in $3$–manifolds, slice, local knot

Mathematical Subject Classification

Primary: 57K10, 57K12

References

Publication

Received: 18 February 2020

Revised: 5 September 2022

Accepted: 23 October 2022

Published: 7 October 2024

Authors

Miriam Kuzbary
School of Mathematics Georgia Institute of Technology Atlanta, GA United States
http://mkuzbary3.math.gatech.edu

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Volume 24, issue 6 (2024)